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On Mieshalkin-Rogozin theorem and some decompositions

Krysicki W.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

2000

[Z] 40

127-138

This paper was inspired by the work by A. Chaturredi, U. Rani, Estimation Procedures for a Family of Density Functions Representing Various Life Testing Models, Metrika, Vol. 46 (9), 1997 (formula (1) of this paper), which included a discussion of the four parametric density family. In this paper I discuss possibilities of decompositions of this family and applications of the Mieshalkin-Rogozin theorem.[…] After proving the convergence of the series [formula] (Theorem 2) to lnX with probability one, we represente the function (1) as a density of an infinite product of independent r.v.'s (2 cases). A similar representation of a r.v. gamma type in the form of an infinite product of r.v.'s with the same distribution was used by Lu and Richards [10] to define the square of the Vandermonde determinant with random elements. Finally, we apply the modified Mieshalkin-Rogozin theorem [14] for an estimation of the difference of some distribution functions (see Theorems 7 and 8).

On some new properties of the exponential distribution

Krysicki W.

Demonstratio Mathematica

1999

Vol. 32, nr 3

629-638