The construction of a criterion for testing hypotheses about the distribution of random vectors (Q2740455)

Let \(\vec\xi\) be a random vector in \(m\)-dimensional Euclidean space with density \(f(\vec x).\) For fixed \(k\in\{1,2,\dots,N-1\}\) set NEWLINE\[NEWLINE\bar\rho_{k}=\left(\prod_{i=1}^{N}\rho_{i,k}\right)^{1/N},NEWLINE\]NEWLINE where \(\rho_{i,k}\) are \(k\)-spacings for the observations of the random vector \(\vec\xi.\) In previous work of the author and his coworkers [\textit{M. Goria, N. Leonenko} and \textit{V. Mergel}, submitted to Aust. N. Z. J. Stat.] it was shown that under some conditions for the function \(f(\vec x)\) the statistical estimation for the entropy NEWLINE\[NEWLINEH:=-\int_{R^{m}}f(\vec x)\ln f(\vec x)d\vec xNEWLINE\]NEWLINE has the form NEWLINE\[NEWLINEH_{k.N}=m\ln\bar\rho_{k}+\ln(N-1)-\psi(k)+\ln c_{1}(m),NEWLINE\]NEWLINE and is asymptotically unbiased and consistent as \(N\to\infty\) for any \(k,\) where \(\psi(k)=(d/dt)\ln\Gamma(t)\) is the digamma function. Properties of such estimators have been studied for the first time by \textit{L.F. Kozachenko} and \textit{N.N. Leonenko} [Probl. Inf. Transm. 23, 95-101 (1987); translation from Probl. Peredachi Inf. 23, No. 2, 9-16 (1987; Zbl 0633.62005)].NEWLINENEWLINENEWLINEIn this paper the entropy hypothesis testing criteria on Pearson type II and Pearson type VII distributions of random vectors and on distributions of general errors of random variables are represented. Results of \textit{T. Taguchi} [Ann. Inst. Stat. Math. 30, 211-242 (1978; Zbl 0444.62022)] and \textit{K. Zografos} [J. Multivariate Anal. 71, No. 1, 67-75 (1999; Zbl 0951.62040)] are exploited.