Determining the lifetime distribution using fractional moments with maximum entropy.

Area of Science:

• Statistics
• Probability Theory
• Inverse Problems

Background:

• Ill-posed inverse problems are common in various scientific fields.
• Determining probability distributions from limited data is challenging.
• Lifetime and survival probability estimation are critical in many applications.

Purpose of the Study:

• To develop a model-free, non-parametric method for solving inverse problems.
• To estimate probability density functions, specifically lifetime and survival probabilities, from fractional moments.
• To analyze the implications for exponential models and model selection.

Main Methods:

• A maximum entropy-based approach was employed.
• The method is model-free and non-parametric.
• It utilizes fractional moments of the probability distribution.

Main Results:

• The proposed method accurately determines probability densities from fractional moments.
• Solutions are compatible with data using a limited number of moments (up to four).
• Different sets of moments can yield accurate but distinct exponential density solutions.

Conclusions:

• The maximum entropy method offers a robust solution for estimating probability densities from fractional moments.
• The necessity of only a few moments simplifies practical application.
• Ambiguity in model selection arises when multiple exponential families fit the data, posing challenges for theoretical interpretation.