Abstract
Let X denote a discrete distribution as Poisson, binomial or negative binomial variable. The score confidence interval for the mean of X is obtained based on inverting the hypothesis test and the central limit theorem is discussed and recommended widely. But it has sharp downward spikes for small means. This paper proposes to move the score interval left a little (about 0.04 unit), called by moved score confidence interval. Numerical computation and Edgeworth expansion show that the moved score interval is analogous to the score interval completely and behaves better for moderate means; for small means the moved interval raises the infimum of the coverage probability and improves the sharp spikes significantly. Especially, it has unified explicit formulations to compute easily.
Highlights
Forming a confidence interval (CI) for the mean of a discrete distribution is one of the most basic problems in statistics, since the discrete lattice nature and skewness make the problem complicated
Let X denote a discrete distribution as Poisson, binomial or negative binomial variable
The score confidence interval for the mean of X is obtained based on inverting the hypothesis test and the central limit theorem is discussed and recommended widely
Summary
Forming a confidence interval (CI) for the mean of a discrete distribution is one of the most basic problems in statistics, since the discrete lattice nature and skewness make the problem complicated. For X~π (λ) Poisson distribution with mean λ, a = 1, b = 0; for X~B(n, p) binomial distribution with mean np, a = 1, b = –1/n; for X~NB(r,p) negative binomial distribution, mean rp/q, a = 1, b = 1/r. The score interval is an approximate interval and has many better properties for moderate mean μ. Those articles concern approximate confidence interval almost refer to the score interval. This paper discusses a moved score interval CI(c) for the means of Poisson, binomial and negative binomial variables.
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