Length-Biased Slash-Exponential-Frechet Distribution: Statistical Properties, Estimation and Application
Benjamin Apam *
Department of Statistics, Bolgatanga Technical University, Sumbrungu, Ghana.
Marycel T. Sajise
Benguet State University, La Trinidad, Philippines.
*Author to whom correspondence should be addressed.
Abstract
Aims: To construct the length-biased slash-exponential-Fréchet (LB-SEFr) distribution, establish its principal distributional and tail properties, and assess its ability to represent positively skewed lifetime observations.
Study Design: A methodological investigation combining mathematical derivation, Monte Carlo simulation, and a comparative application to observed survival times.
Place and duration of study: The theoretical and computational investigations were conducted in Ghana; the empirical application uses historical animal survival observations.
Methodology: The LB-SEFr model was obtained by weighting the SEFr density in proportion to survival duration under the finite-mean constraint \(\lambda>1\). Density, distribution, reliability, moment, quantile, and related functional properties were examined.
Maximum-likelihood estimation was assessed in 1,000-replication Monte Carlo simulations for four parameter settings and sample sizes of 100,300 , and 500 . Empirical fit was compared with six alternative distributions using likelihood-based criteria and empirical-distribution discrepancies for 72 guinea-pig survival observations.
Results: Optimization convergence ranged from \(98.2 \%\) to \(100 \%\). Bias and root mean squared error generally decreased with sample size, although estimation of the additional shape parameter was unstable in some smaller samples. The fitted parameters were \(\widehat{\sigma}=75.4111, \widehat{\lambda}=3.0044\), and \(\widehat{\alpha}=0.4589\). LB-SEFr had the smallest AIC (783.781), AICc (784.134), Kolmogorov-Smirnov discrepancy (0.0708), Cramér-von Mises discrepancy (0.0638), and Anderson-Darling discrepancy (0.4152). The lognormal model had a smaller BIC (789.224 compared with 790.611).
Conclusion: The LB-SEFr distribution is a mathematically well-defined and empirically competitive heavy-tailed lifetime model. The observed sample does not establish a length-biased selection process, and further evaluation under documented length-biased and censored sampling designs is warranted.
Keywords: Length-biased distribution, weighted distribution, slash-exponential-Fr´echet, heavy tails, lifetime modelling, maximum likelihood