New findings in well-trodden spaces: what happens in statistical analysis of finely grouped data?

Abstract

This is planned as free talk on two related but different problems.

1. Divisible statistics are a class of statistics designed for analysis of grouped data — for example, K.Pearson’s chi-square statistic, statistic of empty boxes, spectral statistics, likelihood ratio statistic for discrete distribution, and other equally well-known statistics. What is there new to say about them? Yet, we ask ourselves if they are good for testing contiguous (i.e. local) alternatives in the context of Poisson regression, which is the same as the context of the Large Number of Rare Events, which is also the same as analysis of finely grouped data. What comes out does not look like the result many of us would expect.

2. Unlike directed test statistics, which are very good for detecting one particular deviation from the null, but are very bad for huge number of other deviations, K.Pearson’s chi-square statistic, in the classical situation, is a goodness of fit statistic — a single statistic, with no spectacular power, but sensitive to large number of very different deviations from the null. However, in the situation we pointed out in 1., this property is lost and there is no other goodness of fit statistics in its stead. However, we show that it is possible to restore not one but the class of goodness of fit test statistics, which are also distribution free. The terminology used will include “function-parametric empirical processes”,  “projection operators”, “unitary operators”. However, the outcome — we hope — will not just be a mathematical construction, but a practically usable method.

The hope is that the paper [1] will be published in 2026, and the material will appear in the chapter on regression in the book [2].

The intention is to cover both topics, but we will cover what the time permits.

References;

[1] S. Algeri, E.Khmaladze, On the statistical analysis of grouped data: when Pearson χ2 and other divisible statistics are not goodness-of-fit, Revision submitted to JRSS(B) (2026)

[2] E. Khmaladze, Distribution-free testing in statistics, Chapman and Hall, 2027