[Paper Review] Coverage Probability of Random Intervals

Xinjia Chen|arXiv (Cornell University)|Jul 19, 2007

Bayesian Modeling and Causal Inference3 references4 citations

TL;DR

This paper develops a general theory for computing the minimum coverage probability of random intervals defined by discrete random variables with continuous parameter spaces. It proves that for monotonic interval estimators, the worst-case coverage probability occurs at discrete points—specifically, interval endpoints and parameter values where interval bounds cross the parameter range—enabling more accurate and less conservative statistical inference for binomial, Poisson, negative binomial, and hypergeometric distributions.

ABSTRACT

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.

Motivation & Objective

• To establish a rigorous framework for analyzing the worst-case coverage probability of random intervals in discrete distributions with continuous parameters.
• To address the limitation in classical confidence interval methods that often produce overly conservative coverage due to discrete sampling distributions.
• To identify the exact set of parameter values where minimum coverage probability occurs, enabling precise computation of worst-case performance.
• To generalize results across key discrete distributions including binomial, Poisson, negative binomial, and hypergeometric variables.
• To support more accurate sample size determination and statistical inference by replacing conservative approximations with exact worst-case bounds.

Proposed method

• Derives theoretical conditions under which the coverage probability function of random intervals is continuous and unimodal with respect to the parameter.
• Applies monotonicity constraints on interval bounds $L(k)$ and $U(k)$ as functions of the sufficient statistic $k$ to ensure tractable analysis.
• Identifies critical parameter values—$a$, $b$, and points where $L(k)$ or $U(k)$ lie within $(a,b)$—as the only candidates for minimum coverage probability.
• Uses set-theoretic and probability arguments to show that coverage probability remains unimodal between consecutive critical points in the parameter space.
• Employs auxiliary functions $C(p)$, $C_U(p)$, and $C_L(p)$ to compute exact infimums for closed intervals $[L,U]$.
• Leverages lemmas on interval equivalence and probability mass behavior to prove unimodality of coverage probability across parameter intervals.

Experimental results

Research questions

• RQ1 Where in the parameter space does the coverage probability of a random interval achieve its minimum for discrete distributions with continuous parameters?
• RQ2 How can the worst-case coverage probability be computed exactly rather than conservatively approximated?
• RQ3 Under what conditions is the coverage probability function unimodal with respect to the parameter?
• RQ4 Can the minimum coverage probability be confined to a finite, computable set of parameter values for monotonic interval estimators?
• RQ5 How do the results extend across different discrete distributions such as binomial, Poisson, negative binomial, and hypergeometric?

Key findings

• The minimum coverage probability of a random interval occurs at a discrete, finite set of points: the interval endpoints $a$ and $b$, and all values $p$ such that $L(k) = p$ or $U(k) = p$ for some $k$.
• For monotonic $L(k)$ and $U(k)$, the coverage probability $\Pr\{L(K) < p < U(K) \mid p\}$ is continuous and unimodal in $p$ over intervals between consecutive critical points.
• For closed intervals $[L,U]$, the infimum of the coverage probability is the minimum of $\{C(a), C(b)\} \cup \{C_U(p)\} \cup \{C_L(p)\}$ over specified discrete sets.
• The theory applies to all standard discrete distributions: binomial, Poisson, negative binomial, and hypergeometric, under monotonicity of interval bounds.
• The results allow for exact computation of worst-case coverage, reducing the conservativeness of classical confidence intervals.
• The unimodality of coverage probability ensures that optimization over the parameter space can be efficiently performed by evaluating only finitely many points.