Estimation of parameters in logistic and log-logistic distribution with grouped data

Area of Science:

• Statistics
• Reliability Engineering
• Survival Analysis

Background:

• Grouped failure time data are common in reliability studies.
• Periodic monitoring leads to interval-censored data.
• The log-logistic distribution is a relevant model for such data.

Purpose of the Study:

• To demonstrate the existence and uniqueness of MLEs for log-logistic distribution parameters with grouped data.
• To estimate the time of maximum failure rate and the mode of the probability density function (PDF).

Main Methods:

• Utilized maximum likelihood estimation (MLE) for parameter estimation.
• Applied the log-logistic distribution model to grouped data.
• Performed simulation studies to validate the methodology.
• Exemplified the approach with a real-world dataset from a locomotive life test.

Main Results:

• Established the existence and uniqueness of MLEs for log-logistic distribution parameters under mild conditions for grouped data.
• Successfully estimated the times of maximum failure rate and the PDF mode using MLEs.
• Simulation results supported the robustness of the proposed methodology.

Conclusions:

• The MLE approach is effective for analyzing grouped failure time data from log-logistic distributions.
• The method provides reliable estimates for key reliability characteristics.
• This statistical framework is applicable to various engineering and medical studies with interval-censored data.