The construction of a criterion for testing hypotheses about the distribution of random vectors. (English. Ukrainian original) Zbl 0980.62004

Mergel’, V.

Theory Probab. Math. Stat. 61, 131-135 (2000); translation from Teor. Jmovirn. Mat. Stat. 61, 126-130 (2000).

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 \[ \bar\rho_{k}=\left(\prod_{i=1}^{N}\rho_{i,k}\right)^{1/N}, \] 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 [M. Goria, N. Leonenko and 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 \[ H:=-\int_{R^{m}}f(\vec x)\ln f(\vec x)d\vec x \] has the form \[ H_{k.N}=m\ln\bar\rho_{k}+\ln(N-1)-\psi(k)+\ln c_{1}(m), \] 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 L.F. Kozachenko and N.N. Leonenko [Probl. Inf. Transm. 23, 95-101 (1987); translation from Probl. Peredachi Inf. 23, No. 2, 9-16 (1987; Zbl 0633.62005)].
In 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 T. Taguchi [Ann. Inst. Stat. Math. 30, 211-242 (1978; Zbl 0444.62022)] and K. Zografos [J. Multivariate Anal. 71, No. 1, 67-75 (1999; Zbl 0951.62040)] are exploited.

Reviewer: A.V.Swishchuk (Kyïv)

MSC:

62B10	Statistical aspects of information-theoretic topics
62H15	Hypothesis testing in multivariate analysis
62F03	Parametric hypothesis testing

Keywords:

entropy hypothesis testing criterion; Pearson distributions; random vectors

Citations:

Zbl 0633.62005; Zbl 0444.62022; Zbl 0951.62040