Information for Estimating the Proportions in Mixtures of Exponential and Normal Distributions

Abstract

The Fisher information I ( p ; f 1 , f 2 ) for estimating the proportion p in a mixture λ( x ) = pf 1 ( x ) +(1 − p ) f 2 ( x ) of two densities is investigated. A general power series expansion is obtained, which is then explored in detail for the case of two exponential densities, and for the case of two normal densities with equal scale. Simple approximations are obtained, for example when (μ 1 − μ 2 /σ) is near zero in a mixture of two normal distributions with means μ 1 and μ 2 and common variance σ 2 , and when α/β is near unity in a mixture of two exponential distributions with mean lives (α) −1 and (β) −1 , α < β. Brief tables based on the various approximations are presented, giving an overall picture of the information. The main qualitative conclusion is that extremely large, and often impractical, sample sizes are required to obtain even moderate precision in estimating p unless the mixed distributions are very well separated.