Abstract
Statistical procedures based on the estimated empirical process are well known for testing goodness of fit to parametric distribution families. These methods usually are not distribution free, so that the asymptotic critical values of test statistics depend on unknown parameters. This difficulty may be overcome by the utilization of parametric bootstrap procedures. The aim of this paper is to prove a weak approximation theorem for the bootstrapped estimated empirical process under very general conditions, which allow both the most important continuous and discrete distribution families, along with most parameter estimation methods. The emphasis is on families of discrete distributions, and simulation results for families of negative binomial distributions are also presented.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Al-Saleh MF, Al-Batainah FK (2003) Estimation of the proportion of sterile couples using the negative binomial distribution. J Data Sci 1:261–274 [http://www.sinica.edu.tw/~jds/JDS-130.pdf]
Babu GJ, Rao CR (2004) Goodness-of-fit tests when parameters are estimated. Sankhyā: Indian J Statist 66:63–74
Bartko JJ (1966) Approximating the negative binomial. Technometrics 8:345–350
Bickel PJ, Klaassen CAJ, Ritov Y, Wellner JA (1993) Efficient and adaptive estimation for semiparametric models. Johns Hopkins University Press, Baltimore
Bretagnolle J, Massart P (1989) Hungarian constructions from the nonasymptotic viewpoint. Ann Probab 17:239–256
Burke MD, Csörgő M, Csörgő S, Révész P (1979) Approximations of the empirical process when parameters are estimated. Ann Probab 7:790–810
Csörgő S, Rosalsky A (2003) A survey of limit laws for bootstrapped sums. Int J Math Math Sci 45:2835–2861
Durbin J (1973) Weak convergence of the sample distribution, when parameters are estimated. Ann Statist 1:279–290
Genz M, Haeusler E (2006) Empirical processes with estimated parameters under auxiliary information. J Comput Appl Math 186:191–217
Gürtler N, Henze N (2000) Recent and classical goodness-of-fit tests for the Poisson distribution. J Statist Plan Inference 90:207–225
Hall P (1992) The bootstrap and Edgeworth expansion. Springer, New York
Henze N (1996) Empirical-distribution-function goodness-of-fit tests for discrete models. Can J Statist 24:81–93
Komlós J, Major P, Tusnády G (1975) An approximation of partial sums of independent R.V.’s, and the sample D.F.I. Z., Wahrscheinlichkeitstheorie und verw. Gebiete 32:111–131
Major P (1988) On the tail behavior of the distribution function of multiple stochastic integrals. Probab Theory Relat Fields 78:419–435
Romano JP (1989) Bootstrap and randomization tests of some nonparametric hypotheses. Ann Statist 17:141–159
Rueda R, Pérez-Abreu V, O’Reilly F (1991) Goodness of fit for the Poisson distribution based on the probability generating function. Commun Statist Theory Methods 20:3093–3110 [Corrigendum: (1992) 21:3623]
Stute W, Gonzáles-Manteiga W, Presedo-Quindimil M (1993) Bootstrap based goodness-of-fit-tests. Metrika 40:243–256
Székely GJ, Rizzo ML (2004) Mean distance test of Poisson distribution. Statist Probab Lett 67:241–247
Szűcs G (2005) Approximations of empirical probability generating processes. Stat Decis 23:67–80
Tsirel’son VS (1975) The density of the maximum of a Gaussian process. Theory Probab Appl 20:817–856
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Szűcs, G. Parametric bootstrap tests for continuous and discrete distributions. Metrika 67, 63–81 (2008). https://doi.org/10.1007/s00184-006-0122-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-006-0122-3