Constructing Prediction Intervals Using the Likelihood Ratio Statistic

TL;DR

This paper introduces a likelihood ratio-based method for constructing prediction intervals that offers strong statistical properties and broad applicability, including cases without pivotal quantities.

Contribution

It proposes a general prediction approach using likelihood ratios involving data and future variables, enhancing prediction interval construction.

Findings

Provides a unified framework for prediction intervals

Ensures good coverage properties for various data types

Identifies existing pivotal-based methods as special cases

Abstract

Statistical prediction plays an important role in many decision processes such as university budgeting (depending on the number of students who will enroll), capital budgeting (depending on the remaining lifetime of a fleet of systems), the needed amount of cash reserves for warranty expenses (depending on the number of warranty returns), and whether a product recall is needed (depending on the number of potentially life-threatening product failures). In statistical inference, likelihood ratios have a long history of use for decision making relating to model parameters (e.g., in evidence-based medicine and forensics). We propose a general prediction method, based on a likelihood ratio (LR) involving both the data and a future random variable. This general approach provides a way to identify prediction interval methods that have excellent statistical properties. For example, if a prediction method can be based on a pivotal quantity, our LR-based method will often identify it. For applications where a pivotal quantity does not exist, the LR-based method provides a procedure with good coverage properties for both continuous or discrete-data prediction applications.