Some Characterizations of The Exponential Family

  • Ali Ahmed A-Rahman Professor of Mathematical Statistics,Cairo University,Egypt.
Keywords: Characterization, left, right and double truncated moments, conditional variance, order statistics, recurrence relations, exponentiated Pareto, exponentiated Weibll, generalized exponential, 1^sttype Pearsonian distributions, Burr, Power, beta, uniform distributions


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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How to Cite
A-Rahman, A. A. (2018). Some Characterizations of The Exponential Family. MathLAB Journal, 1(3), 241-250. Retrieved from
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