By Janos Galambos, Samuel Kotz (auth.)
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Extra resources for Characterizations of Probability Distributions: A Unified Approach with an Emphasis on Exponential and Related Models
For a characterization of the uniform distribution by distributional properties of dr: n of the type that it does not depend on r and F(x) is either super-additive or subadditive, see J,S. C. Arnold and M. Ghosh (1977). 3. Independence of functions of order s t a t i s t i c s . We present in this section a number of characterization theorems based on the independence of certain functions of order staitstics. In addition to the actual results, we also wish to draw attention to the methods of proof.
Another meth- od utilizing characteristic functions has the potential of extending some results of the present section to stability theorems. We shall return to this possibility at the end of this section. We start with the following result which was first proved by M. Fisz (1958) under somewhat different assumptions. 1. Let X 1 and X 2 be independent random variables with cormnon con-, tinuous distribution function F(x). increasing for all x > 0. Assume that F(0) = 0 and that F(x) is strictly Then X2:2 - XI: 2 and XI: 2 are independent if, and only if, F(x) = l-e -bx with some b > 0.
Increasing for all x > 0. Assume that F(0) = 0 and that F(x) is strictly Then X2:2 - XI: 2 and XI: 2 are independent if, and only if, F(x) = l-e -bx with some b > 0. 1 implies that X2:2 ~b~l:2 and Xl: 2 are indeed independent for the exponential distribution F(x) = l-e , x e 0, b > 0. Hence, only the converse statement needs proof. If XI: 2 and X2:2 - Xl: 2 are independent then (19) P(X2:2 - Xl: 2 < XlXl: 2 = z) = P(X2:2 - Xl: 2 < x) for almost all z > 0. Here, "almost all" can refer to Lebesgue measure because of the assur~tions on FCx) that it is continuous and strictly increasing for all x > 0.