A Class of Modified Pearson and Neyman Statistics

L. Györfi, I. Vajda

Research output: Contribution to journalArticle

5 Citations (Scopus)


We consider the model with i.i.d. observations Xi, Xn where Xi ~ μ on (X, A). In this model we define a class of statistics Sa,n for a Є [0, 1] as n where Døa denotes the øa-divergence for øa(t) = (t - 1 )2/(a+ (1 - a) t), t >, and are restrictions of the empirical distribution μn and true distribution μ on the events generated by a measurable partition P = Pn = {An1, Anmn} of X. Sa, n is a blend (nonconvex combination) of the Pearson statistic S1,n = n x 2(μP, μP) and Neyman statistic S0.,n = nX2 (μP, μPn) where x2 denotes the chi-square divergence. We describe the goodness of fit tests and minimum distance estimators defined by means of Sa,n with given a Є [0, 1], or by San,n with given sequence an Є [0, 1]. The main results are conditions under which San,n with arbitrary sequence an Є [0, 1] asymptotically coincides with the Pearson S1,n in the sense San,n - S1,n = op (mn) and (San,n - S1,n)/mn → 0 a.s. The results are applied to extension of limit laws proved for S1,n/mn or So,n/mn to all blended statistics San,n/mn.

Original languageEnglish
Pages (from-to)239-252
Number of pages14
JournalStatistics and Risk Modeling
Issue number3
Publication statusPublished - Jan 1 2001


  • Asymptotic distributions
  • Asymptotic equivalence
  • Neyman statistics
  • Pearson statistics
  • Robust versions

ASJC Scopus subject areas

  • Statistics and Probability
  • Modelling and Simulation
  • Statistics, Probability and Uncertainty

Fingerprint Dive into the research topics of 'A Class of Modified Pearson and Neyman Statistics'. Together they form a unique fingerprint.

  • Cite this