Statistical Properties and Applications of <i>k</i> ‐Record Values Based on the Generalized Lindley Distribution

This research explores the characteristics and uses of k ‐record values derived from the generalized Lindley distribution, an adaptable extension of the traditional Lindley distribution. The generalized Lindley distribution is notable for its capacity to model skewed and heavy‐tailed datasets, making it particularly effective for analyses related to time‐to‐event, reliability, and survival data. The study examines the statistical features of k ‐record values, focusing on their distributions, moments, and practical implications. We provide the cumulative distribution function and probability density function for k ‐record values, offering insights into their distributional characteristics and properties. Our theoretical findings are supported by simulation studies that demonstrate the relevance of k ‐record values across various domains, including engineering and healthcare, where the analysis of extreme observations is essential. The results underscore the importance of the generalized Lindley distribution in modeling intricate data patterns and emphasize the role of k ‐record values in the analysis of extreme events. This study enhances the overall understanding of record values in statistical modeling and paves the way for future research in both theoretical and applied areas. The generalized Lindley distribution has garnered significant interest recently due to its diverse applications across multiple disciplines. This paper specifically examines the k ‐record values associated with the generalized Lindley distribution, defined as the k smallest order statistics from a sample drawn from the generalized Lindley distribution. We derive precise formulas for the probability density function, cumulative distribution function, and moments of the k ‐record values. Additionally, we explore several statistical properties of these values, such as their asymptotic behavior, estimation methods, and inference techniques. To illustrate the practical implications of our findings, we provide numerical examples that demonstrate the applicability of k ‐record values in fields like engineering, finance, and environmental science.