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<746F20> Tj 1 0 0 1 374.40 212.58 Tm <4343432E2C32325A> Tj 0.90 Tc 1 0 0 1 47.88 204.84 Tm <526F7365776F6F6420> Tj 1 0 0 1 89.10 204.66 Tm <44726976652C20> Tj 1 0 0 1 115.20 204.66 Tm <44616E766572732C20> Tj 1 0 0 1 151.02 204.12 Tm <4D4120> Tj 1 0 0 1 167.94 204.12 Tm <30313932332E> Tj ET endstream endobj 8 0 obj 13957 endobj 9 0 obj << /Type /Font /Subtype /Type1 /BaseFont /Helvetica /Encoding /WinAnsiEncoding >> endobj 10 0 obj << /Length 2028 >> stream . l, I' \ / i \iolume 7, No. i (1998), pp. 60-77 Allerton Press, lnc. M A T II{ E, M A iI' I] C I. [- TII E,THCIDI S flr E' STATISTICS GOODNESS.OF.FIT TESTS FOR PARAMETRIC HYPOTHESESON THE DISTRIBUTION OF POINT PROCESSESN. O. MncLApERrDzEr , Z. P. TsrcRosrrvrLrl x, AND IvI. vnx pur,?lr\. Razmadze Nlathematical Institute of 鶹ý Academy of SciencesM. Alexidze str., 1, 380093 Tbilisi, Georgiae-mail: zurats@imath.acnet.ge2l{easurement and Computational ApplicationsKonin klij ke/Shell-Laboratorium AmsterdamP.O. Box 38000, 1030 BN Amsterdam, The Netherlands We construct asymptoticalls' distributio n-free goodness-o f-fic tests for testing parametric lts.-pothe-sis on the irrtensicies of point processes. The technique for obtaining asS,mptotical/.r. dis-tributed empirical process 9V,, is based on the theorS'of innou,ation martingales.I( ey w'ords : parametri c hy potheses, innova tion nt artingales, goodness-of-fi t testsAMS I99i Subject Classification: Primary 62G10; SecondarS'62N107,60G44, 6OGSS. 0. IntroductionThe aim pursued in this paper is to construct asymptotically distribution-freegoodness-of-fit tests for testing parametric hypotheses on the distribution of pointprocesses, i.e., our object is to derive goodness-of-fit tests for the hypotheses (i) )"(.) A,,. = {),,(',0),0 e O C R^} on the intensity process )"(.) of a point process.For the particular case of the general multiplicative model of Aalen (see Ander-sen, Borgan, Gill, Keiding (abbreviated belorv as ABGI() [1], p. 128, or Hjort [4]) *Research supported by the International Science Foundation, Grants MXIOoO and MXtzoo.@)1998 by Allerton Press, Inc. 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Tj 1.08 Tc /F3 3.0 Tf 1 0 0 1 88.02 658.44 Tm <6572722E6F7221727629> Tj 0.72 Tc /F3 9.0 Tf 1 0 0 1 63.18 646.56 Tm <4C6174657220> Tj 1 0 0 1 90.90 646.38 Tm <776520> Tj 1 0 0 1 106.56 646.38 Tm <6C76696C6C20> Tj 1 0 0 1 126.72 646.02 Tm <617373756D6520> Tj 1 0 0 1 162.36 645.84 Tm <7468617420> Tj 1 0 0 1 184.86 645.66 Tm <746865726520> Tj 1 0 0 1 210.78 645.48 Tm <65786973747320> Tj 1 0 0 1 239.22 645.30 Tm <616E20> Tj 1 0 0 1 253.80 645.30 Tm <696E6372656173696E6720> Tj 1 0 0 1 300.96 645.12 Tm <73657175656E636520> Tj 1 0 0 1 342.90 644.76 Tm <6F6620> Tj 1 0 0 1 355.32 644.58 Tm <636F6E7374616E747320> Tj 1 0 0 1 400.14 644.22 Tm <6F2C2C> Tj 0.90 Tc 1 0 0 1 50.40 634.50 Tm <7375636820> Tj 1 0 0 1 73.80 634.32 Tm <74686174> Tj 1.26 Tc 1 0 0 1 124.74 617.76 Tm <68495E28742C6F2920> Tj 1 0 0 1 163.98 618.30 Tm <3D20> Tj 1 0 0 1 174.42 617.40 Tm <4E492C28742920> Tj 1 0 0 1 203.76 620.64 Tm <2D20> Tj 1 0 0 1 213.84 617.22 Tm <612C27285E7428722920> Tj 1 0 0 1 263.34 619.20 Tm <2D20> Tj 1 0 0 1 275.04 610.92 Tm <5B5E2720> Tj 1 0 0 1 288.00 616.68 Tm <5E2D28732C20> Tj 1 0 0 1 311.94 616.68 Tm <642920> Tj 1 0 0 1 322.20 616.68 Tm <722F7329> Tj 1.26 Tc 1 0 0 1 272.52 608.40 Tm <4A6F20> Tj 1 0 0 1 331.74 610.56 Tm <2F> Tj 0.90 Tc 1 0 0 1 50.40 595.26 Tm <697320> Tj 1 0 0 1 60.30 595.26 Tm <6120> Tj 1 0 0 1 68.22 595.26 Tm <73717561726520> Tj 1 0 0 1 99.90 594.90 Tm <696E7465677261626C6520> Tj 1 0 0 1 146.34 594.54 Tm <6D617274696E67616C6520> Tj 1 0 0 1 196.20 594.72 Tm <72766869636820> Tj 1 0 0 1 224.28 594.18 Tm <756E64657220> Tj 1 0 0 1 252.00 594.18 Tm <74686520> Tj 1 0 0 1 268.74 594.00 Tm <6879706F74686573697320> Tj 1 0 0 1 317.34 593.64 Tm <636F6E76657267657320> Tj 1 0 0 1 361.26 593.28 Tm <746F20> Tj 1 0 0 1 372.96 593.28 Tm <6120> Tj 1 0 0 1 380.88 592.92 Tm <47617573732D> Tj 0.72 Tc 1 0 0 1 50.40 583.20 Tm <69616E20> Tj 1 0 0 1 67.32 583.02 Tm <6D617274696E67616C6520> Tj 1 0 0 1 118.26 582.66 Tm <4D20> Tj 1 0 0 1 132.66 582.84 Tm <7769746820> Tj 1 0 0 1 155.16 582.48 Tm <726E65616E20> Tj 1 0 0 1 181.98 582.30 Tm <7A65726F20> Tj 1 0 0 1 203.04 582.30 Tm <616E6420> Tj 1 0 0 1 222.30 582.30 Tm <76617269616E6365> Tj 1.08 Tc /F3 11.0 Tf 1 0 0 1 193.14 560.52 Tm <4520> Tj 1 0 0 1 199.98 560.52 Tm <4E727A20> Tj 1 0 0 1 215.28 560.52 Tm <284420> Tj 1 0 0 1 229.50 563.58 Tm <2D20> Tj 1 0 0 1 239.94 560.34 Tm <4620> Tj 1 0 0 1 247.86 560.34 Tm <28742C20> Tj 1 0 0 1 259.56 560.16 Tm <65292E> Tj 0.90 Tc /F3 9.0 Tf 1 0 0 1 50.58 539.64 Tm <466F7220> Tj 1 0 0 1 68.58 539.46 Tm <74686520> Tj 1 0 0 1 86.04 539.28 Tm <646566696E6974696F6E20> Tj 1 0 0 1 130.68 538.92 Tm <6F6620> Tj 1 0 0 1 142.74 538.92 Tm <46286C2C642920> Tj 1 0 0 1 174.42 538.74 Tm <616E6420> Tj 1 0 0 1 194.04 538.74 Tm <636F6E646974696F6E7320> Tj 1 0 0 1 241.38 538.38 Tm <6F6E20> Tj 1 0 0 1 255.42 538.56 Tm <292C2E20> Tj 1 0 0 1 269.82 538.20 Tm <616E6420> Tj 1 0 0 1 289.26 538.20 Tm <6F2C2C20> Tj 1 0 0 1 302.76 538.20 Tm <73656520> Tj 1 0 0 1 319.14 537.84 Tm <53656374696F6E20> Tj 1 0 0 1 354.42 537.66 Tm <312E> Tj 0.72 Tc 1 0 0 1 62.46 527.40 Tm <496620> Tj 1 0 0 1 74.16 527.76 Tm <776520> Tj 1 0 0 1 90.72 527.22 Tm <6B6E657720> Tj 1 0 0 1 118.62 527.22 Tm <74686520> Tj 1 0 0 1 137.70 527.04 Tm <7472756520> Tj 1 0 0 1 160.56 527.04 Tm <76616C756520> Tj 1 0 0 1 187.74 526.86 Tm <6F6620> Tj 1 0 0 1 201.06 526.68 Tm <74686520> Tj 1 0 0 1 219.60 526.86 Tm <706172616D6574657220> Tj 1 0 0 1 268.92 526.32 Tm <642C20> Tj 1 0 0 1 282.06 526.14 Tm <74616B696E6720> Tj 1 0 0 1 314.28 526.14 Tm <766172696F757320> Tj 1 0 0 1 349.38 525.78 Tm <66756E6374696F6E616C7320> Tj 1 0 0 1 401.94 525.24 Tm <6F66> Tj 0.72 Tc 1 0 0 1 50.22 515.52 Tm <74686520> Tj 1 0 0 1 69.12 515.70 Tm <70726F6365737320> Tj 1 0 0 1 106.02 514.98 Tm <4D6E20> Tj 1 0 0 1 126.54 515.16 Tm <776520> Tj 1 0 0 1 143.10 514.80 Tm <636F756C6420> Tj 1 0 0 1 171.18 514.62 Tm <6F627461696E20> Tj 1 0 0 1 203.94 514.62 Tm <6173796D70746F746963616C6C7920> Tj 1 0 0 1 272.52 514.08 Tm <646973747269627574696F6E2D6672656520> Tj 1 0 0 1 346.86 513.90 Tm <676F6F646E6573732D6F662D666974> Tj 0.90 Tc 1 0 0 1 50.04 503.28 Tm <746573747320> Tj 1 0 0 1 73.80 503.28 Tm <6C696B6520> Tj 1 0 0 1 92.52 501.30 Tm <58322C20> Tj 1 0 0 1 109.44 503.10 Tm <6372322C20> Tj 1 0 0 1 127.80 502.92 Tm <49696F6C6D6F676F726F762D536D69726E6F762C20> Tj 1 0 0 1 227.70 502.38 Tm <6574632E20> Tj 1 0 0 1 248.40 502.38 Tm <496C6F7276657665722C20> Tj 1 0 0 1 291.42 502.20 Tm <696E20> Tj 1 0 0 1 302.76 502.02 Tm <6F757220> Tj 1 0 0 1 320.94 502.02 Tm <6361736520> Tj 1 0 0 1 342.00 502.02 Tm <28727668656E20> Tj 1 0 0 1 372.24 501.48 Tm <74657374696E6720> Tj 1 0 0 1 405.18 501.30 Tm <61> Tj 0.90 Tc 1 0 0 1 49.86 492.12 Tm <706172616D657472696320> Tj 1 0 0 1 100.80 491.40 Tm <6879706F746865736973292C20> Tj 1 0 0 1 156.96 491.22 Tm <776520> Tj 1 0 0 1 171.54 490.86 Tm <646F20> Tj 1 0 0 1 185.04 490.86 Tm <6E6F7420> Tj 1 0 0 1 202.32 490.68 Tm <6B6E6F727620> Tj 1 0 0 1 228.42 490.68 Tm <74686520> Tj 1 0 0 1 244.98 490.68 Tm <7472756520> Tj 1 0 0 1 265.50 490.68 Tm <76616C756520> Tj 1 0 0 1 290.70 490.32 Tm <6F6620> Tj 1 0 0 1 301.86 490.14 Tm <6420> Tj 1 0 0 1 309.78 490.14 Tm <616E6420> Tj 1 0 0 1 328.86 489.96 Tm <697420> Tj 1 0 0 1 338.04 489.96 Tm <7365656D7320> Tj 1 0 0 1 366.12 489.60 Tm <6E61747572616C20> Tj 1 0 0 1 400.86 489.60 Tm <746F> Tj 0.72 Tc 1 0 0 1 49.86 479.88 Tm <636F6E736964657220> Tj 1 0 0 1 89.28 479.70 Tm <74686520> Tj 1 0 0 1 106.56 479.88 Tm <70726F63657373> Tj 1.08 Tc /F3 14.0 Tf 1 0 0 1 122.76 450.90 Tm <6672224020> Tj 1 0 0 1 152.10 451.62 Tm <3D20> Tj 1 0 0 1 162.90 450.72 Tm <7475495E282E2C6929> Tj 0.90 Tc /F3 9.0 Tf 1 0 0 1 49.50 421.56 Tm <776865726520> Tj 1 0 0 1 78.66 421.20 Tm <6420> Tj 1 0 0 1 87.30 421.20 Tm <697320> Tj 1 0 0 1 97.38 421.02 Tm <616E20> Tj 1 0 0 1 111.24 421.02 Tm <657374696D61746520> Tj 1 0 0 1 151.56 420.66 Tm <6F6620> Tj 1 0 0 1 163.26 420.66 Tm <6420> Tj 1 0 0 1 171.90 420.84 Tm <2873656520> Tj 1 0 0 1 191.88 420.48 Tm <636F6E646974696F6E7320> Tj 1 0 0 1 239.58 420.48 Tm <2863352920> Tj 1 0 0 1 259.74 419.94 Tm <696E20> Tj 1 0 0 1 271.26 419.76 Tm <53656374696F6E20> Tj 1 0 0 1 306.72 419.76 Tm <31292E20> Tj 1 0 0 1 322.38 419.58 Tm <486F77657665722C20> Tj 1 0 0 1 365.04 419.22 Tm <756E6C696B6520> Tj 1 0 0 1 394.74 419.04 Tm <4975496E> Tj 0.90 Tc 1 0 0 1 49.50 408.06 Tm <74686520> Tj 1 0 0 1 66.06 408.24 Tm <70726F6365222220> Tj 1 0 0 1 101.34 407.70 Tm <4D6E20> Tj 1 0 0 1 119.16 407.52 Tm <697320> Tj 1 0 0 1 128.70 407.52 Tm <6E6F20> Tj 1 0 0 1 142.38 407.34 Tm <6C6F6E67657220> Tj 1 0 0 1 171.90 407.16 Tm <6120> Tj 1 0 0 1 179.46 407.16 Tm <6D617274696E67616C6520> Tj 1 0 0 1 228.78 406.98 Tm <616E6420> Tj 1 0 0 1 247.68 406.80 Tm <6D6F72656F7665722C20> Tj 1 0 0 1 292.68 406.62 Tm <74686520> Tj 1 0 0 1 309.42 406.44 Tm <636C6173736963616C20> Tj 1 0 0 1 346.86 406.08 Tm <6368616E676520> Tj 1 0 0 1 379.08 405.90 Tm <6F6620> Tj 1 0 0 1 390.24 405.72 Tm <74696D65> Tj 0.90 Tc 1 0 0 1 49.32 396.00 Tm <7520> Tj 1 0 0 1 58.68 396.90 Tm <2D20> Tj 1 0 0 1 69.84 396.00 Tm <4628742C20> Tj 1 0 0 1 89.82 396.00 Tm <642920> Tj 1 0 0 1 102.24 395.82 Tm <646F657320> Tj 1 0 0 1 125.28 395.64 Tm <6E6F7420> Tj 1 0 0 1 143.46 395.46 Tm <6C65616420> Tj 1 0 0 1 165.06 395.28 Tm <746F20> Tj 1 0 0 1 177.66 395.28 Tm <616E20> Tj 1 0 0 1 191.70 395.28 Tm <6173796D70746F746963616C6C7920> Tj 1 0 0 1 258.84 394.74 Tm <646973747269627574696F6E2D6672656520> Tj 1 0 0 1 332.28 394.74 Tm <70726F636573733A20> Tj 1 0 0 1 371.52 394.02 Tm <74686520> Tj 1 0 0 1 388.80 393.84 Tm <6C696D6974> Tj 1.26 Tc 1 0 0 1 49.32 382.68 Tm <646973747269627574696F6E20> Tj 1 0 0 1 104.40 382.32 Tm <6F6620> Tj 1 0 0 1 116.46 382.14 Tm <4D5E402D7428272C64292920> Tj 1 0 0 1 181.62 381.78 Tm <646570656E647320> Tj 1 0 0 1 219.96 381.60 Tm <6F6E20> Tj 1 0 0 1 234.36 381.60 Tm <462820> Tj 1 0 0 1 246.42 383.58 Tm <2E2C30292C3020> Tj 1 0 0 1 276.30 380.52 Tm <8020> Tj 1 0 0 1 285.66 380.88 Tm <4F2E20> Tj 1 0 0 1 300.60 381.06 Tm <5468657265666F726520> Tj 1 0 0 1 346.14 380.70 Tm <74686520> Tj 1 0 0 1 363.42 380.52 Tm <747261646974696F6E616C> Tj 0.90 Tc 1 0 0 1 49.14 370.62 Tm <77617920> Tj 1 0 0 1 69.66 370.26 Tm <746F20> Tj 1 0 0 1 82.08 370.26 Tm <636F6E73747275637420> Tj 1 0 0 1 126.54 369.72 Tm <6173796D7074723E746963616C6C7920> Tj 1 0 0 1 194.22 369.36 Tm <646973747269627574696F6E2D6672656520> Tj 1 0 0 1 267.84 369.00 Tm <746573747320> Tj 1 0 0 1 290.88 368.82 Tm <697320> Tj 1 0 0 1 300.78 368.82 Tm <6E6F20> Tj 1 0 0 1 314.46 368.64 Tm <6D6F726520> Tj 1 0 0 1 339.30 368.64 Tm <76616C69642E> Tj 0.72 Tc 1 0 0 1 61.02 358.38 Tm <496E20> Tj 1 0 0 1 74.70 358.38 Tm <7468697320> Tj 1 0 0 1 95.04 358.56 Tm <706170657220> Tj 1 0 0 1 124.38 358.20 Tm <776520> Tj 1 0 0 1 139.86 358.20 Tm <70726F706F736520> Tj 1 0 0 1 177.84 357.66 Tm <616E6F7468657220> Tj 1 0 0 1 215.28 357.48 Tm <77617920> Tj 1 0 0 1 235.98 357.12 Tm <6F6620> Tj 1 0 0 1 248.22 357.12 Tm <61766F6964696E6720> Tj 1 0 0 1 288.54 356.94 Tm <7468697320> Tj 1 0 0 1 308.52 357.12 Tm <70726F626C656D20> Tj 1 0 0 1 347.94 356.40 Tm <616E6420> Tj 1 0 0 1 367.92 356.04 Tm <6F627461696E696E67> Tj 0.72 Tc 1 0 0 1 48.78 346.32 Tm <6173796D70746F746963616C6C7920> Tj 1 0 0 1 116.64 345.78 Tm <646973747269627574696F6E2D6672656520> Tj 1 0 0 1 190.80 345.24 Tm <6E6F6E706172616D657472696320> Tj 1 0 0 1 256.68 345.24 Tm <676F6F646E6573732D6F662D66697420> Tj 1 0 0 1 322.92 344.52 Tm <74657374732E20> Tj 1 0 0 1 349.38 344.16 Tm <54686520> Tj 1 0 0 1 369.90 344.16 Tm <67756964656C696E65> Tj 0.72 Tc 1 0 0 1 48.78 334.62 Tm <696E20> Tj 1 0 0 1 61.56 334.62 Tm <7468697320> Tj 1 0 0 1 82.26 334.62 Tm <72766F726B20> Tj 1 0 0 1 108.00 334.08 Tm <697320> Tj 1 0 0 1 119.34 334.08 Tm <74686520> Tj 1 0 0 1 137.52 333.90 Tm <6964656120> Tj 1 0 0 1 159.48 333.72 Tm <6F6620> Tj 1 0 0 1 172.26 333.72 Tm <4B686D616C61647A6520> Tj 1 0 0 1 225.90 331.02 Tm <31362C375D20> Tj 1 0 0 1 252.18 333.18 Tm <746F20> Tj 1 0 0 1 265.32 333.00 Tm <6261736520> Tj 1 0 0 1 288.18 333.18 Tm <676F6F646E6573732D6F662D66697420> Tj 1 0 0 1 355.50 332.46 Tm <746573747320> Tj 1 0 0 1 379.62 332.10 Tm <6E6F7420> Tj 1 0 0 1 398.16 331.92 Tm <6F6E> Tj 0.90 Tc 1 0 0 1 48.60 322.92 Tm <74686520> Tj 1 0 0 1 66.42 323.10 Tm <706172616D657472696320> Tj 1 0 0 1 117.72 322.20 Tm <656D7069726963616C20> Tj 1 0 0 1 161.82 322.38 Tm <70726F6365737320> Tj 1 0 0 1 197.28 321.84 Tm <28696E20> Tj 1 0 0 1 212.76 321.66 Tm <74686520> Tj 1 0 0 1 230.22 321.48 Tm <636C6173736963616C20> Tj 1 0 0 1 268.74 321.12 Tm <692E692E642E20> Tj 1 0 0 1 291.60 321.12 Tm <636173652920> Tj 1 0 0 1 316.62 320.94 Tm <62757420> Tj 1 0 0 1 334.98 320.76 Tm <6F6E20> Tj 1 0 0 1 348.84 320.40 Tm <69747320> Tj 1 0 0 1 363.24 320.40 Tm <696E6E6F766174696F6E> Tj 0.90 Tc 1 0 0 1 48.60 311.22 Tm <70726F6365737320> Tj 1 0 0 1 84.24 310.68 Tm <2873656520> Tj 1 0 0 1 104.22 310.32 Tm <616C736F20> Tj 1 0 0 1 124.74 310.14 Tm <4E696B616261647A652C20> Tj 1 0 0 1 176.04 309.78 Tm <4B686D616C61647A6520> Tj 1 0 0 1 228.42 307.08 Tm <5B675D292E> Tj 0.90 Tc 1 0 0 1 60.84 298.80 Tm <4F6E6520> Tj 1 0 0 1 81.00 298.80 Tm <63616E20> Tj 1 0 0 1 98.46 298.62 Tm <66696E6420> Tj 1 0 0 1 117.90 298.44 Tm <74686520> Tj 1 0 0 1 134.28 298.44 Tm <6465736372697074696F6E20> Tj 1 0 0 1 184.50 297.90 Tm <6F6620> Tj 1 0 0 1 195.30 297.90 Tm <4B686D616C61647A65277320> Tj 1 0 0 1 252.72 297.54 Tm <6964656120> Tj 1 0 0 1 272.70 297.36 Tm <696E20> Tj 1 0 0 1 283.32 297.18 Tm <74686520> Tj 1 0 0 1 299.16 297.18 Tm <636F6E7465787420> Tj 1 0 0 1 333.18 296.82 Tm <6F6620> Tj 1 0 0 1 343.98 297.18 Tm <706F696E7420> Tj 1 0 0 1 368.82 297.00 Tm <70726F636573736573> Tj 0.90 Tc 1 0 0 1 48.42 287.46 Tm <69727220> Tj 1 0 0 1 59.94 287.46 Tm <74686520> Tj 1 0 0 1 76.86 287.28 Tm <6E726F6E6F677261706820> Tj 1 0 0 1 129.24 286.74 Tm <4142474B20> Tj 1 0 0 1 162.90 284.22 Tm <5B315D2C20> Tj 1 0 0 1 178.56 286.74 Tm <70702E3436342D3436392E20> Tj 1 0 0 1 237.24 286.02 Tm <416E726F6E6720> Tj 1 0 0 1 271.26 285.84 Tm <726563656E7420> Tj 1 0 0 1 300.60 285.84 Tm <70617065727320> Tj 1 0 0 1 331.20 285.48 Tm <6F6E20> Tj 1 0 0 1 344.88 285.30 Tm <6173796D70746F746963616C6C79> Tj 0.90 Tc 1 0 0 1 48.42 275.76 Tm <646973747269627574696F6E2D6672656520> Tj 1 0 0 1 123.30 275.22 Tm <746573747320> Tj 1 0 0 1 146.70 274.86 Tm <666F7220> Tj 1 0 0 1 162.18 275.22 Tm <706172616D657472696320> Tj 1 0 0 1 212.76 274.50 Tm <6879706F74686573657320> Tj 1 0 0 1 263.70 274.32 Tm <6C766520> Tj 1 0 0 1 278.28 273.96 Tm <6D656E62696F6E20> Tj 1 0 0 1 316.80 273.78 Tm <486A6F727420> Tj 1 0 0 1 343.44 271.08 Tm <5B20> Tj 1 0 0 1 350.64 271.08 Tm <5D2E20> Tj 1 0 0 1 360.54 273.24 Tm <54686520> Tj 1 0 0 1 381.24 273.06 Tm <726561646572> Tj 0.90 Tc 1 0 0 1 48.42 263.88 Tm <6D617620> Tj 1 0 0 1 69.84 263.70 Tm <646973636F76657220> Tj 1 0 0 1 108.18 263.52 Tm <6120> Tj 1 0 0 1 116.46 263.70 Tm <7665727920> Tj 1 0 0 1 138.24 261.18 Tm <666C697265637420> Tj 1 0 0 1 166.14 262.98 Tm <636F6E6E656374696F6E20> Tj 1 0 0 1 215.64 262.80 Tm <626574727665656E20> Tj 1 0 0 1 253.80 262.44 Tm <74686520> Tj 1 0 0 1 270.18 260.28 Tm <583220> Tj 1 0 0 1 284.22 262.26 Tm <746573747320> Tj 1 0 0 1 306.72 262.08 Tm <6F6620> Tj 1 0 0 1 318.06 262.08 Tm <486A6F727420> Tj 1 0 0 1 344.34 259.38 Tm <5B615D20> Tj 1 0 0 1 357.48 261.54 Tm <616E636C20> Tj 1 0 0 1 376.74 261.36 Tm <4E696B756C696E> Tj 0.90 Tc 1 0 0 1 48.60 249.48 Tm <5B31305D20> Tj 1 0 0 1 67.50 251.82 Tm <616E6420> Tj 1 0 0 1 87.48 251.64 Tm <74686520> Tj 1 0 0 1 104.76 251.46 Tm <696E6E6F766174696F6E20> Tj 1 0 0 1 154.26 251.10 Tm <6F6620> Tj 1 0 0 1 166.14 251.10 Tm <74686520> Tj 1 0 0 1 183.42 251.28 Tm <70726573656E7420> Tj 1 0 0 1 218.34 251.10 Tm <70617065722E20> Tj 1 0 0 1 250.20 250.38 Tm <5468697320> Tj 1 0 0 1 273.24 250.20 Tm <636F6E6E656374696F6E20> Tj 1 0 0 1 323.28 250.38 Tm <77696C6C20> Tj 1 0 0 1 342.18 249.84 Tm <626520> Tj 1 0 0 1 355.50 249.66 Tm <64657363726962656420> Tj 1 0 0 1 400.14 249.30 Tm <696E> Tj 0.90 Tc 1 0 0 1 48.06 239.94 Tm <6D6F726520> Tj 1 0 0 1 73.44 239.76 Tm <64657461696C20> Tj 1 0 0 1 101.52 239.58 Tm <696E20> Tj 1 0 0 1 113.04 239.40 Tm <54736967726F736876696C6920> Tj 1 0 0 1 168.12 236.70 Tm <5B31325D20> Tj 1 0 0 1 187.02 239.22 Tm <28746F20> Tj 1 0 0 1 202.86 238.86 Tm <617070656172292E> Tj 0.90 Tc 1 0 0 1 60.12 228.06 Tm <496E20> Tj 1 0 0 1 72.54 227.88 Tm <53656374696F6E20> Tj 1 0 0 1 108.36 227.88 Tm <3120> Tj 1 0 0 1 115.92 227.88 Tm <72766520> Tj 1 0 0 1 130.50 227.52 Tm <696E74726F6475636520> Tj 1 0 0 1 174.78 227.16 Tm <736F6D6520> Tj 1 0 0 1 199.98 227.16 Tm <6E6F746174696F6E20> Tj 1 0 0 1 239.40 226.80 Tm <616E6420> Tj 1 0 0 1 258.48 226.98 Tm <6769766520> Tj 1 0 0 1 278.82 226.44 Tm <636F6E646974696F6E7320> Tj 1 0 0 1 326.34 226.44 Tm <28636C292D2863352920> Tj 1 0 0 1 368.28 226.08 Tm <72766869636820> Tj 1 0 0 1 396.72 225.90 Tm <7765> Tj 0.72 Tc 1 0 0 1 47.88 216.18 Tm <6E65656420> Tj 1 0 0 1 71.10 216.00 Tm <7468726F7567686F757420> Tj 1 0 0 1 123.84 215.64 Tm <74686520> Tj 1 0 0 1 140.58 215.64 Tm <70617065722E20> Tj 1 0 0 1 172.98 215.10 Tm <467572746865726D6F726520> Tj 1 0 0 1 230.40 215.10 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The hypothesis (1) meansthat the process A={o(.,0),deOCr?-}.that there exist d and a sequence of ),., A,, suchftA*" (r) - I l"(s, d) dsJo is I rnrrtinc'"lo' 11s1g and everl'n'here Afr, clenoles a sequence of point processes.err.or!rv)Later we lvill assume that there exists an increasing sequence of constants o,,such thathI^(t,o) = NI,(t) - a,'(^t(r) - [^' ^-(s, d) r/s)Jo /is a square integrable martingale rvhich under the hypothesis converges to a Gauss-ian martingale M with rnean zero and varianceE Nrz (D - F (t, e).For the definition of F(l,d) and conditions on ),. and o,, see Section 1.If we knew the true value of the parameter d, taking various functionals ofthe process Mn we could obtain asymptotically distribution-free goodness-of-fittests like X2, cr2, Iiolmogorov-Smirnov, etc. Ilorvever, in our case (rvhen testing aparametric hypothesis), we do not knorv the true value of d and it seems natural toconsider the process fr"@ = tuI^(.,i) where d is an estimate of d (see conditions (c5) in Section 1). However, unlike IuInthe proce"" Mn is no longer a martingale and moreover, the classical change of timeu - F(t, d) does not lead to an asymptotically distribution-free process: the limitdistribution of M^@-t(',d)) depends on F( .,0),0 O. Therefore the traditionalway to construct asymptr>tically distribution-free tests is no more valid.In this paper we propose another way of avoiding this problem and obtainingasymptotically distribution-free nonparametric goodness-of-fit tests. The guidelinein this rvork is the idea of Khmaladze 16,7] to base goodness-of-fit tests not onthe parametric empirical process (in the classical i.i.d. case) but on its innovationprocess (see also Nikabadze, Khmaladze [g]).One can find the description of Khmaladze's idea in the context of point processesirr the nronograph ABGK [1], pp.464-469. Anrong recent papers on asymptoticallydistribution-free tests for parametric hypotheses lve menbion Hjort [ ]. The readermav discover a very flirect connection betrveen the X2 tests of Hjort [a] ancl Nikulin[10] and the innovation of the present paper. This connection will be described inmore detail in Tsigroshvili [12] (to appear).In Section 1 rve introduce some notation and give conditions (cl)-(c5) rvhich weneed throughout the paper. 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O. Maglaperidze, Z. P. Tsigroshuili, ond M. aan Pul and its limiting process Lt . In Section 2 the Doob-lvleyer decomposition is given forthe limiting process .Lr as a linear transformation Q of this process and sorne of itsproperties are studied. In Section 3 we consider the transformation Q of the processMn and prove a limit theorem for QM". In Section 4 rve construct goodness-of-fitstatistics in trvo cases: when the function F(.,0) is knorvn (only 0 is unknown) andwhen the function F is unknorvn. Finally, in Section 5 lve present some examplesof the situations described in Section 4.1. Description of the Limiting Process for M^(.,d)Consider the process (2) fr^(t) - Mn(.,i) - ";'(N"(r) - l' ^",r, D'y ar) , where F i, ,o*. estimate of the true value of the parameter 0.In this section rve rvant to clarify the nature of this process and shorv that M^(',d)is asymptotically a projection of Mn(.,d) for a wide class of estimates 0 (see (cb)belorv).First, let us formulate regularity assumptions on the intensity process )r,(.,0):(c1) There exists a neighbourhood Oo of d such that for all t [0, "] there existfirst and second continuous derivatives of ),.(.,0) and log)r,(.,0) w.r.t. d (see alsoABGK [1], p. 420).(c2) There exist functions f(t,e) and h1(t,0),...,h^(t,0) and a sequence ofincreasing nonnegative constants {o"}, n = L,2, . .., such that for all e > 0 and forall 0,d e Oo ll"r'*$ -'ll, -' = oo(,), Fn(t,o) - a^2 | ^^@,0)ds, hn(s,o) -Jo2Tc^(t,0) = | nG,qhr @,0)d,Fn(r,0),Jt ll"^r#r,.(s, d).- *r,(s, a) ilt = or(r) as n _+ oo.ll*' f(.,0) llo-"ot^(c3) for all i = L,2,...,ffi ll"t'W-'ll; ,=op(t), c(t,o) - 1," h(s,o)hr (s,o)dF(s,o). log ),, (.,0) - hr(. op(r) Here and everywhere ll/(.)lll is the sup norm of thesidered on the interval [o,D].Introduce some notation: as t? --+ oo.real-valued function / con- log),.(r, d), F(t,0) = f(s,O)ds, ll#' t: oa0 The superscript ? denotes the transposition. 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O. Magla.peridze, Z. P. Tsigroshaili, and M. tan Puland(6) L(t) - M(t) - [' nr61dF(s)c-r(q [' h@) tur(ds)Jo Joare Gaussian processes with zero rnean and coaariance functionsR1,(t,u) = r(r ^ u) - [' nr61dr(s) [" ,G)dF(s)Jo Jo- f nt(r) dr'(s) [' ,@dr(s)Jo \ / Jo* Ir'n'(r) dF(s) lo' \,)t'(s) dF(s) lo" o'r,rdF(s);andRn(t,u) =r(r ^ q - [^ n'(r) dF(s)c-r@ l:h(s) dr(s).JoHere M is a Wiener process with covariance function n(U1t1U(")) = FU Au),and 3 denotes the convergence in distribution. All functions above depend on theparameter 0 but for the sake of brevity we omit it from notation. Hopefully thiswill not lead to confusion.Denote and 1tI nG) d,F(s) - e,JOfiu, - L'(t), fiu, - L(t), so that we have(T) frtnt, - Mt - sT [' ,@dM(s),. Jo(8) fiu, - M1 - sTc-'ri, [' h@)dM(s).JoIntroduce the class C'[0,"] of functions 6 e n1O,?] such that the derivativew.r.t. r'(.) exists and (rrorr")':- fo' (ffi)'011,1 (oo. Then, as it is well known, the bilinear functional on C2[0 ,4, D[O,T]of the form(e) (6; xl = [' ( ad?) \= Jo (ffiJdx(s) endstream endobj 70 0 obj 1035 endobj 71 0 obj << /Length 27 >> stream % CANON_PFINF_TYPE2_TEXTON endstream endobj 72 0 obj << /Filter /FlateDecode /Length 124 >> stream x10 ݧx'x8W rTP1H=|V oV> /XObject << /Obj2 62 0 R /Obj3 63 0 R /Obj4 64 0 R /Obj5 65 0 R >> /Subtype 69 0 R /ProcSet [ /PDF /Text /ImageB /ImageC /ImageI ] >> /Contents [ 71 0 R 72 0 R 66 0 R ] /CropBox [ 0 0 595.26 841.86 ] /Rotate 0 >> endobj 74 0 obj << /Kids [ 73 0 R ] /Count 1 /Type /Pages /Parent 15 0 R >> endobj 75 0 obj << /Pages 74 0 R /Type /Catalog >> endobj 76 0 obj << /Creator (Canon ) /CreationDate (D:20240429175029+04'00') /Producer ( ) >> endobj 77 0 obj << /Type /XObject /Subtype /Image /Width 1653 /Height 2338 /BitsPerComponent 8 /ColorSpace /DeviceRGB /Filter /DCTDecode /Length 88792 >> stream JFIF (B+(%%(Q:=0B`Ued_U][jxjqs[]gɺǙ(#(N++Nn]n  }!1AQa"q2#BR$3br %&'()*456789:CDEFGHIJSTUVWXYZcdefghijstuvwxyzw!1AQaq"2B #3Rbr $4%&'()*56789:CDEFGHIJSTUVWXYZcdefghijstuvwxyz "u! 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Goodness-o!-Fit Tests for Paran etric Hypotheses is well defined for almost all trajectories X of the Gaussian martingale M, whileon C2[0, T] x C2[0, ?] this functional is a scalar product. Now, we are ready todescribe the structure of the limiting processes (7) and (8):Lemma 1. The transformatio, fi,M, is a projection and the transformalionfit t, is an orthogonal projection if anil only ;f (gl holds. Projections are parallelto.the function g(t) on the subspace which is orthogonal w.r.t. (9) to the function"fr t(r) dr(s) o, [: h(s) dr'(s) respectiuety.Proof. Indeed, for all 0 e R t(s)hr (s) dF(s)B = o, OD frgTp-sTp-nT I,' i...,and fifi,u, - fiM, l(s)h"(s) dr(s) - I^ r(s)h" (,) dFG) lor (10) nt = rlo v "{ Ir" n@) atu(")} = rlo v ,{ I,'a(s) au(s)}. nr 61c-'(q I, h(s) dM(s) dF(t), W l,t(s)dr(s);fiu1) = g' l,' ,Urd,M(s) t(s) dM(s) = 0. tr Note that in the case of fr, the proce*.r frM, and g"(t)c-t(0) ff n@dM(s)are independent. By the nature of this lemma we will call the estimates whichsatisfy (c5) "projecting" estimates (see Khmaladze [b]).2. Doob-Meyer Decornposition for the Limiting Processes and ItsPropertiesIn this section we derive a one-to-one transformation Q of the processes ^Ll and.[ into a Wiener process and give some properties of this transformation.Consider the filtration {7tt}0c(?, We would like to derive the Doob-Meyer decomposition for the process M (.)w.r.t. {?|t}ostsz. To do this, we proceed as follorvs: since M(dt) and ff h$)dM(s)are independent of flr, by redundant conditioning (see, e.g., Bremaud [3], p.2g1)we haveE(M(dt) ln,) - nlugtll [,' o1,1oM(,)]t I JtNow using the fact'that for all t [0,7], M(dt) and f n!)d,M(s) are jointlyGaussian random variables and nlu4tll l,' h(s)dM(,)] = endstream endobj 83 0 obj 1732 endobj 84 0 obj << /Length 27 >> stream % CANON_PFINF_TYPE2_TEXTON endstream endobj 85 0 obj << /Length 99 >> stream q 595.26 0 0 841.86 0.00 0.00 cm /Obj2 Do Q q 0.196 g 362.88 0 0 587.52 51.84 204.66 cm /Obj3 Do Q endstream endobj 86 0 obj << /Type /Page /MediaBox [ 0 0 595.26 841.86 ] /Parent 87 0 R /Resources << /Font << /F3 81 0 R >> /XObject << /Obj2 77 0 R /Obj3 78 0 R >> /Subtype 82 0 R /ProcSet [ /PDF /Text /ImageB /ImageC /ImageI ] >> /Contents [ 84 0 R 85 0 R 79 0 R ] /CropBox [ 0 0 595.26 841.86 ] /Rotate 0 >> endobj 87 0 obj << /Kids [ 86 0 R ] /Count 1 /Type /Pages /Parent 15 0 R >> endobj 88 0 obj << /Pages 87 0 R /Type /Catalog >> endobj 89 0 obj << /Creator (Canon ) /CreationDate (D:20240429175331+04'00') /Producer ( ) >> endobj 90 0 obj << /Type /XObject /Subtype /Image /Width 1653 /Height 2338 /BitsPerComponent 8 /ColorSpace /DeviceRGB /Filter /DCTDecode /Length 85688 >> stream JFIF (B+(%%(Q:=0B`Ued_U][jxjqs[]gɺǙ(#(N++Nn]n  }!1AQa"q2#BR$3br %&'()*456789:CDEFGHIJSTUVWXYZcdefghijstuvwxyzw!1AQaq"2B #3Rbr $4%&'()*56789:CDEFGHIJSTUVWXYZcdefghijstuvwxyz "u! 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O. Maglaperidze, Z, P. Tsigroshaili, and M. ton Pal we obtain the following Doob-Meyer decomposition for M (.):(t) = M(t) - lr' nr(s)c-,(i l, h@)dM(u)dr(s). ,Equation (11) defines a linear transformation I of M into W. Let us apply thesame transformation to processes .Ll and .L. Then the following equalities hold.Property L. Qltt(t) = QL(t) = QLI(t).Proof. The proof of these equalities rvould follorv fromgx(r)=0 itr #-hr|)p forall geR^.To prove this fact, denote so that QX(t) = 0 is equivalent to-dv(t) = h(qhr \c-t 1t7v plarplor dp-r1r;rz1r11 = 9,which leads to C-t(t)y(n) - I or V(t) - C(t)p.This equality is valid if and onlyir #E = hr (t)p. trNote that this property clarifies our choice of the filtration. As a consequence ofit, an empirical analogu e of Q Lt (.) (which we will consider in the following section)will be independent of the estimate, and will have the same form for all projectingestimates. This is certainly an advantage from the computational viewpoint.By the following property we obtain a one-teone relationship betrveen the pro.cesses I(.) and ALO.Property 2. 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<73756666696369656E7420> Tj 1 0 0 1 166.68 694.08 Tm <746F20> Tj 1 0 0 1 178.56 693.90 Tm <73686F727620> Tj 1 0 0 1 203.94 693.72 Tm <7468617420> Tj 1 0 0 1 225.72 693.54 Tm <74686520> Tj 1 0 0 1 243.00 693.36 Tm <6B65726E656C20> Tj 1 0 0 1 272.16 693.00 Tm <6F6620> Tj 1 0 0 1 284.22 693.00 Tm <74686520> Tj 1 0 0 1 301.50 692.82 Tm <7472616E73666F726D6174696F6E20> Tj 1 0 0 1 369.36 689.94 Tm <5120> Tj 1 0 0 1 380.52 691.56 Tm <636F6E7461696E73> Tj 0.72 Tc 1 0 0 1 55.62 683.64 Tm <6F72726C7920> Tj 1 0 0 1 79.20 683.28 Tm <66756E6374696F6E7320> Tj 1 0 0 1 123.66 682.92 Tm <6F6620> Tj 1 0 0 1 136.62 682.74 Tm <69686520> Tj 1 0 0 1 154.62 682.38 Tm <666F726D20> Tj 1 0 0 1 178.56 682.56 Tm <677220> Tj 1 0 0 1 190.62 679.68 Tm <55297020> Tj 1 0 0 1 212.04 681.84 Tm <666F7220> Tj 1 0 0 1 228.24 681.66 Tm <61696920> Tj 1 0 0 1 242.46 679.86 Tm <4920> Tj 1 0 0 1 253.08 680.58 Tm <6520> Tj 1 0 0 1 264.06 680.94 Tm <5222722E20> Tj 1 0 0 1 288.90 681.12 Tm <6C696F7276657665722C20> Tj 1 0 0 1 332.64 680.40 Tm <7268697320> Tj 1 0 0 1 352.80 680.22 Tm <697320> Tj 1 0 0 1 363.96 680.04 Tm <742C6C696520> Tj 1 0 0 1 381.06 679.86 Tm <73616D6520> Tj 1 0 0 1 407.16 679.68 Tm <6173> Tj 0.72 Tc 1 0 0 1 55.80 671.94 Tm <727668617420> Tj 1 0 0 1 81.36 671.58 Tm <77617320> Tj 1 0 0 1 100.62 671.04 Tm <73686F776E20> Tj 1 0 0 1 131.40 670.86 Tm <696E20> Tj 1 0 0 1 143.64 670.50 Tm <74686520> Tj 1 0 0 1 160.56 670.86 Tm <70726F6F6620> Tj 1 0 0 1 186.66 670.14 Tm <6F6620> Tj 1 0 0 1 198.36 669.96 Tm <50726F706572747920> Tj 1 0 0 1 241.02 669.42 Tm <6C2E20> Tj 1 0 0 1 257.94 669.42 Tm <44> Tj 0.72 Tc 1 0 0 1 68.22 653.58 Tm <5265777269746520> Tj 1 0 0 1 106.02 653.58 Tm <7472616E73666F726D6174696F6E20> Tj 1 0 0 1 174.96 652.68 Tm <2831322920> Tj 1 0 0 1 195.84 652.50 Tm <696E20> Tj 1 0 0 1 207.54 652.32 Tm <74686520> Tj 1 0 0 1 224.46 652.14 Tm <666F6C6C6F77696E6720> Tj 1 0 0 1 266.40 651.78 Tm <666F726D3A> Tj 1.44 Tc /F3 11.0 Tf 1 0 0 1 55.62 623.34 Tm <2820> Tj 1 0 0 1 59.94 622.98 Tm <313329> Tj 0.72 Tc /F3 9.0 Tf 1 0 0 1 54.90 594.72 Tm <7768657265> Tj 1.44 Tc 1 0 0 1 55.08 567.00 Tm <28313429> Tj 1.08 Tc /F3 13.0 Tf 1 0 0 1 114.12 622.26 Tm <7728742920> Tj 1 0 0 1 139.14 622.80 Tm <3D20> Tj 1 0 0 1 149.58 613.26 Tm <496F2720> Tj 1 0 0 1 167.94 621.72 Tm <6B28742C7329644C283420> Tj 1 0 0 1 220.86 621.90 Tm <3D20> Tj 1 0 0 1 231.30 612.36 Tm <6C6F20> Tj 1 0 0 1 250.02 620.82 Tm <2872282C20> Tj 1 0 0 1 270.54 618.48 Tm <5320> Tj 1 0 0 1 280.44 620.46 Tm <722920> Tj 1 0 0 1 290.70 622.44 Tm <2D20> Tj 1 0 0 1 300.42 620.28 Tm <7228722C73292920> Tj 1 0 0 1 331.56 619.92 Tm <645E632873292C> Tj 0.90 Tc 1 0 0 1 152.82 565.74 Tm <7228722C20> Tj 1 0 0 1 169.74 565.56 Tm <732920> Tj 1 0 0 1 181.26 566.28 Tm <3D20> Tj 1 0 0 1 196.92 556.38 Tm <496F27222720> Tj 1 0 0 1 218.88 565.02 Tm <74727220> Tj 1 0 0 1 228.06 564.84 Tm <287529632D4C39316172362968287229> Tj 0.72 Tc /F3 9.0 Tf 1 0 0 1 54.72 537.48 Tm <616E6420> Tj 1 0 0 1 74.70 537.30 Tm <2F28412920> Tj 1 0 0 1 98.64 536.94 Tm <64656E6F74657320> Tj 1 0 0 1 135.54 536.58 Tm <74686520> Tj 1 0 0 1 152.64 536.40 Tm <696E64696361746F7220> Tj 1 0 0 1 194.58 535.86 Tm <6F6620> Tj 1 0 0 1 206.46 535.86 Tm <74686520> Tj 1 0 0 1 223.20 535.50 Tm <6576656E7420> Tj 1 0 0 1 249.84 535.32 Tm <2E342E> Tj 0.54 Tc 1 0 0 1 66.96 525.60 Tm <4E6F7720> Tj 1 0 0 1 90.00 525.42 Tm <776520> Tj 1 0 0 1 104.76 525.06 Tm <737461746520> Tj 1 0 0 1 130.14 524.70 Tm <736F6D6520> Tj 1 0 0 1 155.52 524.52 Tm <656173696C7920> Tj 1 0 0 1 183.42 524.34 Tm <766572696669656420> Tj 1 0 0 1 218.52 524.16 Tm <70726F7065727469657320> Tj 1 0 0 1 265.50 523.08 Tm <6F6620> Tj 1 0 0 1 277.20 523.08 Tm <74686520> Tj 1 0 0 1 294.12 522.72 Tm <6B65726E656C20> Tj 1 0 0 1 323.82 522.54 Tm <7228742C20> Tj 1 0 0 1 340.56 522.18 Tm <732920> Tj 1 0 0 1 352.62 522.36 Tm <2863662E20> Tj 1 0 0 1 371.70 521.64 Tm <4B686D616C61642D> Tj 0.90 Tc 1 0 0 1 54.18 513.90 Tm <7A6520> Tj 1 0 0 1 67.14 511.20 Tm <5B365D293A> Tj 0.90 Tc 1 0 0 1 66.24 496.08 Tm <50726F706572747920> Tj 1 0 0 1 116.10 495.54 Tm <332E20> Tj 1 0 0 1 129.42 495.18 Tm <496620> Tj 1 0 0 1 140.40 495.18 Tm <74686520> Tj 1 0 0 1 156.78 495.18 Tm <6B65726E656C20> Tj 1 0 0 1 186.48 494.64 Tm <7228742C732920> Tj 1 0 0 1 215.82 494.46 Tm <697320> Tj 1 0 0 1 226.26 494.10 Tm <646566696E656420> Tj 1 0 0 1 260.10 493.74 Tm <627920> Tj 1 0 0 1 272.52 493.74 Tm <283134292C20> Tj 1 0 0 1 297.18 493.38 Tm <7468656E> Tj 1.08 Tc /F3 7.0 Tf 1 0 0 1 157.86 472.86 Tm <315420> Tj 1 0 0 1 252.72 471.78 Tm <3174> Tj 0.90 Tc /F3 13.0 Tf 1 0 0 1 152.28 455.22 Tm <4A2C20> Tj 1 0 0 1 170.64 463.68 Tm <723231722C20> Tj 1 0 0 1 191.70 463.50 Tm <733B20> Tj 1 0 0 1 201.78 463.32 Tm <644628732920> Tj 1 0 0 1 230.22 463.86 Tm <3D2C20> Tj 1 0 0 1 246.96 454.14 Tm <4A6F20> Tj 1 0 0 1 262.26 462.60 Tm <7228722C20> Tj 1 0 0 1 279.00 462.42 Tm <732920> Tj 1 0 0 1 288.90 462.24 Tm <64722873292E> Tj 0.54 Tc /F3 9.0 Tf 1 0 0 1 65.88 430.56 Tm <50726F6F662E20> Tj 1 0 0 1 96.30 430.20 Tm <497420> Tj 1 0 0 1 107.10 430.02 Tm <697320> Tj 1 0 0 1 117.54 430.02 Tm <616E20> Tj 1 0 0 1 131.58 429.84 Tm <6561737920> Tj 1 0 0 1 153.90 429.48 Tm <636F6E73657175656E636520> Tj 1 0 0 1 210.06 428.76 Tm <6F6620> Tj 1 0 0 1 221.76 428.76 Tm <74686520> Tj 1 0 0 1 238.50 428.40 Tm <6661637420> Tj 1 0 0 1 258.48 428.22 Tm <74686174> Tj 0.72 Tc 1 0 0 1 176.22 406.80 Tm <6654> Tj 0.90 Tc /F3 13.0 Tf 1 0 0 1 170.64 390.24 Tm <4A2C20> Tj 1 0 0 1 188.82 398.88 Tm <6B2728742C7329647228732920> Tj 1 0 0 1 247.50 399.06 Tm <3D20> Tj 1 0 0 1 257.94 397.98 Tm <722872292E20> Tj 1 0 0 1 289.62 397.62 Tm <7472> Tj 0.72 Tc /F3 9.0 Tf 1 0 0 1 64.80 365.76 Tm <50726F706572747920> Tj 1 0 0 1 114.30 365.04 Tm <342E20> Tj 1 0 0 1 127.98 364.86 Tm <7228742C20> Tj 1 0 0 1 144.90 364.68 Tm <732920> Tj 1 0 0 1 156.96 364.50 Tm <647320> Tj 1 0 0 1 168.12 364.32 Tm <6120> Tj 1 0 0 1 178.38 364.14 Tm <48696C626572742D5363686D69647420> Tj 1 0 0 1 250.74 363.42 Tm <6B65726E656C2E> Tj 0.54 Tc 1 0 0 1 64.80 347.94 Tm <50726F6F662E20> Tj 1 0 0 1 95.40 347.58 Tm <46726F6D20> Tj 1 0 0 1 122.04 347.22 Tm <50726F706572747920> Tj 1 0 0 1 164.34 346.32 Tm <4220> Tj 1 0 0 1 172.98 346.50 Tm <627920> Tj 1 0 0 1 187.02 346.32 Tm <74686520> Tj 1 0 0 1 203.94 345.78 Tm <4361756368792D7363687761727A20> Tj 1 0 0 1 279.54 345.06 Tm <696E657175616C69747920> Tj 1 0 0 1 325.80 344.70 Tm <776520> Tj 1 0 0 1 340.38 344.16 Tm <68617665> Tj 30.24 Tc 1 0 0 1 75.96 324.90 Tm <66545E6C742C7354> Tj 0.90 Tc /F3 13.0 Tf 1 0 0 1 70.56 309.42 Tm <4A2C20> Tj 1 0 0 1 88.92 317.70 Tm <2C3231742C20> Tj 1 0 0 1 110.70 316.98 Tm <732920> Tj 1 0 0 1 120.96 316.98 Tm <647228732920> Tj 1 0 0 1 149.58 317.34 Tm <3D20> Tj 1 0 0 1 160.20 316.44 Tm <2C20> Tj 1 0 0 1 166.86 308.16 Tm <4A6F20> Tj 1 0 0 1 182.16 316.26 Tm <7228722C20> Tj 1 0 0 1 198.90 315.90 Tm <732920> Tj 1 0 0 1 208.98 315.90 Tm <647228732920> Tj 1 0 0 1 237.42 313.38 Tm <7320> Tj 1 0 0 1 247.68 315.18 Tm <322872282C292920> Tj 1 0 0 1 279.90 318.96 Tm <2C2C2C20> Tj 1 0 0 1 293.22 313.20 Tm <2820> Tj 1 0 0 1 299.16 306.36 Tm <492C20> Tj 1 0 0 1 319.32 314.64 Tm <313231742C20> Tj 1 0 0 1 341.10 314.10 Tm <737964462872292920> Tj 1 0 0 1 378.54 311.04 Tm <22272C> Tj 0.72 Tc /F3 10.0 Tf 1 0 0 1 51.48 289.98 Tm <616E6420> Tj 1 0 0 1 71.28 289.62 Tm <7468657265666F7265> Tj 1.08 Tc 1 0 0 1 178.92 267.84 Tm <5B2720> Tj 1 0 0 1 194.04 271.44 Tm <2C7231742C7329722F7228732920> Tj 1 0 0 1 252.18 270.90 Tm <3C20> Tj 1 0 0 1 265.50 270.54 Tm <34462874292C> Tj 1.80 Tc /F3 7.0 Tf 1 0 0 1 138.24 261.54 Tm <2E20> Tj 1 0 0 1 175.86 263.16 Tm <4A30> Tj 0.72 Tc /F3 9.0 Tf 1 0 0 1 50.58 249.84 Tm <736F20> Tj 1 0 0 1 63.54 249.66 Tm <7468617420> Tj 1 0 0 1 85.14 249.12 Tm <66696E616C6C79> Tj 1.26 Tc /F3 10.0 Tf 1 0 0 1 164.52 218.16 Tm <723228742C7329644628732964462874293C324632283F2920> Tj 1 0 0 1 294.12 215.82 Tm <3C20> Tj 1 0 0 1 304.20 216.18 Tm <2A2E20> Tj 1 0 0 1 326.70 215.82 Tm <7472> Tj ET endstream endobj 112 0 obj 10215 endobj 113 0 obj << /Type /Font /Subtype /Type1 /BaseFont /Helvetica /Encoding /WinAnsiEncoding >> endobj 114 0 obj << /Length 1253 >> stream G ecdn.ess- ol- Fit Tests tor Paremetric Hypctheses 6TProof. Being a linear transform of the Gaussian process.[(.), the process W(.)is also Gaussian. A direct calculation gives usE(vv(t)W(")) = F(t^r).To prove that L is the only inverse on the space of functions orthogonal to g(.)rv.r.t. (9), it is sufficient to shorv that the kernel of the transformation Q containsorrly functions of ihe form gr U)p for aii I e R"r. liorvever, rhis is t,lie same asrvhat was shown in the proof of Property l. DRewrite transformation (12) in the following form: ( 13)where(14) w(t) = Io' k(t,s)dL(4 = lo (r(, S r) - r(r,s)) d^c(s), r(r, s) = Io'"' trr (u)c-L91ar6)h(r) and /(A) denotes the indicator of the event .4.Now we state some easily verified properties of the kernel r(t, s) (cf. Khmalad-ze [6]):Property 3. If the kernel r(t,s) is defined by (14), then1T 1tJ, r21r, s; dF(s) =, Jo r(r, s) dr(s).Proof. It is an easy consequence of the fact thatfTJ, k'(t,s)dr(s) = r(r). trProperty 4. r(t, s) ds a Hilbert-Schmidt kernel.Proof. From Property B by the Cauchy-schwarz inequality we havefT^lt,sTJ, ,21t, s) dr(s) = , Jo r(r, s) dr(s) s 2(r(,)) ,,, ( I, 121t, sydF(r)) "',and therefore[' ,r1t,s)r/r(s) < 4F(t),. 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O. IVuglaperidze, Z. P. Tsigroshaili, and. iLI. aan Pult Property 5. On the functional space (C'P,f]; ll . ll") the linear transformationQ is continuous.Proof. Consider(15) |Q6l," - (1,'(#8) ' 0r1,',1)'''/fTtd.6(s\ -,v..- fT .12.-..\t/'= u, (ffi -hr(s)c-'(')l n@)d$(u)) ar(s))= (ttott3 * I,' r,'(u)d,s@)c-'$) I, h@)ds@,)"'.Consider now the quadratic form1T 1T,l,G) = J , n' @) dg(u)C- '(') /, h(t) dS@).By (c5), the matrix C(r) is positive definite, so r/(s) is positive for all s [0,4,and we havefTrlT+G) = J, U, n, @) dq@)c-r(s)rr(u)) d6@)s (1, (1,' n'@)dg(u)c-'1,yr,1,,y) 'ao@))''' (1,'#8) 'arpl)'t'= (/(,)) ','( I,' (#E) ' ar1u1)'t' ,i.e.,/(,) s I,' (ffi)',ar61for all s [0,7] and therefore ,h(f) = 0.Now from (15) we havelloll,=r.lllB,", ffi 5rt.So the linear transformation Q is bounded on CzlT,Tl. tr3. Weak Convergence of The Process QM^(.,'d).' 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Tests lor Po.rn,melrie HypotbesesI o, equivalently,( li) w^(t,i1 - [' r(r, s) ctNrn(s, o-) = [' U(, S r) - r(r, s)) dNr^(s,d).Jo .loTheorem 2. Under the conditions (cl)-(c5), u,hen the hgpothesis holds,tvn(.,d1 3 vv (.)on lhe space D[0,7].To prove this theorem lve need an auxiliary lemma.Lemma 2. Let d"(.) e C2[0,"] andllS^(.)ll. * 0, as n -+ oo. ThenllQL"(')llf + Q. Proof. By Property 5 of the previous section, lld"(.)ll" * 0 implies lled"(.)ll, =o(1), i.e.,[' ( dd^(') r'J, (#CJ dF(s)'-* Q as n + o"'Now ned^olll = ll,'(#ffi) ,",',11, sted^()tt,r(")'t2 -0 asn+oo. trProof of Theorem 2. By the Taylor expansionM^(t,7y - M^(t,0) - o^(V - 0)' [' o^(r, d) dFn(s d) + e^(t),JOwheree^(t) - anli - ,l1r ar, I: (#^"(,, d) - #^^r,, e;) a,and 9 is some point between 0 and d.By the linearity of Q and the fact thataU, h@,0)dF(s d)) = d rve havew,(t,T) - wn(t, 0) + Qe^(t)- o^(d - 0)' a( l, h,,(s, 0) dF^(r, d) - Ir' h(s,o) d,r(s,o)). endstream endobj 143 0 obj 950 endobj 144 0 obj << /Length 27 >> stream % CANON_PFINF_TYPE2_TEXTON endstream endobj 145 0 obj << /Filter /FlateDecode /Length 126 >> stream x1 AE{O :N~b!Ed!E yύ|84X"]4-AgHFJ}4D!쭣k5.87WDTj$}Y?? 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194.40 563.76 Tm <76616C756520> Tj 1 0 0 1 219.60 563.76 Tm <6F6620> Tj 1 0 0 1 230.22 564.12 Tm <706172616D6574657220> Tj 1 0 0 1 277.20 563.58 Tm <302C20> Tj 1 0 0 1 286.56 563.76 Tm <7228742C20> Tj 1 0 0 1 303.30 563.58 Tm <732920> Tj 1 0 0 1 314.64 564.66 Tm <3D20> Tj 1 0 0 1 325.08 563.76 Tm <7228742C732C302920> Tj 1 0 0 1 362.34 563.58 Tm <286F7220> Tj 1 0 0 1 377.46 563.58 Tm <2628742C20> Tj 1 0 0 1 394.92 563.40 Tm <732920> Tj 1 0 0 1 406.44 564.30 Tm <3D> Tj 0.72 Tc 1 0 0 1 53.64 551.88 Tm <6B28742C732C3929292E20> Tj 1 0 0 1 101.34 551.52 Tm <53696E636520> Tj 1 0 0 1 128.34 551.70 Tm <74686520> Tj 1 0 0 1 145.80 551.70 Tm <7472756520> Tj 1 0 0 1 167.04 551.88 Tm <76616C756520> Tj 1 0 0 1 193.50 551.70 Tm <6F6620> Tj 1 0 0 1 205.38 551.70 Tm <3020> Tj 1 0 0 1 214.02 551.70 Tm <72656D61696E7320> Tj 1 0 0 1 251.28 551.52 Tm <756E6B6E6F776E20> Tj 1 0 0 1 294.30 551.70 Tm <776520> Tj 1 0 0 1 309.24 551.52 Tm <63616E6E6F7420> Tj 1 0 0 1 342.00 551.52 Tm <75736520> Tj 1 0 0 1 359.82 551.70 Tm <2831372920> Tj 1 0 0 1 380.88 551.34 Tm <6469726563746C79> Tj 0.54 Tc 1 0 0 1 52.92 539.82 Tm <666F7220> Tj 1 0 0 1 69.84 540.00 Tm <74686520> Tj 1 0 0 1 88.56 539.82 Tm <74657374696E6720> Tj 1 0 0 1 123.12 540.18 Tm <707572706F7365732E20> Tj 1 0 0 1 172.26 539.82 Tm <497420> Tj 1 0 0 1 184.50 539.82 Tm <697320> Tj 1 0 0 1 195.48 540.18 Tm <706F737369626C6520> Tj 1 0 0 1 234.36 539.82 Tm <7468617420> Tj 1 0 0 1 257.04 539.82 Tm <74686520> Tj 1 0 0 1 275.40 539.64 Tm <6C696D6974696E6720> Tj 1 0 0 1 313.38 539.64 Tm <66756E6374696F6E20> Tj 1 0 0 1 353.88 539.64 Tm <46282E2C642920> Tj 1 0 0 1 385.74 539.46 Tm <697320> Tj 1 0 0 1 397.26 539.46 Tm <616C736F> Tj 0.54 Tc 1 0 0 1 53.28 527.76 Tm <756E6B6E6F776E2E20> Tj 1 0 0 1 100.98 527.76 Tm <417320> Tj 1 0 0 1 116.28 527.76 Tm <616E20> Tj 1 0 0 1 130.50 527.76 Tm <6578616D706C6520> Tj 1 0 0 1 169.92 527.58 Tm <6F6620> Tj 1 0 0 1 181.62 527.58 Tm <7375636820> Tj 1 0 0 1 204.48 527.58 Tm <736974756174696F6E20> Tj 1 0 0 1 246.60 527.76 Tm <776520> Tj 1 0 0 1 261.54 527.58 Tm <636F6E736964657220> Tj 1 0 0 1 300.24 527.58 Tm <74686520> Tj 1 0 0 1 317.52 527.58 Tm <41616C656E277320> Tj 1 0 0 1 352.80 527.76 Tm <67656E6572616C20> Tj 1 0 0 1 387.36 527.22 Tm <6D756C74692D> Tj 0.54 Tc 1 0 0 1 53.10 516.24 Tm <706C6963617469766520> Tj 1 0 0 1 93.24 515.88 Tm <6D6F64656C20> Tj 1 0 0 1 123.12 515.88 Tm <2873656520> Tj 1 0 0 1 142.56 515.52 Tm <53656374696F6E20> Tj 1 0 0 1 177.48 515.34 Tm <3520> Tj 1 0 0 1 185.04 515.70 Tm <62656C6F77292E20> Tj 1 0 0 1 221.04 515.70 Tm <496E20> Tj 1 0 0 1 233.10 515.70 Tm <7468697320> Tj 1 0 0 1 251.82 515.70 Tm <73656374696F6E20> Tj 1 0 0 1 284.76 515.70 Tm <776520> Tj 1 0 0 1 298.80 515.70 Tm <77696C6C20> Tj 1 0 0 1 316.98 515.52 Tm <73686F7720> Tj 1 0 0 1 341.28 515.52 Tm <686F7720> Tj 1 0 0 1 361.62 515.52 Tm <746F20> Tj 1 0 0 1 373.32 515.34 Tm <636F6E737472756374> Tj 0.54 Tc 1 0 0 1 53.10 504.18 Tm <676F6F646E6573732D6F662D66697420> Tj 1 0 0 1 120.42 503.82 Tm <7374617469737469637320> Tj 1 0 0 1 162.54 503.64 Tm <696E20> Tj 1 0 0 1 174.42 503.64 Tm <626F746820> Tj 1 0 0 1 198.18 503.64 Tm <63617365732E> Tj 0.72 Tc 1 0 0 1 65.52 491.76 Tm <46697273742C20> Tj 1 0 0 1 92.34 491.76 Tm <6C657420> Tj 1 0 0 1 106.74 491.94 Tm <757320> Tj 1 0 0 1 119.34 491.76 Tm <636F6E736964657220> Tj 1 0 0 1 158.22 491.76 Tm <74686520> Tj 1 0 0 1 174.96 491.76 Tm <736974756174696F6E20> Tj 1 0 0 1 216.54 491.76 Tm <7768656E20> Tj 1 0 0 1 242.64 491.58 Tm <46282E2C302920> Tj 1 0 0 1 272.88 491.58 Tm <697320> Tj 1 0 0 1 282.60 491.58 Tm <6B6E6F776E20> Tj 1 0 0 1 313.56 491.58 Tm <616E6420> Tj 1 0 0 1 332.46 491.58 Tm <6F6E6C7920> Tj 1 0 0 1 354.06 491.40 Tm <3020> Tj 1 0 0 1 361.80 491.40 Tm <697320> Tj 1 0 0 1 371.52 491.40 Tm <756E6B6E6F776E2E> Tj 0.54 Tc 1 0 0 1 53.28 480.06 Tm <416E20> Tj 1 0 0 1 70.56 479.88 Tm <6578616D706C6520> Tj 1 0 0 1 111.06 479.88 Tm <6F6620> Tj 1 0 0 1 123.48 479.88 Tm <7375636820> Tj 1 0 0 1 147.06 479.88 Tm <736974756174696F6E20> Tj 1 0 0 1 190.08 479.88 Tm <697320> Tj 1 0 0 1 201.42 479.88 Tm <616C736F20> Tj 1 0 0 1 222.12 480.06 Tm <676976656E20> Tj 1 0 0 1 248.94 479.70 Tm <696E20> Tj 1 0 0 1 261.54 479.52 Tm <53656374696F6E20> Tj 1 0 0 1 297.36 479.52 Tm <352C20> Tj 1 0 0 1 309.42 479.70 Tm <7468697320> Tj 1 0 0 1 329.58 479.70 Tm <697320> Tj 1 0 0 1 340.74 479.70 Tm <4D757361277320> Tj 1 0 0 1 375.12 479.52 Tm <6D6F64656C20> Tj 1 0 0 1 405.54 479.52 Tm <6F66> Tj 0.72 Tc 1 0 0 1 53.10 467.82 Tm <4142474B20> Tj 1 0 0 1 87.30 465.48 Tm <5B315D2C20> Tj 1 0 0 1 103.32 468.18 Tm <702E20> Tj 1 0 0 1 115.56 467.82 Tm <3133332C20> Tj 1 0 0 1 135.90 467.82 Tm <6F7220> Tj 1 0 0 1 147.96 467.82 Tm <76616E20> Tj 1 0 0 1 166.50 467.64 Tm <50756C20> Tj 1 0 0 1 185.04 465.30 Tm <5B31326620> Tj 1 0 0 1 207.18 467.64 Tm <496E20> Tj 1 0 0 1 219.60 467.64 Tm <7468697320> Tj 1 0 0 1 238.68 467.64 Tm <6361736520> Tj 1 0 0 1 259.38 467.64 Tm <697420> Tj 1 0 0 1 268.74 467.64 Tm <7365656D7320> Tj 1 0 0 1 297.00 467.64 Tm <6E61747572616C20> Tj 1 0 0 1 331.20 467.46 Tm <746F20> Tj 1 0 0 1 343.08 467.46 Tm <7265706C61636520> Tj 1 0 0 1 376.56 467.46 Tm <6420> Tj 1 0 0 1 384.66 467.46 Tm <627920> Tj 1 0 0 1 398.34 467.46 Tm <616E79> Tj 0.54 Tc 1 0 0 1 53.28 454.50 Tm <612C6F2D636F6E73697374656E7420> Tj 1 0 0 1 114.30 454.32 Tm <657374696D61746520> Tj 1 0 0 1 155.34 454.50 Tm <286C696B6520> Tj 1 0 0 1 177.30 454.14 Tm <3020> Tj 1 0 0 1 185.40 454.14 Tm <6F7220> Tj 1 0 0 1 198.00 454.32 Tm <302920> Tj 1 0 0 1 210.06 454.14 Tm <696E20> Tj 1 0 0 1 221.58 454.14 Tm <6B65726E656C20> Tj 1 0 0 1 251.28 454.14 Tm <26282E2C20> Tj 1 0 0 1 268.56 455.94 Tm <2E2C20> Tj 1 0 0 1 275.40 454.14 Tm <642920> Tj 1 0 0 1 286.92 453.96 Tm <616E6420> Tj 1 0 0 1 306.36 453.96 Tm <636F6E736964657220> Tj 1 0 0 1 345.06 453.96 Tm <74686520> Tj 1 0 0 1 361.98 454.32 Tm <70726F63657373> Tj 1.08 Tc /F3 10.0 Tf 1 0 0 1 53.46 423.90 Tm <28313829> Tj 0.54 Tc 1 0 0 1 53.10 394.74 Tm <4F6E6520> Tj 1 0 0 1 74.52 394.92 Tm <63616E20> Tj 1 0 0 1 92.70 394.92 Tm <73656520> Tj 1 0 0 1 109.26 394.92 Tm <74686174> Tj 0.72 Tc /F3 16.0 Tf 1 0 0 1 164.34 423.36 Tm <6672225129> Tj 0.90 Tc /F3 10.0 Tf 1 0 0 1 222.30 423.54 Tm <6B28742C20> Tj 1 0 0 1 239.76 423.36 Tm <732C20> Tj 1 0 0 1 248.76 423.36 Tm <302920> Tj 1 0 0 1 259.20 423.36 Tm <644D2228722C20> Tj 1 0 0 1 292.32 423.36 Tm <64292E> Tj 1.08 Tc /F3 13.0 Tf 1 0 0 1 113.76 647.82 Tm <6F5E286420> Tj 1 0 0 1 135.90 649.80 Tm <2D20> Tj 1 0 0 1 145.98 647.64 Tm <30292720> Tj 1 0 0 1 162.54 646.02 Tm <286C722720> Tj 1 0 0 1 184.32 647.82 Tm <6F5E282C2C302920> Tj 1 0 0 1 221.76 647.82 Tm <64466E28732C4520> Tj 1 0 0 1 260.28 649.98 Tm <2D20> Tj 1 0 0 1 269.82 639.36 Tm <493A20> Tj 1 0 0 1 287.82 647.82 Tm <68652C6F2964722728732C20> Tj 1 0 0 1 339.84 647.82 Tm <642929> Tj 1.08 Tc /F3 14.0 Tf 1 0 0 1 112.50 365.76 Tm <776E28742C693120> Tj 1 0 0 1 150.66 368.10 Tm <2D667222747429> Tj 0.90 Tc /F3 12.0 Tf 1 0 0 1 219.42 365.94 Tm <287228722C20> Tj 1 0 0 1 239.76 365.76 Tm <732C20> Tj 1 0 0 1 248.76 365.76 Tm <543120> Tj 1 0 0 1 259.92 367.74 Tm <2D20> Tj 1 0 0 1 269.64 365.76 Tm <2C31742C722C20> Tj 1 0 0 1 295.56 365.58 Tm <61292920> Tj 1 0 0 1 310.32 365.58 Tm <644D5E402C64292C> Tj 0.72 Tc /F3 9.0 Tf 1 0 0 1 52.74 334.44 Tm <776865726520> Tj 1 0 0 1 82.44 334.08 Tm <576E28742C462920> Tj 1 0 0 1 121.50 334.08 Tm <697320> Tj 1 0 0 1 131.76 334.26 Tm <676976656E20> Tj 1 0 0 1 157.86 334.08 Tm <627920> Tj 1 0 0 1 172.26 334.08 Tm <283137292E20> Tj 1 0 0 1 197.10 333.90 Tm <546F20> Tj 1 0 0 1 211.86 334.26 Tm <70726F766520> Tj 1 0 0 1 238.86 333.90 Tm <74686174> Tj 1.62 Tc /F3 17.0 Tf 1 0 0 1 202.86 306.90 Tm <6672226F2177282E29> Tj 0.90 Tc /F3 9.0 Tf 1 0 0 1 52.56 283.86 Tm <6F6E20> Tj 1 0 0 1 67.32 283.86 Tm <445B302C3F5D2C20> Tj 1 0 0 1 104.58 283.86 Tm <776520> Tj 1 0 0 1 119.70 283.68 Tm <6E65656420> Tj 1 0 0 1 143.46 283.68 Tm <736F6D6520> Tj 1 0 0 1 168.66 283.68 Tm <736D6F6F74686E65737320> Tj 1 0 0 1 222.84 283.50 Tm <636F6E646974696F6E7320> Tj 1 0 0 1 270.18 283.32 Tm <6F6E20> Tj 1 0 0 1 284.58 283.32 Tm <7228742C2E2C302920> Tj 1 0 0 1 320.76 283.32 Tm <772E722E742E20> Tj 1 0 0 1 347.58 283.14 Tm <3020> Tj 1 0 0 1 356.58 283.14 Tm <666F7220> Tj 1 0 0 1 372.96 282.96 Tm <616C6C7420> Tj 1 0 0 1 392.58 282.42 Tm <3C372E> Tj 0.54 Tc 1 0 0 1 52.38 271.80 Tm <546865736520> Tj 1 0 0 1 81.90 271.80 Tm <636F6E646974696F6E7320> Tj 1 0 0 1 130.68 271.62 Tm <61726520> Tj 1 0 0 1 147.06 271.80 Tm <676976656E20> Tj 1 0 0 1 173.16 271.62 Tm <62656C6F7720> Tj 1 0 0 1 202.32 271.62 Tm <2873656520> Tj 1 0 0 1 222.30 271.62 Tm <2863362920> Tj 1 0 0 1 242.64 271.44 Tm <616E6420> Tj 1 0 0 1 261.90 271.08 Tm <52656D61726B20> Tj 1 0 0 1 299.34 270.90 Tm <33292E> Tj 0.72 Tc 1 0 0 1 64.80 260.10 Tm <496620> Tj 1 0 0 1 74.52 259.92 Tm <6E6F7720> Tj 1 0 0 1 95.58 259.92 Tm <4628742C652920> Tj 1 0 0 1 127.44 259.92 Tm <697320> Tj 1 0 0 1 137.34 259.92 Tm <616E20> Tj 1 0 0 1 151.02 259.74 Tm <756E6B6E6F776E20> Tj 1 0 0 1 193.32 259.74 Tm <66756E6374696F6E20> Tj 1 0 0 1 232.74 259.74 Tm <28692E652E2C20> Tj 1 0 0 1 255.06 259.56 Tm <74686520> Tj 1 0 0 1 271.62 259.38 Tm <6C696D6974696E6720> Tj 1 0 0 1 308.52 259.38 Tm <2928722C20> Tj 1 0 0 1 326.16 259.38 Tm <412920> Tj 1 0 0 1 337.68 259.38 Tm <697320> Tj 1 0 0 1 347.58 259.38 Tm <756E6B6E6F776E292C20> Tj 1 0 0 1 396.72 259.02 Tm <697420> Tj 1 0 0 1 406.44 259.20 Tm <6973> Tj 0.72 Tc 1 0 0 1 52.56 248.04 Tm <726561736F6E61626C6520> Tj 1 0 0 1 102.06 247.86 Tm <746F20> Tj 1 0 0 1 114.48 247.86 Tm <75736520> Tj 1 0 0 1 132.12 247.86 Tm <74686520> Tj 1 0 0 1 149.22 247.68 Tm <6B65726E656C20> Tj 1 0 0 1 179.28 247.86 Tm <726E28742C732C20> Tj 1 0 0 1 210.42 247.68 Tm <642920> Tj 1 0 0 1 222.30 247.68 Tm <696E737465616420> Tj 1 0 0 1 256.50 247.50 Tm <6F6620> Tj 1 0 0 1 268.20 247.50 Tm <7228742C732C30292C20> Tj 1 0 0 1 308.52 247.50 Tm <7768657265> Tj 1.08 Tc /F3 12.0 Tf 1 0 0 1 53.28 366.30 Tm <2820> Tj 1 0 0 1 57.42 366.12 Tm <316529> Tj 1.08 Tc /F3 11.0 Tf 1 0 0 1 52.74 217.26 Tm <28323029> Tj 0.90 Tc /F3 7.0 Tf 1 0 0 1 181.08 225.36 Tm <377420> Tj 1 0 0 1 188.64 228.06 Tm <4173> Tj 1.26 Tc /F3 10.0 Tf 1 0 0 1 121.86 217.08 Tm <726E28742C20> Tj 1 0 0 1 144.00 216.90 Tm <732C652920> Tj 1 0 0 1 164.88 220.14 Tm <2D20> Tj 1 0 0 1 178.56 213.84 Tm <4920> Tj 1 0 0 1 199.98 216.90 Tm <742728222C3029633B2728752C302920> Tj 1 0 0 1 277.74 216.54 Tm <64466E28752C30296828732C6729> Tj 1.26 Tc /F3 8.0 Tf 1 0 0 1 175.32 208.08 Tm <4A6F> Tj ET endstream endobj 153 0 obj 15764 endobj 154 0 obj << /Type /Font /Subtype /Type1 /BaseFont /Helvetica /Encoding /WinAnsiEncoding >> endobj 155 0 obj << /Length 2018 >> stream 70 N. O. Maglaperidze, Z. P. Tsigroshuili, and fu[. aan PulHere Wn(t,0) - QM"(t,0) is a martingale w.r.t. {Fr}oSrS7 with jumps o;1 andquadratic variation (QM,,Xt,0) - F"(t, d) which converges in probability to F(t,0)for all t [0,7], i.e., the conditions of the CLT for martingales are fulfilled (see,e.g., Liptser and Shiryaev [8]) and so, on the space Dl},Tl, we have eM,(.,q Z AM(.) By the condition (c2) both and e,,(t) are the functions from C'[0,7] converging in probability in ll .ll.-normto 0 and an application of Lemma 2 gives the assertion. tr4. Construction of The Goodness-of-Fit StatisticsThe kernel of the transformation (16) depends on the limiting distribution func-tion F and also on the unknown value of parameter 0, r(t, s) = r(t,s,0) (or &(t, s) =k(t,s,9)). Since the true value of 0 remains unknown we cannot use (17) directlyfor the testing purposes. It is possible that the limiting function F(.,d) is alsounknown. As an example of such situation we consider the Aalen's general multi-plicative model (see Section 5 below). In this section we will show how to constructgoodness-of-fit statistics in both cases.First, let us consider the situation when F(.,0) is known and only 0 is unknown.An example of such situation is also given in Section 5, this is Musa's model ofABGK [1], p. 133, or van Pul [12f In this case it seems natural to replace d by anya,o-consistent estimate (like 0 or 0) in kernel &(., ., d) and consider the process (18)One can see that fr"Q) k(t, s, 0) dM"(r, d). o^(d - 0)' (lr' o^(,,0) dFn(s,E - I: he,o)dr'(s, d)) wn(t,i1 -fr"tt) (r(r, s, T1 - ,1t,r, a)) dM^@,d), where Wn(t,F) is given by (17). To prove thatfr"o!w(.)on D[0,?], we need some smoothness conditions on r(t,.,0) w.r.t. 0 for allt <7.These conditions are given below (see (c6) and Remark 3).If now F(t,e) is an unknown function (i.e., the limiting )(r, A) is unknown), it isreasonable to use the kernel rn(t,s, d) instead of r(t,s,0), where ( 1e) (20) 7t Asrn(t, s,e) - I t'(",0)c;'(u,0) dFn(u,0)h(s,g)Jo endstream endobj 156 0 obj 2018 endobj 157 0 obj << /Length 27 >> stream % CANON_PFINF_TYPE2_TEXTON endstream endobj 158 0 obj << /Length 99 >> stream q 595.26 0 0 841.86 0.00 0.00 cm /Obj2 Do Q q 0.196 g 362.88 0 0 588.24 51.84 207.54 cm /Obj3 Do Q endstream endobj 159 0 obj << /Type /Page /MediaBox [ 0 0 595.26 841.86 ] /Parent 160 0 R /Resources << /Font << /F3 154 0 R >> /XObject << /Obj2 150 0 R /Obj3 151 0 R >> /Subtype 155 0 R /ProcSet [ /PDF /Text /ImageB /ImageC /ImageI ] >> /Contents [ 157 0 R 158 0 R 152 0 R ] /CropBox [ 0 0 595.26 841.86 ] /Rotate 0 >> endobj 160 0 obj << /Kids [ 159 0 R ] /Count 1 /Type /Pages /Parent 15 0 R >> endobj 161 0 obj << /Pages 160 0 R /Type /Catalog >> endobj 162 0 obj << /Creator (Canon ) /CreationDate (D:20240429181907+04'00') /Producer ( ) >> endobj 163 0 obj << /Type /XObject /Subtype /Image /Width 1653 /Height 2338 /BitsPerComponent 8 /ColorSpace /DeviceRGB /Filter /DCTDecode /Length 89520 >> stream JFIF (B+(%%(Q:=0B`Ued_U][jxjqs[]gɺǙ(#(N++Nn]n  }!1AQa"q2#BR$3br %&'()*456789:CDEFGHIJSTUVWXYZcdefghijstuvwxyzw!1AQaq"2B #3Rbr $4%&'()*56789:CDEFGHIJSTUVWXYZcdefghijstuvwxyz "u! ?騠((((((((((((((((((((((((((((((((((((((((((((((((((( :zǠ((((((((JZ((((((((((((((((((((((((((((()(h Z((( Z(JZJZJZ( ( (|'Z1}'Z|_ju܁FsUtPQ@ EPEPEPEPEPREPEPEPE%-PI@ EPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPI@ EPEPEPEPE%-PEPEPEPEPEPEPEPEPE%-PQ@ EP c=WdPMEPI@ EPI@ EPEPEPRRI@ E%-PEPEPEPEPEPEPEPRE%-PEPEPEPEPEPEPREPEPEPEPEPEPEPEPEPREPEPEPEPEPEPEPEPEPEPEPEPQ@ EPEQ?=tԔPI@ EPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPEPI@ EPEPEPEPREPEPEPEPEPEPEPEPI@ EPEREPEPEPREPE%REPEP?EtPQ@-PEPEPEPEPI@ EPQ@ I@ I@ EPEPEPI@ EPEPEPEPEPEPEPEPE%-PEPEPEPEPEPI@ EPEPEPEPEPEPEPEPEPEPEPEPEPEPEPE%-PEPH31ڀ(JZ(9PEPEP?EtPEPK@ EPK@PPK@%PK@%-PEPEPE-%-Q@ K@ EQ@RP@Q@ E-Q@Q@Q@Q@RP@Q@Q@ K@Q@Q@Q@ E-Q@%-Q@Q@Q@Q@Q@RP@ E-%-%-Q@Q@RPK@%Q@ EPK@%PK@%-Q@5_5@-Q@Q@Q@PRI@ I@RQ@Q@Q@Q@Q@Q@Q@ ERQ@ I@Q@-RRI@ E%PI@ I@-%-PI@ EPEPE%RE%Q@ I@RQ@Q@Q@Q@Q@ I@Q@Q@Q@PQ@Q@ I@Q@Q@Q@Q@Q@PIM-GXU|#X@Q))8 -RQ@Q@ I@RI@Q@ I@Q@Q@Q@-%Q@-PRQ@Q@ I@PE%-PE%PRQ@-PRERI@RQ@ I@ EPRRQ@Q@Q@Q@Q@Q@RQ@-Q@Q@Q@RE%Q@Q@ I@Ph ( (h( :zǠ(u܌V7zP@ K@Q@ $@ E%RI@ E%-%RQ@ E%RRRE%-PI@ EPRREPREPRI@ EPEPEPREPEPEPEPERE%PI@ EPRE%-PERREPERI@ I@PERRE%REPI@u_5@5%-%P@%-PPPE-RPK@ K@%-%-%P@RP@%PK@%-Q@ EQ@%P@Q@ K@Q@Q@ EQ@Q@Q@Q@Q@Q@Q@RPK@%-%-Q@RP@Q@Q@Q@Q@ K@RPE-Q@%-Q@Q@ K@ K@ E-RPK@ K@RPEց-RP@%Q?EtPEPEPE-%Q@%PK@ EPK@ EP@%PEPE-RPK@%-RPE-%-%-PEQ@ E-%-%RPK@%-RPPK@%-Q@%-Q@RPK@PK@ E-%PK@%PP@ E-Q@RP@ E-%PPK@ K@RPE-%-%Q@ cpJgrmOzWԤ% 9 ( (ߙʡp=*chJܧ* ((k:Z( ( (h ( (JZJ(J(i( ( ( ()( ZJ(J((JZ(()(i()(h( ZJ( ZJ(J(i( (i(h)( ZJ( ZJ(h)( ( ([(?QJ(hJZJ( (i(i(h :zǠZJ()h ( JZJZ( ()h)h)h( (( ( ()h)h)h)h(( (()h ( JZJZ( ( ( ()h ()h(ZJ)h( (( ( J)h ( ( J(Z(()hZJZ( ((ZJZ((k:ZZ)(h()(hJZ(()(h()(h(JZ(((((( ZJ(i(i(((()(h)(h((((( Z((()(h(((()(i(h)(h)()(hǪ?zih J)h JZ( ()h ((Z( ( JZ( JZ( ( J)h JZJZ( (((Z(( JZ(Z(( (()h J)h ( (( ( ()h ()h ( ( ( ()h ( JZ( JZ( JZ((k:Z(J()(h( (i(J( Z( Z( ZJ( ( ZJ( ( ;CȠJZ(4f) )PEf (i( @ (Q(ހ ZCERPEPEPEPhZNQ@Q@ E%Q@ E%-PE%Q@PI@ EPERQ@Q@ EPEQ?=tPEPEPEP@ K@Q@ E-RP@ K@Q@ K@%PEPEPEPEPEPEPEPHz()JZLPzR(@ ((@P((hi(Ew(( Q@Q@Q@ ( ZJ(( (i( ( Z()( ()(hǪ?zh)h)h ()h ( ()h)h ()h ( JZ( (((((()h((((((((((Z((((ZJ((Z( (( J)h( JZJ)h ( ()h(( ( ((k:ZZJZJZ( Z)(h(((JZ((JZJ(h)(hJZ()(h ( ()((J(h((((JZ()((((((((J(h((JZJ(h (h(Ǫ?zh( JZ( ((ZJZ((ZJ)h ( J)h(Z( JZ(ZJ)h (( ( ( ( ()h ()h JZ( JZ( JZ( JZ( ( ( ( ( ()h ((Z(( (F ( (i(i( (i(J( ZJ(h)(J( Z(( ZJ( ( (JZ()(h Z( Z( (i( (h Z)(hJ(h ZJZ( Z)(h ( ZJ(i( Z( ZJ( ZJZJ(hǪ?zh*Phw ;GZ1E? 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Hence, it seems that the followingprocess is suitable in thiscase:(zL) fr^1t1 - [' r^(r, r,i1atvt^1r,i1 - enMn4,d),Jowherekn(t , t, e) - 1(r < t) - ,n(t , s,0).However, it is possibile to simplify the transformation Q first. Instead of calculatingE(M(dt) tn,) - n(ugrl l:n(s)au(s))in Section 2, let us consider the (^ * l)-dimensional vector-function H'(.) =(1, ht(.)). Then(22) nlugll l,' H(,) dM@l - HrU)c-'(r) [.' ,rurdM(u)dF(t)Jtand instead of (11) of Section 2 we obtain the process(28) w(t)- M(t)- [' nr1,1c-,(r) [' ,@)d,M(u)d,F(,)= eM(t),Jo J sor equivalently,1T(24) w(t) = Jo /((r, s) dM(s) - QM(t)with/((t, s) = r{s S t} - r?(t, s)andrtAs(25) r?(1, s, 4 = Jo nr @)c-L p1 arp)H(r).Here C-'(t) is the inverse of the (** 1) x (m* 1) matrixc(t,o) - [' ,61r"1";ar1,; - f (*1 hlk) \rt rt ,.,,G) rrlryni( s1 ) ar61'It is easy to show thaf W(t) still has all properties of Section 2. At the sametime the pre-limiting process has a computational advantage. Namely, os I suitableprocess to base the testing procedure on we will take the process(26) fr^1t1 - [' o^(r,r,d1dMn(r,i) = enMn(t,i)Jo endstream endobj 169 0 obj 1165 endobj 170 0 obj << /Length 27 >> stream % CANON_PFINF_TYPE2_TEXTON endstream endobj 171 0 obj << /Length 99 >> stream q 595.26 0 0 841.86 0.00 0.00 cm /Obj2 Do Q q 0.196 g 362.88 0 0 609.84 53.28 206.10 cm /Obj3 Do Q endstream endobj 172 0 obj << /Type /Page /MediaBox [ 0 0 595.26 841.86 ] /Parent 173 0 R /Resources << /Font << /F3 167 0 R >> /XObject << /Obj2 163 0 R /Obj3 164 0 R >> /Subtype 168 0 R /ProcSet [ /PDF /Text /ImageB /ImageC /ImageI ] >> /Contents [ 170 0 R 171 0 R 165 0 R ] /CropBox [ 0 0 595.26 841.86 ] /Rotate 0 >> endobj 173 0 obj << /Kids [ 172 0 R ] /Count 1 /Type /Pages /Parent 15 0 R >> endobj 174 0 obj << /Pages 173 0 R /Type /Catalog >> endobj 175 0 obj << /Creator (Canon ) /CreationDate (D:20240429182251+04'00') /Producer ( ) >> endobj 176 0 obj << /Type /XObject /Subtype /Image /Width 1653 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O. Maglaperidze, Z. P. Tsigroshuili, and M. tan PulwithKn(t, s,0) - /(r < t) - R"(t , s, 0)andTtAs(27) Rn(t,',0) = Jo nr fu)c;t(r) ar"(u)//(s),wherecn(t,D= I, H@)Hr(s)dr,(s) - I,'(rl, h(:;ql,)) a+r,).Then unlike the process defined by (21), the process (26) has the followingProperty 6.fr^(t) - a,t [' Kn(t, ,,i1 dN^(r) = onr enN" (r).JOProof. The kernel of transformation Q, contains the functions of the form,6rl(") d,Fn(s d;, *ith ry(s) = Hr(t,t,7) for any (* * l)-dimensional vector (.If we choose T = (1, 0, . . . ,0), then we get(28) [' ,'(r, r,d1ar,1r,i)e - F^Q,V),Jowhich means that Q" annulates the compensator of ff". trR,emark 1. 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If th,e conditions (cl)-(c6) are fulf;lled, lhert,, 'under the hypolhesis,on the functional space D[0, "]w"03w(.),where W(.) is a Wiener process uith couariance functionE(w(t)w(")) =F(rnu).To prove this theorem we need an auxiliary lemma.Lemma 3. If lhe assumption (c2) holds, then for all ) 0,os?lg-, lt;'(,) t$ffil - c-'(,)l = op(l) as ,l -) oo.Proof. Consider the (f , j)th component of the matrix C"(u),[.' noti*i@)a^2.t,(r) d,s = [.' u,tirr, (r){# d.(s)Ju JufT-e r. /,s\ 1'o,,2,\n(s) _ r_ J, r/;(s)t_,._,, /(s) _ rJ dr(s)fT -,+ | uu(s)Hi(s) dF(s)Ju* l:_,H;(s)H1(') [#] ,.(")Now by (c2) the first and third sufnmands in the right-hand side are small, whilethe second summand tends to the (i, j)th component of the matrix C(u), whenever eis small. Therefore all components of the matrix Cn are close to the correspondingcomponents of the matrix C, and hence the determinants and all correspondingminors of these matrices are close to each other, i.e., we haveo.?ll_, lc;t(t) - c-t(t)l = op(r) as tl --+ oo'An application of assumption (c2) again gives the assertion. DProof of Theorern 3. By the Taylor series expansion we obtain using Property 6fr^Q) - a^'[lr,(r) - lo' ^^(t,s,d)d/r"(r)] endstream endobj 195 0 obj 1457 endobj 196 0 obj << /Length 27 >> stream % CANON_PFINF_TYPE2_TEXTON endstream endobj 197 0 obj << /Length 99 >> stream q 595.26 0 0 841.86 0.00 0.00 cm /Obj2 Do Q q 0.196 g 391.68 0 0 614.16 21.60 205.38 cm /Obj3 Do Q endstream endobj 198 0 obj << /Type /Page /MediaBox [ 0 0 595.26 841.86 ] /Parent 199 0 R /Resources << /Font << /F3 193 0 R >> /XObject << /Obj2 189 0 R /Obj3 190 0 R >> /Subtype 194 0 R /ProcSet [ /PDF /Text /ImageB /ImageC /ImageI ] >> /Contents [ 196 0 R 197 0 R 191 0 R ] /CropBox [ 0 0 595.26 841.86 ] /Rotate 0 >> endobj 199 0 obj << /Kids [ 198 0 R ] /Count 1 /Type /Pages /Parent 15 0 R >> endobj 200 0 obj << /Pages 199 0 R /Type /Catalog >> endobj 201 0 obj << /Creator (Canon ) /CreationDate (D:20240429183134+04'00') /Producer ( ) >> endobj 202 0 obj << /Type /XObject /Subtype /Image /Width 1653 /Height 2338 /BitsPerComponent 8 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O. Maglaperidze, Z. P. Tsigroshuili, and M. uan PalfT- Mn(r) - I R"(t, s,0) dM"(s)JO- o,(d - o)'o;' Ir' #R,(t,s,o)ax"(s) * oo(1)fT= QMn(t) - | lR"(t, s,0) - R(t, s,0)l dM"(s)JO- o,(7 - o\r n-z [' ! o^) oi" Jo UR"(t, s,0)dN"(r) + oo (1).By Property 5 of Section 2, Mn(.) converges to the Wiener process and it is sufficientto show that the other two terms are small (see, e.g., Billingsley [2], Theorem 4.I).ButrTI (n U, s, o) - ft(f , s, o))dM"(s)Jo 1t ,= Io' n'(,) (rr'(')1# - c-'\ (')) / H@)d'M^(u)d'r(,)and by Lemma 3 it suffices to consider only t > T - e and to show that the randomvariables [,lu'1,1c;'(4 I, H@) d,M^(u) | a+(,)[,ln' 1,7c-' (q l,' H (u)dM^1,y I ar1"y are small in probability. But the first term is small because of Remark I above. Toshow that the second term is small, we only need to apply Chebyshev's inequality.Now consider1T A t+^ a-'r.r^r /^\ -^-2 f' ( o o .' :' a -"r, J: frn"(t, s,F)dNn(r) = - lo \aa,u,,(r, ,,e) - fin,{t,r, d)) dN,(")* a,, Io' {rn^{r, s,0) dMn(r) + ,;, lo ft*,{r, s,0) d,Fn(r, g).Assumption (c6) guarantees that the first two integrals in the right-hand sideare small in sup-norm, while the third integral is equal to 0. Indeed, by Property 6, QnFn(t,O) = 0, ora fTafu r"(t,0) = Jo. fr o"(t, s,0) dF"(s,0)with fi'^P''1 - and * on' lo' *^{r, s,o)#^^rr,o) ds Io' o^rr,q dF^(", d), endstream endobj 208 0 obj 1333 endobj 209 0 obj << /Length 27 >> stream % CANON_PFINF_TYPE2_TEXTON endstream endobj 210 0 obj << /Length 99 >> stream q 595.26 0 0 841.86 0.00 0.00 cm /Obj2 Do Q q 0.196 g 361.44 0 0 587.52 48.96 207.54 cm /Obj3 Do Q endstream endobj 211 0 obj << /Type /Page /MediaBox [ 0 0 595.26 841.86 ] /Parent 212 0 R /Resources << /Font << /F3 206 0 R >> /XObject << /Obj2 202 0 R /Obj3 203 0 R >> /Subtype 207 0 R /ProcSet [ /PDF /Text /ImageB /ImageC /ImageI ] >> /Contents [ 209 0 R 210 0 R 204 0 R ] /CropBox [ 0 0 595.26 841.86 ] /Rotate 0 >> endobj 212 0 obj << /Kids [ 211 0 R ] /Count 1 /Type /Pages /Parent 15 0 R >> endobj 213 0 obj << /Pages 212 0 R /Type /Catalog >> endobj 214 0 obj << /Creator (Canon ) /CreationDate (D:20240429183520+04'00') /Producer ( ) >> endobj 215 0 obj << /Type /XObject /Subtype 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/BaseFont /Helvetica /Encoding /WinAnsiEncoding >> endobj 221 0 obj << /Length 1472 >> stream Goodness-of-Fif Tests lor Parametric HyTtotheses so that Remark 2. Without condition (c6) lve can still prove thatw"o!*(.)on D[0, T - e] for any e > 0.Remark 3. In the case of a known limiting F(t,O) we need R(t,r,0) to satisfy(30) and (31), and also (30) with a;2dNn(r) replaced by dF(r,d). This additionalrequirement is needed because now QFn(t,A) + 0. Thus we have an additionalcondition, but on the essentially simpler kernel.5. ExamplesIn this section we will derive the forms of the transformations Q for two differentmodels: for the so-called Musa's model, in which the limiting function F is known,and for Aalen's multiplicative model, where this limiting function is unknown.o Mus.q.'s MoDEL. In this model en = nrlz and the intensity of the pointprocess is),,(t, 0r,02) - n0t(0, - "-tlr"(t- ))(see ABGK [1], p. 133, also van Pul [11]). According to this model the largernumber ff(r) of "errors" we have detected, the smaller is the intensity of detectinga nelv one.It is easv to see that in this case andh(t,01,02) = (oit, @, - F(t,01,0r))) ,where the limiting function F is known and has quite a simple form:F (t, 0t, gr) = gr0 - exp(-011)).The kernel ^R(t, r,0r,02) of the transformation Q in this model has the followingform: Ir' *Rn(r,q dF^(r,0) = en( Ir' n^(r,0) clFn(s,ay) = o. n hT,(t,h,oz) = (tit ,(0, - n-L Nn(r-))-t) R(t,r,0t,02) - 0r( fo'n' Gr(1t(u)) d,u * ,-u(') lo'n' Gz(u@)) au), whereGr(r) eu-l-u ^, Gr(u) eu-l-,eu(erlz -e-u/2)2 -r.r2' G'(")= ry' endstream endobj 222 0 obj 1472 endobj 223 0 obj << /Length 27 >> stream % CANON_PFINF_TYPE2_TEXTON endstream endobj 224 0 obj << /Filter /FlateDecode /Length 118 >> stream xE= BA > d^ a*7 d)WTN>pt?m F6Za\2a7JnvQvC*?'! endstream endobj 225 0 obj << /Type /Page /MediaBox [ 0 0 595.26 841.86 ] /Parent 226 0 R /Resources << /Font << /F3 220 0 R >> /XObject << /Obj2 215 0 R 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O. lv'Ia.glaperidze, Z. P. Tsigroshrili. and M. aan Pall undt-r(u) = p(u,01) = 0r(T - ").Norv rve see that for all t < T aud r ( T l,he funcliou R(l, r,0 t,0z) does uot dependon 02 and is sufficiently smooth w.r.t. parameter d1,n 1r# n(t,r,01,02) = " lp(O)(Gr(r(O)) a e-pt Gr(p(O)))oat ut L- p(tn ';(c,(p(t n r)) + e-p'Gz(trfr n r11) - rt(r)e-,' ["0' G2({dt].J p, JClearly, condition (cO) is satisfied for .R(t,r,01,02).o AIIBN's MULTIpLIoATIVE MoDEL. In this model en = nLlz ag:ain and theintensity of the point process N"(t) is),,(t, 0) = a(t,0)Y"(t)(see, e.g., Hjort [4], ABGK [U). In this casehn(t,0) = h(t,q = #log a(r,0)andHr (t,41 = (r ,*rouo(r, a)).However, unlike the previous example, we do not know the limiting function forn-t\n(t,0), i..., for n-tY;(r) (we only assume that this limit exists). Now thekernel of the transformation has the following form:Rn(t,r,0) = ['n' ,' (u)B;r (u)Y"(u) a(u) duH (r),Jowhere1TB"(u) = J, Y"(')d/(s)and/(s)=- f'( r hr(^\ \= Jo ( ni"l r,1'1r,F (") ) a(o) dois an "extended partial information matrix".Let us demonstrate the construction of the goodness.of-fit process in the specialcase when the parameter d is one-dimensional and the function a(t,0) - r/(9) doesnot depend on t. In such situation the extension of function h to function // givesnothing new, becausg the components 1 and {(0) of H arc linearly dependent (forpossible degeneration of the matrix.I see Tsigroshvili [12]. In this ca^se rve havcB^(u) = t(o)l#^t,!@) l,' ,^1i o, endstream endobj 236 0 obj 1495 endobj 237 0 obj << /Length 27 >> stream % CANON_PFINF_TYPE2_TEXTON endstream endobj 238 0 obj << /Filter /FlateDecode /Length 118 >> stream xEA Q C=ENi{Wp 2a)%9n̲׻Ls(c&J˿nx-l 6eZGSJ  ! endstream endobj 239 0 obj << /Type /Page /MediaBox [ 0 0 595.26 841.86 ] /Parent 240 0 R /Resources << /Font << /F3 234 0 R >> /XObject << /Obj2 229 0 R /Obj3 230 0 R /Obj4 231 0 R >> /Subtype 235 0 R /ProcSet [ /PDF /Text /ImageB /ImageC /ImageI ] >> /Contents [ 237 0 R 238 0 R 232 0 R ] /CropBox [ 0 0 595.26 841.86 ] /Rotate 0 >> endobj 240 0 obj << /Kids [ 239 0 R ] /Count 1 /Type /Pages /Parent 15 0 R >> endobj 241 0 obj << /Pages 240 0 R /Type /Catalog >> endobj 242 0 obj << /Creator (Canon ) /CreationDate (D:20240429184145+04'00') /Producer ( ) >> endobj 243 0 obj << /Type /XObject /Subtype /Image /Width 1653 /Height 2338 /BitsPerComponent 8 /ColorSpace /DeviceRGB /Filter /DCTDecode /Length 86928 >> stream JFIF 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248 0 obj 15746 endobj 249 0 obj << /Type /Font /Subtype /Type1 /BaseFont /Helvetica /Encoding /WinAnsiEncoding >> endobj 250 0 obj << /Length 2236 >> stream La't- Goodness-of-Fit Tesls lor Pararnetric Hypotheses and the kernel R"(1,r,0) becomes free of d:/Ar, fT ,,\-t-,, \, /- iln'y',r(s)ds1R^(t,r,0)= | ( I y"(s)ds) y"(u)du--los(t-$----,Jo \Ju '-\' / '-" "\ .f,)'r,(s) dslIt is easy to verify that Q,, annulates the compensatonhQ) I;)'"(t) r/s of /\;(r)and finally the process I,l'" has the following form:.,^r rr ' f:'%(r)dr\,^...lV'{"(t) - n-"'lNn(r) + / tog (t - ltt----:::-:-- ldN^(t)lL .' Jo -\ fiv"(s)ds/ JHence W"O does not depend on the value of the parameter d.Acknowledgernents: The authors would like to thank Professors E. V. Khmal-adze and R. D. Gill for their helpful comments and constant attention to their work.References[1] P.K. Andersen, O. Borgan, R. D. Gill, and N. I(eiding, Statislical lvlod.els Based onCounting Processes, Springer, New York, 1993.[2J P.Billingsley, Conrergence ol Probability Measures, Wiley, New York, 1966.[3] P. Bremaud, Point Processes and, Queues: Martingale Dynamdcs, Springer, New York,1981.[4] N.L. Hjort, Goodness-o!-fit tests in models lor lile history ilata based on curnulatiaehozard. rotes, Ann. Statist., 18 (1990), 122l-1258.[5] E.V. Khmaladze, The use o! w2 tests lor testing para,rnetric hgpotheses, Theory Probab.Appl., 24 (1979), 283-30r.[6] E. V. Khmaladze, Martingale approach lo the theory of goodness-ol-fit lesls, TheoryProbab. Appl., 26 (1981), 240-257.[7] E. V. Khmaladze, Goodness oJ fit problem and scanning innoaation marlingcle,r, Ann.Statist., 21 (f993), 798-829.[8] R. S. Liptser and A. N. Shiryaev, A tunctional Central Limit Theorem Jor semimartin-gales, Theory Probab. Appl., 25 (1980), 667-688.[9] A. M. Nikabadze and E. V. Khmaladze, On goodness o! fil tests tor porometric hypothesisin R^ , Soviet Math. Dokl., 35 (1987) , 627-629.[tO] M. S. Nikulin, Chi-square tesl tor continuous distributions uith shifl and, scale parom-elers, Theory Probab. Appl., 18 (1973), 559-568.[11] Ivl. C. J. van Pul,.Asyzrptotic properties of a class o! statisticol models in soJtworereliobility, Scand. J. Statist., r9 (1992), 1-23.ItZ) Z. P. Tsigroshvili, Sorne notes on innoaotion martingales ond goodness-oJ-fit lesls, Proc.A. Razmadze Math. 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