# 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

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

Preview not available

## 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.

• [223]
• 224
• 225
• 226
• 227
• 228
• 229
• 230
• 231
• 232
• 233
• 234
• 235
• 236
• 237
• 238
• 239
• 240
• 241
• 242
• 243