The problem of testing hypotheses concerning the parameters of the Pólya distribution
The Pólya distribution has the following form: \[ {\mathbf P}_{p,\lambda} (\xi =x/n) ={n\choose x} {p^{[x;\lambda]} (1-p)^{[n-x; \lambda]} \over 1^{[n; \lambda]}}; \tag{1} \] here \(a^{[r;\gamma]} =\prod^{r-1}_{h=0} (a+\gamma h)\) and \(a^{[0; \gamma]}=1\); \(\lambda\) satisfies the condition \(\min\{p+ n\lambda; 1-p+n \lambda\} >0\) (for \(\lambda\geq 0\) this condition is always fulfilled). When \(\lambda=0\), distribution (1) reduces to the binomial law. For \(\lambda= -1/N\); \(p=M/N\) with \(M,N\) being some natural numbers and \(0<M<N\), distribution (1) reduces to the hypergeometric law. The problem of testing hypotheses concerning the parameter \(p\) of the binomial distribution is classical. This problem is a particular case of the problem of testing hypotheses concerning the parameter \(p\) of the Pólya distribution with the parameter \(\lambda\) assumed known. There are a few papers devoted to distinguishing the binomial and hypergeometric distributions. This problem, as well as many others, is also a particular case of a more general problem of testing hypotheses concerning the value of the parameter \(\lambda\) of the Pólya distribution. We solve the problems of testing hypotheses concerning the parameters \(p\) and \(\lambda\) of the Pólya distribution. Numerical examples are also considered.
scientific article; zbMATH DE number 1213174