Vol. 16, No. 1, 1966

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ISSN: 0030-8730
Some characterizations of exponential-type distributions

Edward Martin Bolger and William Leonard Harkness

Vol. 16 (1966), No. 1, 5–11

Let = {f(x;δ) = exp[+q(δ)], δ (a,b)} be a family of exponential-type probability density-functions (exp. p.d.f.’s) with respect to a σ-finite measure μ. Let M(t;δ),a δ < t < b δ, denote the moment generating function (m.g.f.) corresponding to f(x;δ) ∈ℱ, and let c(t;δ) = lnM(t;δ) = k=1λk(δ)tk∕k! be the cumulative generating function. The main results pertain to characterizations of certain exp. p.d.f.’s in terms of the cumulants λk(δ). First, it is shown that if M(t;δ0) is the m.g.f., respectively, of a degenerate, Poisson, or normal law for some δ0 (a,b), then M(t;δ) is the m.g.f. of the given law for all δ (a,b), and that infinite divisibility (inf. div) of M(t;δ0) for some δ0 implies inf. div. for all δ. Further, it is shown that if φ(t) is a nondegenerate, inf. div. characteristic function (ch. f.) with finite fourth cumulant λ4, then λ4 = 0 if and only if φ(t) is the ch.f. of a normal law, while if λ4 = 3 = a2λ20, then φ(t) is the ch. f. of a Poisson law. Combining these results, it follows that if M(t;δ0) is inf. div., and nondegenerate, with λ4(δ0) = 0, then M(t;δ) is the m.g.f. of a normal law for all δ (a,b). A similar result characterizes the Poisson law. Finally, it is proved that the normal law is the unique exp. p.d.f. which is symmetric.

Mathematical Subject Classification
Primary: 60.20
Secondary: 62.10
Received: 12 March 1964
Revised: 27 July 1964
Published: 1 January 1966
Edward Martin Bolger
William Leonard Harkness