On Tests of Fit Based on Grouped Data

  • Sherzod M. Mirakhmedov
  • Saidbek S. Mirakhmedov


The problem of testing the goodness-of-fit of a continuous distribution for a set of n observations grouped into N equal probability intervals is considered. It is assumed that N as n. Let η1, , η N be the numbers of observations in the intervals. We show that within the class of tests based on statistics of the form fη1 + ⋯ + fη N the classical chi-square test is optimal in terms of the Pitman’s and the Kalenberg’s intermediate asymptotic efficiencies but it is much inferior to tests satisfying Cramer condition in terms of the Kalenberg’s strong intermediate and the Bahadur’s exact asymptotic efficiencies. For the chi-square statistic, a probability of large deviation result, likely to be of its own interest, is proved.


Spacing Test Stat Plan Inference Large Deviation Result Large Deviation Probability Lattice Random Variable 
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© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Sherzod M. Mirakhmedov
    • 1
  • Saidbek S. Mirakhmedov
  1. 1.Ghulam Ishaq Khan Institute of Engineering Sciences and TechnologyTopiPakistan

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