Ja n 20 09 Optimal Explicit Binomial Confidence Interval with Guaranteed Coverage Probability ∗

In this paper, we develop an approach for optimizing the explicit binomial confidence interval recently derived by Chen et al. The optimization reduces conservativeness while guaranteeing prescribed coverage probability. 1 Explicit Formula of Chen et al. Let X be a Bernoulli random variable defined in probability space (Ω,F ,Pr) with distribution Pr{X = 1} = 1 − Pr{X = 0} = p ∈ (0, 1). It is a frequent problem to construct a confidence interval for p based on n i.i.d. random samples X1, · · · ,Xn of X. Recently, Chen et al. have proposed an explicit confidence interval in [2] with lower confidence limit Ln,δ = K n + 3 4 1− 2K n − √ 1 + 9 2 ln 2 δ K(1− Kn ) 1 + 9n 8 ln 2 δ (1) and upper confidence limit Un,δ = K n + 3 4 1− 2K n + √ 1 + 9 2 ln 2 δ K(1− Kn ) 1 + 9n 8 ln 2 δ (2) where K = ∑n i=1 Xi. Such confidence interval guarantees that the coverage probability Pr{Ln,δ < p < Un,δ | p} is greater than 1− δ for any p ∈ (0, 1). Clearly, the explicit binomial confidence interval is conservative and it is desirable to optimize the confidence interval by tuning the parameter δ. This is objective of the next section. ∗The author is currently with Department of Electrical Engineering, Louisiana State University at Baton Rouge, LA 70803, USA, and Department of Electrical Engineering, Southern University and A&M College, Baton Rouge, LA 70813, USA; Email: [email protected]