Shorter Confidence Intervals Using Prior Observations

The purpose of this paper is to make the reader aware of the applicability and advantage of a particular mathematical model. The application is typified by an example and the advantage is via confidence intervals; that is, shorter confidence intervals are possible using the model than if one ignores it, providing the applicability is valid. It is also shown that an improved estimate can be obtained through use of the model. Let f(y|μ,σ) be a normal density with mean μ and variance σ2 and let g(μ|λ, β) be a normal density with mean λ and variance β2. A sequence y 1, y 2, …, y n+1 of independent observations from the mixture of f and g can be considered as follows: An unobservable μi is first drawn from g(μ|λ, β) and then yi which can be observed is drawn from f(y|μi, σ). Confidence intervals on μ n+1 are obtained which are based on the observations y 1, …, y n+1 and which are shorter than the standard interval based on y n+1 only for any n. Shorter intervals are obtained for two cases: (i) λ unkno...