×

Invariance principles for von Mises and U-statistics. (English) Zbl 0552.60030

Let \(\{X_ j,j\geq 1\}\) be a sequence of i.i.d. random variables with a common distribution function F. Let \(F_ n\) be the empirical distribution function of a sample of size n. The empirical process R is defined as \(R(s,t)=t(F_{[t]}(s)-F(s))\), \(s\in {\mathbb{R}}\), \(t\geq 0\), where [t] denotes the largest integer not exceeding t. Let \(h:R^ 2\to R\) be a measurable function and define \(\| h\|_ p=(\iint | h(x,y)|^ pdF(x)dF(y))^{1/p}+(\int | h(x,x)|^ pdF(x))^{1/p}.\) If \(\| h\|_ 1<\infty\) then the stochastic double integral \(V_ n(h)=\iint h(x,y)R(dx,n)R(dy,n)\) is called a von- Mises statistic. The U-statistic is defined as follows (disregarding normalizing constants) \(U_ n(h)=\sum_{1\leq j\neq i\leq n}h(X_ i,X_ j)\), where h satisfies the usual symmetry condition.
The authors obtain sure approximation of \(V_ n(h)\) and \(U_ n(h)\) by \(W_ t(h)=\iint h(x,y)K(dx,t)K(dy,t).\) Here K(s,t) is a standard Kiefer process (a separable Gaussian process \(\{\) K(s,t), \(0\leq s\leq 1\), \(t\geq 0\}\) with the following conditions: K(s,0)\(\equiv 0\), \(K(0,t)=K(1,t)\equiv 0\), E K(s,t)\(=0\), \(EK(s,t)K(s',t')=(t\wedge t')s(1- s')\), \(0\leq s\leq s'\leq 1\), t,t’\(\geq 0)\). As applications almost sure versions for the estimator of the variance and for the \(\chi^ 2\) test of goodness of fit are obtained.
Reviewer: N.Gamkrelidze

MSC:

60F17 Functional limit theorems; invariance principles
62G10 Nonparametric hypothesis testing
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Berkes, I.; Morrow, G. J., Strong invariance principles for mixing random field, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 57, 15-37 (1981) · Zbl 0443.60029 · doi:10.1007/BF00533712
[2] Berkes, I.; Philipp, W., Approximation theorems for independent and weakly dependent random vectors, Annals Probability, 7, 29-54 (1979) · Zbl 0392.60024 · doi:10.1214/aop/1176995146
[3] Csáki, E., On the standardized empirical distribution function, Coll. Mathem. Soc. János Bolyai, 32, 123-138 (1980) · Zbl 0511.62049
[4] Denker, M., Grillenberger, C., Keller, G.: A note on invariance principles for v. Mises’ statistics. Metrika, to appear · Zbl 0609.62029
[5] de Wet, T.; Venter, J. H., Asymptotic distributions for quadratic forms with applications to tests of fit, Ann. Statist., 1, 380-387 (1973) · Zbl 0256.62018 · doi:10.1214/aos/1176342378
[6] Doob, J. L., Stochastic Processes (1953), New York: Wiley, New York · Zbl 0053.26802
[7] Fernique, X., Intégrabilité des vecteurs gaussiens, C.R. Acad. Sci. Paris Sér. A, 270, A1698-1699 (1970) · Zbl 0206.19002
[8] Fillipova, A. A., Mises’ theorem on the asymptotic behavior of functionals of empirical distribution functions and its statistical applications, Theory Probability Applications, 7, 24-57 (1962) · Zbl 0118.14501 · doi:10.1137/1107003
[9] Gregory, G. G., Large sample theory for U-statistics and tests of fit, Ann. Statist., 5, 110-123 (1977) · Zbl 0371.62033 · doi:10.1214/aos/1176343744
[10] Hall, P., On the invariance principle for U-statistics, Stochastic Processes Applications, 9, 163-174 (1979) · Zbl 0422.62019 · doi:10.1016/0304-4149(79)90028-0
[11] Hoeffding, W., A class of statistics with asymptotically normal distribution, Ann. Math. Statist., 19, 293-325 (1948) · Zbl 0032.04101 · doi:10.1214/aoms/1177730196
[12] Kuelbs, J., Kolmogorov’s law of the iterated logarithm for Banach space valued random variables, Illinois J. Math., 21, 784-800 (1977) · Zbl 0392.60010
[13] Loynes, R. M., On the weak convergence of U-statistic processes and of the empirical processes, Math. Proc. Cambridge Philos. Soc., 83, 269-272 (1978) · Zbl 0403.60034 · doi:10.1017/S0305004100054530
[14] Miller, R. G. Jr.; Sen, P. K., Weak convergence of U-statistics and von Mises’ differentiable statistical functions, Ann. Math. Statist., 43, 31-41 (1972) · Zbl 0238.62057 · doi:10.1214/aoms/1177692698
[15] von Mises, R., On the asymptotic distribution of differentiable statistical functions, Ann. Math. Statist., 18, 309-348 (1947) · Zbl 0037.08401 · doi:10.1214/aoms/1177730385
[16] Moricz, F., Moment inequalities for the maximum of partial sums of random fields, Acta Sci. Math. Hungar., 39, 353-366 (1977) · Zbl 0363.60022
[17] Morrow, G.; Philipp, W., An almost sure invariance principle for Hilbert space valued martingales, Trans. Amer. Math. Soc., 273, 231-251 (1982) · Zbl 0508.60014 · doi:10.1090/S0002-9947-1982-0664040-3
[18] Neuhaus, G., Functional limit theorems for U-statistics in the degenerate case, J. Multivariate Analysis, 7, 424-439 (1977) · Zbl 0368.60034 · doi:10.1016/0047-259X(77)90083-5
[19] Philipp, W., Almost sure invariance principles for sums of B-valued random variables, Probability in Banach spaces, II, (Proc. Conf. Oberwolfach, 1978), Lecture Notes in Math. 709, 171-193 (1979), Berlin-Heidelberg-New York: Springer, Berlin-Heidelberg-New York · Zbl 0418.60013
[20] Sen, P. K., Almost sure behaviour of U-statistics and von Mises’ differentiable statistical functions, Ann. Statist., 2, 387-396 (1974) · Zbl 0276.60009 · doi:10.1214/aos/1176342675
[21] Stout, W. F., Almost Sure Convergence (1974), New York: Academic Press, New York · Zbl 0321.60022
[22] Yurinskii, V. V., On the error of the Gaussian approximation for convolutions, Theory Probability Appl., 22, 236-247 (1977) · Zbl 0378.60008 · doi:10.1137/1122030
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.