The Unit Power‐Log Distribution for Bounded Data: Properties, Inference, and an Application to Educational Performance in Nepal
Laxmi Prasad Sapkota, Nirajan Bam, Pankaj Kumar, Vijay Kumar
Source abstract
In real‐world applications, data are often naturally bounded within the unit interval (0, 1), including boundary values. In particular, success rates of the School Leaving Certificate (SLC) examinations in Nepal across schools are bound to this range and may include both 0 and 1 values. However, existing unit‐interval bounded distributions, such as the beta and Kumaraswamy distributions, are not always adequate for accommodating such data structures. Therefore, a new one‐parameter unit‐interval bounded distribution, known as the unit power–log distribution (UPLD), is proposed. Important properties of the proposed distribution, including Bonferroni and Lorenz curves, entropy, and the Cramér–Rao inequality, are presented. The derivation of several inferential statistical expressions is presented, including the sequential probability ratio test, likelihood ratio test, and uniformly most powerful test based on the Neyman–Pearson lemma. Multiple parameter estimation techniques such as maximum likelihood estimation, methods of moments, and the least squares estimation method are used. The performance of the proposed estimation methods is validated using a Monte Carlo simulation study. Finally, the model is applied to SLC results bounded on [0, 1] from different schools across selected geographical regions of Nepal. To compare its performance with existing bounded models, it is further applied to another bounded dataset of the tensile strength of polyester fibers. The proposed UPLD demonstrates better performance compared to existing distributions.
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