Some Characterizations of The Exponential Family
This paper introduces some characterizations concerning the exponential family. Recurrence relation between two consecutive conditional moments of h(z) given x<z<y is presented. In addition, an expression of V[h(Z]x<Z< y)as well as a closed form of E[hr(Z)x<Z< y] in terms of the failure rate and the reversed failure rate is deduced. Finally, the left rth truncated moment of h(Yk) ( where Yk is the Kth order statistic) is expressed in terms of a polynomial, h(-) , of degree r. Some results concerning the exponentiated Pareto, exponentiated Weibull, the Modified Weibull, Weibull, generalized exponential, Linear failure rate,1st type Pearsonian distributions, Burr, power and the uniform distributions are obtained as special cases.
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