Se p 20 09 Coverage Probability of Random Intervals ∗

In this paper, we develop a general theory on the coverage probability of random intervals defined in terms of discrete random variables with continuous parameter spaces. The theory shows that the minimum coverage probabilities of random intervals with respect to corresponding parameters are achieved at discrete finite sets and that the coverage probabilities are continuous and unimodal when parameters are varying in between interval endpoints. The theory applies to common important discrete random variables including binomial variable, Poisson variable, negative binomial variable and hypergeometrical random variable. The theory can be used to make relevant statistical inference more rigorous and less conservative. 1 Binomial Random Intervals Let X be a Bernoulli random variable defined in a probability space (Ω,F ,Pr) such that Pr{X = 1} = p and Pr{X = 0} = 1− p where p ∈ (0, 1). Let X1, · · · ,Xn be n identical and independent samples of X. In many applications, it is important to construct a confidence interval (L,U) such that Pr{L < p < U | p} ≈ 1 − δ with δ ∈ (0, 1). Here L = L(n, δ,K) and U = U(n, δ,K) are multivariate functions of n, δ and random variable K = ∑n i=1 Xi. To simply notations, we drop the arguments and write L = L(K) and U = U(K). Also, we use notation Pr{L(K) < p < U(K) | p} to represent the probability when the binomial parameter assumes value p. Such notation is used in a similar way throughout this paper. We would thus advise the reader to distinguish this notation from conventional notation of conditional probability. Clearly, the construction of confidence interval is independent of the binomial parameter p. But, for fixed n and δ, the quantity Pr{L(K) < p < U(K) | p} is a function of p and is conventionally referred to as the coverage probability. In many situations, it is desirable to know what is the worst-case coverage probability for p belonging to interval [a, b] ⊆ [0, 1]. For this purpose, we have ∗The author had been previously working with Louisiana State University at Baton Rouge, LA 70803, USA, and is now with Department of Electrical Engineering, Southern University and A&M College, Baton Rouge, LA 70813, USA; Email: [email protected]