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Asymptotic Behaviour of the Variance Function

Bent Jørgensen, José Raúl Martínez and Min Tsao
Scandinavian Journal of Statistics
Vol. 21, No. 3 (Sep., 1994), pp. 223-243
Stable URL: http://www.jstor.org/stable/4616314
Page Count: 21
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Asymptotic Behaviour of the Variance Function
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Abstract

We investigate the asymptotic behaviour of the variance function V of a natural exponential family with support S R. If inf S = 0, we show that V(0) = 0 and that the right derivative at zero is $V^{\prime}(0^{+})=\text{inf}\{S\ \{0\}\}$. Using a theorem by Mora (1990) we show that if lim $c^{-p}V(c\mu)=\mu ^{p}$ uniformly on compact subsets in μ for either c → ∞ or c → 0, then p ∉ (0, 1), and the corresponding exponential dispersion model, suitably scaled, converges to a member of the Tweedie family of exponential dispersion models, corresponding to the variance function $V(\mu)=\mu ^{p}$. This gives a kind of central limit theory for exponential dispersion models. In the case p = 2, the limiting family is gamma, and the result essentially follows from Tauber theory. For p = 1, we obtain a version of the Poisson law of small numbers, generalizing a result for discrete models due to Jørgensen (1986). For 1 < p < 2, the limiting family is compound Poisson, and for p > 2 or p ≤ 0 the limiting families are generated by respectively positive stable distributions or extreme stable distributions, in the latter case inf S = - ∞. A number of illustrative examples are considered.

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