The Multivariate Normal Mean with Intraclass Correlated Components: Estimation of Urban Fire Alarm Probabilities

Abstract

Assume that X has a k-dimensional normal distribution (k ≥ 4) with mean vector θ and known diagonal covariance matrix. Empirical Bayes estimators are given for θ under the further assumption that θ has a conjugate prior distribution with intraclass correlation structure. For var(x_i | θ _i ) = σ², i = 1, …, k, it is shown that the estimators are minimax. Our methods are applied to Carter and Rolph's (1974) Bronx fire alarm data: an urban neighborhood contains k alarm boxes, and we require the estimated probability that a given alarm indicates “structural fire.” Estimates based upon yearly (1967–69) data are evaluated against actual 1970 outcomes.

Key Words:

• Intraclass correlated means
• Conjugate prior distributions
• Structural fire probabilities
• Minimax estimators