Estimation of stress strength reliability using log-symmetric kernel methods

{A nonparametric kernel method based on log-symmetric distributions is proposed for estimating stress strength reliability}. This paper introduces a nonparametric kernel approach based on asymmetric kernels generated from log-symmetric (LS) distributions. These kernels provide flexibility and boundary adapted tools for estimating distributions supported on the positive real line. We derive the main statistical properties of the proposed estimator, including its bias, variance, mean squared error, and asymptotic normality. A projection-based representation is introduced as an analytical tool to study the smoothing bias and the first-order asymptotic behaviour of the estimator, while the original two-sample estimator is retained throughout the numerical implementation. For bandwidth selection, a rule-of-thumb (RT) procedure is derived from the asymptotic properties of the associated LS kernel, while an unbiased cross-validation (UCV) criterion is formulated directly for the stress–strength reliability estimator. Simulation results based on several lifetime distributions and sample-size configurations show that the proposed LS estimators generally provide competitive finite-sample performance, with the RT method frequently yielding lower RMSE values than UCV. Comparisons with classical symmetric kernels indicate that the LS kernels often perform favourably for positive data, although the differences between competing estimators can be modest in some settings. The practical performance of the proposed estimators is further illustrated through applications to real positive data sets.