Bayesian estimation of survival functions under stochastic precedence

Area of Science:

• Statistics
• Biostatistics
• Statistical Modeling

Background:

• Estimating distributions of random variables X and Y often involves prior knowledge of their relationship.
• Stochastic ordering (Pr(X ≤ z) ≥ Pr(Y ≤ z)) is a strong assumption, often too restrictive for practical applications.
• Weaker constraints, like stochastic precedence (Pr(X ≤ Y) ≥ 0.5), offer more flexibility.

Purpose of the Study:

• To develop a Bayesian estimation method for distribution functions of X and Y under the stochastic precedence constraint.
• To address the limitations of the more restrictive stochastic ordering assumption.
• To provide a computational framework for analyzing data where one variable is generally expected to be larger than another.

Main Methods:

• Bayesian estimation of distribution functions for two random variables X and Y.
• Utilizing the stochastic precedence constraint Pr(X ≤ Y) ≥ 0.5.
• Developing a Gibbs sampling algorithm for posterior computation with categorical variables.
• Generalizing the method for survival time data.

Main Results:

• A flexible Bayesian approach for estimating distributions under stochastic precedence was developed.
• The Gibbs sampling algorithm provides a viable method for posterior computation.
• The methodology was successfully applied to both categorical and survival time data.

Conclusions:

• The proposed Bayesian method effectively incorporates the weaker stochastic precedence constraint.
• This approach offers a practical alternative to strict stochastic ordering when prior information suggests one variable tends to be larger.
• The application to malignant melanoma survival data demonstrates the method's utility in biostatistical research.