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Infinitely divisible distributions arising from first crossing times and related results

Abstract

Results of Jain and Khan (1979) and Khan and Jain (1978) on the time to first emptiness of a reservoir are generalized to include the case of defective random variables where the mass at ∞ can be positive. The assumption of an underlying exponential family is not needed — the general condition is infinite divisibility and closure under convolutions. The support of the distributions can be nonnegative reals or nonnegative integers. Examples are given to illustrate the general theory and show the bivariate extension.

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References

  • Consul, P.C. (1989) Generalized Poisson distribution: Properties and applications. Marcel Dekker, New York.

    MATH  Google Scholar 

  • Hassairi, A. (1999). Generalized variance and exponential families. Ann. Statist., 27, 374–385.

    MathSciNet  MATH  Article  Google Scholar 

  • Jain, G.C. and Consul, P.C. (1971). A generalized negative binomial distribution. SIAM J. Appl. Math., 21, 501–513.

    MathSciNet  MATH  Article  Google Scholar 

  • Jain, G.C. and Khan, M.S.H. (1979). On an exponential family. Math. Operationsforsch. Statist., Ser. Statist., 10, 153–168.

    MathSciNet  MATH  Google Scholar 

  • Johnson, N.L., Kotz, S. and Kemp, A.W. (1994). Univariate discrete distributions. Wiley, New York.

    Google Scholar 

  • Kendall, D.G. (1957). Some problems in the theory of dams. J. Roy. Statist. Soc. Ser. B, 2, 207–212.

    MathSciNet  Google Scholar 

  • Khan, M.S.H. and Jain, G.C. (1978). A class of distributions in the first emptiness of a semi-infinite reservoir. Biometrical J. Zeitschrift, 20, 243–252.

    MathSciNet  MATH  Article  Google Scholar 

  • Letac, G. and Mora, M. (1990). Natural exponential families with cubic variance functions. Ann. Statist., 18, 1–37.

    MathSciNet  MATH  Article  Google Scholar 

  • Marshall, A.W. and Olkin, I. (2007). Life distributions: Structure of nonparametric, semiparametric, and parametric families. Springer, New York.

    MATH  Google Scholar 

  • Morris, C.N. (1982). Natural exponential families with quadratic variance functions. Ann. Statist., 10, 65–80.

    MathSciNet  MATH  Article  Google Scholar 

  • Seshadri, V. (1993). The inverse Gaussian distribution. A case study in exponential families. Clarendon, Oxford.

    Google Scholar 

  • Seshadri, V. (1999). The inverse Gaussian distribution. Statistical theory and applications. Lecture Notes in Statistics, v. 137, Springer, New York.

    MATH  Book  Google Scholar 

  • Vandal, A.C. (1994). A compendium of variance functions for real natural exponential families. M.Sc. thesis, McGill University.

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Acknowledgement

Thanks to Alain Vandal for providing his M.Sc. thesis.

This research was supported by an NSERC Discovery Grant.

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Correspondence to Harry Joe.

Appendix

Appendix

A.1 Natural exponential family; an example

Let f 0(z) be a density (say, with respect to Lebesgue or counting measure) with cumulant generating function ψ(t) over a range t ∈ Θ where the moment generating function e ψ is finite. Then, as in Morris (1982), the exponential family

$$ e^{\theta z-\psi(\theta)} f_0(z), \quad \theta\in \Theta, $$

can be derived. Its mean and variance are ψ′(θ) and ψ′′(θ) respectively. If f 0(z) = f 0(z;λ) is infinitely divisible with convolution parameter λ, then the cumulant generating function has the form of ψ(t) = ψ λ (t) = λψ 1(t), where ψ 1(t) is the cumulant generating function of f 0(z;1). The exponential family with “natural parameter” θ and convolution parameter λ > 0 is

$$ e^{\theta z-\lambda\psi_1(\theta)} f_0(z;\lambda), \quad \theta\in \Theta, \quad \lambda>0. $$
(A.1)

For non-Poisson families, a non-convolution parameter appears in f 0 and it combines with θ in (A.1). If f 0(z;λ) is Poisson(λ), then (A.1) is Poisson(λe θ). From its definition, it follows that (A.1) has cumulant generating function λ[ψ 1(θ + t) − ψ 1(t)] for t such that θ ∈ Θ, θ + t ∈ Θ; furthermore its mean and variance are respectively λψ1(θ) and λψ′′1(θ) for θ ∈ Θ.

This is illustrated for the negative binomial distribution NB(λ,ξ) with 0 < ξ < 1 fixed and q = 1 − ξ. The probability mass function is

$$ f_0(z;\lambda)=\frac{\Gamma(\lambda+z)}{\Gamma(\lambda)\,z!} \xi^\lambda q^z, \quad z=0,1,2,\ldots, $$

and its cumulant generating function is \(\psi_\lambda(t)= \lambda[\log \xi-\log(1-qe^t)]\) and Θ = {t: qe t < 1}. The natural exponential family in (A.1) is

$$ e^{\theta z-\lambda[\log \xi-\log(1-qe^\theta)]} \frac{\Gamma(\lambda+z)}{\Gamma(\lambda)\,z!} \xi^\lambda q^z = \frac{\Gamma(\lambda+z)}{\Gamma(\lambda)\,z!} (1-qe^\theta)^\lambda (qe^\theta)^x. $$
(A.2)

That is, this is the NB(λ,1 − qe θ) distribution and the natural exponential family generated from NB(λ,ξ) is NB with a different parametrization of the non-convolution parameter. For the parametrization of (A.2), the mean is μ(θ) = λqe θ/(1 − qe θ), the variance is V(θ) = λqe θ/(1 − qe θ)2 and V(θ) = μ(θ) + μ 2(θ)/λ for all θ ∈ Θ.

A.2 Results on derivatives of univariate and bivariate Laplace transforms

For the first two results, let Y be a nonnegative random variable with finite mean μ Y and LT L Y (s).

Result A1. |L Y (s)/L Y (s)| ≤ μ Y for all s ≥ 0.

Proof

Note that \(|L'_Y(s)|/L_Y(s)=\int_0^\infty ye^{-sy}dF_Y(y)/L_Y(s)\). Consider the family of densities with \(g(y;s)=e^{-sy}f_Y(y)/L_Y(s)\) on Lebesgue or counting measure. As s increases, the density g(·;s) puts less weight on large values of y. Hence g(·;s) is stochastic decreasing as s increases (from page 261 of Marshall and Olkin 2007, this family is decreasing in the stronger likelihood ratio order). Hence the expected value of Y * with density g(·;s) is decreasing in s. The expected value for s = 0 is μ Y and hence this is a bound.

Result B1. lim s→ ∞  |L Y (s)/L Y (s)| = 0.

Proof

For densities g(·;s) defined above, the family converges in distribution to the degenerate random variable 0 as s→ ∞.

The above results can be extended to bivariate LTs with similar proofs. Let (Y 1,Y 2) be a nonnegative random pair with finite means \(\mu_{Y_1},\mu_{Y_2}\) and LT \(L_{Y_1,Y_2}(s_1,s_2)\).

Result A2.

$$ \left|\frac{\partial L_{Y_1,Y_2}(s_1,s_2)}{\partial s_j}\right|\left /\vphantom{\frac{\partial L_{Y_1,Y_2}(s_1,s_2)}{\partial s_j}}\right. L_{Y_1,Y_2}(s_1,s_2)\le \mu_{Y_j}, $$

j = 1,2, for all s 1,s 2 ≥ 0.

Proof

$$ \frac{|\frac{\partial L_{Y_1,Y_2}(s_1,s_2)}{ \partial s_j}| }{ L_{Y_1,Y_2}(s_1,s_2)} =\frac{\int_0^\infty y_je^{-s_1y_1-s_2y_2}dF_{Y_1,Y_2}(y_1,y_2) }{ L_{Y_1,Y_2}(s_1,s_2)}. $$

Consider the family of densities with

$$ g(y_1,y_2;s_1,s_2)=e^{-s_1y_1-s_2y_2}f_{Y_1,Y_2}(y_1,y_2) /L_{Y_1,Y_2}(s_1,s_2) $$

on Lebesgue or counting measure. As s 1,s 2 increase, the density g(·;s 1,s 2) puts less weight on large values of y 1,y 2. Hence g(·;s 1,s 2) is stochastic decreasing as s 1,s 2 increase. Hence the expected values of random variables \(Y_1^*,Y_2^*\) with density g(·;s 1,s 2) are decreasing in s 1,s 2. The expected value for s 1 = s 2 = 0 are \(\mu_{Y_1},\mu_{Y_2}\) and hence these are bounds.

Result B2.

$$ \lim\limits_{s\to\infty} \left|\frac{\partial L_{Y_1,Y_2}(sr_1,sr_2)}{\partial s_j}\left/\vphantom{\frac{\partial L_{Y_1,Y_2}(sr_1,sr_2)}{\partial s_j}}\right. L_{Y_1,Y_2}(sr_1,sr_2)\right|=0. $$

Proof

For densities g(·;s 1,s 2) defined above, the family converges in distribution to the degenerate random vector (0,0) as s→ ∞ with s 1 = sr 1 and s 2 = sr 2.

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Joe, H., Seshadri, V. Infinitely divisible distributions arising from first crossing times and related results. Sankhya A 74, 222–248 (2012). https://doi.org/10.1007/s13171-012-0002-z

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Keywords and phrases

  • Convolution-closed
  • defective distribution
  • generalized negative binomial
  • generalized Poisson
  • Kendall-Ressel distribution
  • natural exponential family
  • strict arcsine distribution

AMS (2000) subject classification

  • Primary 60E07; Secondary 62E19