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New statistical tests in the multinomial scheme and their asymptotic properties. (Russian) Zbl 0645.62050
The following generalization of the chi-square goodness-of-fit test is considered:
Let the i.i.d. rv’s \(X_ 1,X_ 2,...,X_ n\) be grouped into N equiprobable classes and let \(\nu_ 1,\nu_ 2,...,\nu_ n\) be the appropriate frequencies. Consider the statistic defined by \[ L_{N,s}(f)=\sum^{N}_{i=1}f(\nu_ i,\nu_{i\oplus 1},...,\nu_{i\oplus s}) \] where \(i\oplus j=(i+j) mod N\) and f is a (nonlinear) function. Asymptotic theory (as N,n\(\to \infty\), n/N\(\to \alpha\), \(0<\alpha <\infty)\) of tests based on statistics of the above type is presented. A class of contiguous alternatives is formulated and optimality of the generalized tests is discussed.
Reviewer: R.Zielinski

62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference