THE DISTRIBUTION OF COOK'S D STATISTIC

Area of Science:

• Statistics
• Regression Analysis

Background:

• Cook's diagnostic (1977) quantifies the influence of individual observations on regression coefficients in General Linear Univariate Models (GLUM).
• Previous assessments indicated that comparing Cook's diagnostic to the median F-statistic for overall regression captures a variable proportion of influential values.

Purpose of the Study:

• To derive the exact distribution of Cook's statistic for GLUM with Gaussian predictors and response.
• To develop accurate computational forms, approximations, and asymptotic results for Cook's statistic.
• To evaluate the performance of these methods for single and maximum diagnostic values.

Main Methods:

• Derivation of the exact distribution of Cook's statistic under specific model assumptions.
• Development of computational formulas and approximations.
• Monte Carlo simulations to validate the accuracy of the derived distributions and approximations.

Main Results:

• The exact distribution of Cook's statistic for GLUM with Gaussian response and predictors is described.
• Accurate computational forms, simple approximations, and asymptotic results are presented.
• Simulations confirm the accuracy of the derived methods, which allow precise evaluation of single or maximum diagnostic values.

Conclusions:

• The proposed methods provide accurate evaluation of influential observations in GLUM.
• Approximations are effective for single diagnostic values but less so for maximum values.
• Cook's original cut-point yields highly variable tail probabilities, emphasizing the need for careful interpretation and scientific judgment in data analysis.