Abstract
In survival analysis, the two-parameter inverse Lomax distribution is an important lifetime distribution. In this study, the estimation of R = P Y < X is investigated when the stress and strength random variables are independent inverse Lomax distribution. Using the maximum likelihood approach, we obtain the R estimator via simple random sample (SRS), ranked set sampling (RSS), and extreme ranked set sampling (ERSS) methods. Four different estimators are developed under the ERSS framework. Two estimators are obtained when both strength and stress populations have the same set size. The two other estimators are obtained when both strength and stress distributions have dissimilar set sizes. Through a simulation experiment, the suggested estimates are compared to the corresponding under SRS. Also, the reliability estimates via ERSS method are compared to those under RSS scheme. It is found that the reliability estimate based on RSS and ERSS schemes is more efficient than the equivalent using SRS based on the same number of measured units. The reliability estimates based on RSS scheme are more appropriate than the others in most situations. For small even set size, the reliability estimate via ERSS scheme is more efficient than those under RSS and SRS. However, in a few cases, reliability estimates via ERSS method are more accurate than using RSS and SRS schemes.
Highlights
The inverse Lomax (ILo) distribution is considered as the reciprocal of the Lomax distribution
This article tackles the estimation of the stress strength (S-S) reliability R = P1⁄2Y < X when the strength X and stress Y are independent inverse Lomax distributed random variables
It is observed that the mean squared errors (MSEs) of reliability estimates based on simple random sample (SRS) data are bigger than the comparable based on ranked set sampling (RSS) and extreme ranked set sampling (ERSS) data, respectively
Summary
The inverse Lomax (ILo) distribution is considered as the reciprocal of the Lomax distribution. The RSS scheme is used in situations when it is difficult and expensive to measure a large number of elements, but visually (without inspection) ranking some of them is easier and cheaper This sampling design is both a cost-effective and powerful alternative to the commonly used SRS. For even set size (ESZ), we chose from ðm1/2Þ samples the smallest ranked unit and from the other ðm1/2Þ samples the largest ranked unit for actual measurement This procedure can be repeated q times to obtain m1q units from ERSS data. Due to the importance of the ILo distribution in reliability research, we propose to evaluate the reliability estimator of the S-S model where the strength X ~ ILoðρ, ωÞ and stress Y ~ ILoðρ, φÞ are both independent.
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