Distributions of the Kullback-Leibler divergence with applications

Area of Science:

• Statistics
• Psychometrics
• Data Analysis

Background:

• The Kullback-Leibler divergence (KLD) is a standard metric for assessing the similarity between two probability distributions.
• The exact statistical distribution of KLD is often unknown in practical applications.
• Assumptions common in psychometrics are key to determining the KLD distribution.

Purpose of the Study:

• To determine the asymptotic distribution of the Kullback-Leibler divergence under psychometric assumptions.
• To explore the utility of KLD in identifying heterogeneous response data.
• To highlight the application of KLD in enhancing test security.

Main Methods:

• Asymptotic statistical analysis was employed to derive the distribution of KLD.
• The derived distribution was shown to follow a scaled chi-square or a scaled F-distribution.
• The study discusses practical applications of these findings.

Main Results:

• The distribution of KLD is asymptotically approximated by a scaled non-central chi-square distribution with one degree of freedom.
• Alternatively, it can be approximated by a scaled doubly non-central F-distribution.
• These approximations provide a basis for statistical inference using KLD.

Conclusions:

• The KLD distribution can be reliably approximated under common psychometric conditions.
• These findings offer a statistical framework for using KLD to detect response heterogeneity.
• KLD analysis is a valuable tool for maintaining test security and data integrity.