Asymptotic Behaviour of the Variance Function
研究了自然指数族方差函数的渐近行为，证明在特定缩放下指数色散模型收敛到Tweedie族，为指数色散模型提供了中心极限理论。
We investigate the asymptotic behaviour of the variance function V of a natural exponential family with support S c R. If inf S = 0, we show that V(O) = 0 and that the right derivative at zero is V'(O+) = inf {S\{0}}. Using a theorem by Mora (1990) we show that if lim c -P V(cp) = uP uniformly on compact subsets in p for either c -+ oo or c -+0, then p 0 (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(p) = pP. 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 Jorgensen (1986). For 1 2 or p < 0 the limiting families are generated by respectively positive stable distributions or extreme stable distributions, in the latter case inf S =-oo. A number of illustrative examples are considered.