Nikulin, M. S. Some recent results on chi-squared tests. (English) Zbl 0752.62038 Queen’s Papers in Pure and Applied Mathematics. 86. Kingston: Queen’s University Press, 74 p. (1991). This paper presents a list of the author’s previous results on \(\chi^ 2\) tests starting from 1973. Paragraph 2 (and §3) deals with \(\chi^ 2\) tests for i.i.d random variables (random vectors, respectively) with discrete exponential distributions of rank one, that is \[ p_ x(\theta)=p_ \theta(X_ i=x)=a(x)\psi^ x(\theta)/B(\theta),\qquad x\in\mathbb{N}. \] The class of these distributions contains many discrete distributions such as the binomial, Poisson, generalized binomial, generalized Poisson, and so on.Many papers were written on chi-square goodness of fit tests with estimated parameters for specific discrete distributions. The method presented in this paper could be viewed as a certain generalization of the results of this sort. The method consists in the following:Consider \(w_ n(x)=p_ \theta(x\mid S_ n)\) which is the minimum variance unbiased estimator for \(p_ x(\theta)\) based on the complete sufficient statistic \(S_ n=\sum^ n_{i=1} X_ i\), then choose any nonrandom partition of \(\mathbb{N}\) for given \(S_ n\) into \(r\) nonempty classes \(J_ 1,\dots,J_ r\) and consider the vector \(\nu=(\nu_ 1,\dots,\nu_ r)^ T\) of frequencies arising as the result of grouping the data into these classes and the corresponding vector of probabilities \(p=(p_ 1,\dots,p_ r)^ T\), where \(p_ i=p_ \theta(X_ j\in J_ i\mid S_ n)\). Finally, construct the quadratic form \(Y^ 2_ n=n^{- 1}(\nu-np)^ T B^ -_ n(\nu-np)\) where \(B^ -_ n\) is a generalized inverse matrix of \(B_ n\) so that \(Y^ 2_ n\overset\mathcal {D}\longrightarrow\chi^ 2_{n-1}\) (Theorem 2, §2).The author considered some examples to which he applied his theory. In Paragraph 1, he constructed a modified \(\chi^ 2\) test for continuously distributed random variables. Reviewer: E.Khmaladze (Moskva) Cited in 2 Documents MSC: 62G10 Nonparametric hypothesis testing 62G99 Nonparametric inference 62F03 Parametric hypothesis testing Keywords:modified chi-square test; discrete exponential distribution of rank one; discrete distributions; binomial; Poisson; generalized binomial; generalized Poisson; chi-square goodness of fit tests; minimum variance unbiased estimator; complete sufficient statistic; grouping; quadratic form; generalized inverse matrix × Cite Format Result Cite Review PDF