Asymptotic distribution for a discrete version of integrated square error of multivariate density kernel estimators. (English) Zbl 0931.60015

Tenreiro, Carlos

J. Stat. Plann. Inference 69, No. 1, 133-151 (1998).

Summary: We consider the weighted average square error \[ A_n(\pi)= (1/n) \sum^n_{j=1} \{f_n(X_j)- f(X_j)\}^2 \pi(X_j), \] where \(f\) is the common density function of the independent and identically distributed random vectors \(X_1,\dots, X_n\), \(f_n\) is the kernel estimator based on these vectors and \(\pi\) is a weight function. Using U-statistics techniques and the results of C. Gouriéroux and the author [Preprint 9617, Departamento de Matemática, Universidade de Coimbra, 1996], we establish a central limit theorem for the random variable \(A_n(\pi)- EA_n(\pi)\). This result enables us to compare the stochastic measures \(A_n(\pi)\) and \(I_n(\pi\cdot f)= \int\{f_n(x)- f(x)\}^2(\pi\cdot f)(x) dx\) and to deduce an asymptotic expansion in probability for \(A_n(\pi)\) which extends a previous one, obtained, in a real context with \(\pi= 1\), by P. Hall [J. Multivariate Anal. 14, 1-16 (1984; Zbl 0528.62028)]. The approach developed in this paper is different from the one adopted by Hall, since he uses Komlós-Major-Tusnády-type approximations to the empiric distribution function. Finally, applications to goodness-of-fit tests are considered. More precisely, we present a consistent test of goodness-of-fit for the functional form of \(f\) based on a corrected bias version of \(A_n(\pi)\), and we study its local power properties.

MSC:

60F05	Central limit and other weak theorems
62G10	Nonparametric hypothesis testing

Keywords:

kernel estimators; average square error; asymptotic distribution; U-statistics; goodness of fit

References:

[1]	Bickel, P.J.; Rosenblatt, M., On some global measures of the deviations of density function estimates, Ann. statist., 1, 1071-1095, (1973) · Zbl 0275.62033
[2]	Fan, Y., Testing the goodness of fit of a parametric density function by kernel method, Econometric theory, 10, 316-356, (1994)
[3]	Gouriéroux, C.; Tenreiro, C., Local power properties of kernel based goodness of fit tests, () · Zbl 1081.62529
[4]	Hall, P., Limit theorems for stochastic measures of the accuracy of density estimators, Stochastic process. appl., 13, 11-25, (1982) · Zbl 0486.60022
[5]	Hall, P., Central limit theorem for integrated square error properties of multivariate nonparametric density estimators, J. multivariate anal., 14, 1-16, (1984) · Zbl 0528.62028
[6]	Hoeffding, W., A class of statistics with asymptotically normal distribution, Ann. math. statist., 19, 293-325, (1948) · Zbl 0032.04101
[7]	Parzen, E., On estimation of a probability density function and mode, Ann. math. statist., 33, 1065-1076, (1962) · Zbl 0116.11302
[8]	Rosenblatt, M., Remarks on some non-parametric estimates of a density function, Ann. math. statist., 27, 832-837, (1956) · Zbl 0073.14602
[9]	Tenreiro, C., Théorèmes limites pour LES erreurs quadratiques intégrées des estimateurs à noyau de la densité et de la régression sous des conditions de dépendance, C. R. acad. sci. Paris, 320, 1535-1538, (1995), Séri. I · Zbl 0836.62033
[10]	Tenreiro, C., Estimation fonctionnelle: applications aux tests d’adéquation et de paramètre constant, (1995), Département de Mathématiques de l’Université de Coimbra, (French version of Ph.D. Thesis)
[11]	Wegman, E.J., Nonparametric probability density estimation: A comparison of density estimation methods, J. statist. comput. simulation, 1, 225-245, (1972) · Zbl 0243.62029
[12]	Yoshihara, K., Limiting behavior of U-statistics for stationary, absolutely regular processes, Z. wahrsch. verw. gebiete, 35, 237-252, (1976) · Zbl 0314.60028