Alpha-Power Exponentiated Inverse Rayleigh distribution and its applications to real and simulated data

Area of Science:

• Statistics
• Probability Theory
• Reliability Engineering

Background:

• The existing literature on probability distributions lacks models capable of effectively handling data with non-monotonic failure rates.
• Accurate modeling of failure rates is crucial in various fields, including engineering, finance, and survival analysis.

Purpose of the Study:

• To introduce a novel probability distribution, the Alpha Power Exponentiated Inverse Rayleigh (APEIR) distribution.
• To demonstrate the APEIR distribution's capability in modeling data exhibiting non-monotonic hazard rates.
• To establish the superiority of the APEIR distribution in terms of goodness-of-fit compared to existing models.

Main Methods:

• The Alpha Power Family of distributions was utilized to generate the new APEIR distribution.
• Key statistical properties, including order statistics, moments, residual life, mean waiting time, quantiles, entropy, and stress-strength parameter, were derived.
• Parameter estimation was performed using the maximum likelihood estimation (MLE) method.
• Theoretical analysis was conducted to prove the distribution's fitting capabilities for both monotonic and non-monotonic hazard rates.

Main Results:

• The APEIR distribution was successfully formulated and its statistical properties were thoroughly investigated.
• Theoretical proofs confirmed the distribution's effectiveness for data with both monotonic and non-monotonic hazard rate shapes.
• Empirical evaluation using two real-world data sets demonstrated the APEIR distribution's superior fit and flexibility over other established distributions.

Conclusions:

• The newly proposed APEIR distribution offers a valuable addition to the field of probability distributions.
• The APEIR distribution is particularly effective for modeling complex data, especially where non-monotonic failure rates are present.
• The flexibility and improved fit of the APEIR distribution were validated through theoretical analysis and real-world data applications.