This article focuses on improving the estimation of population variances for natural exponential family distributions, drawing inspiration from the innovative idea presented by Stein. The estimation is based on utilizing sample information on kurtosis. We derive the general form of the shrinkage estimator for the quadratic variance function of natural exponential family under the Scaled mean squared error loss function. Also, our theoretical calculations show that the new shrinkage estimator for the quadratic variance function, which dominates the standard sample variance estimator. Consequently, this article extends the method for obtaining the new variance estimator in natural exponential family distributions such as binomial, Poisson, negative binomial, gamma, and generalized hyperbolic secant distributions. A simulation study was conducted to evaluate the performance of the proposed data-driven shrinkage variance estimators. Results demonstrate significant improvements compared to the existing shrinkage standard sample variance estimator. Finally, applying these estimators to real data of binomial, Poisson, and negative binomial distributions, shows superior performance over existing estimators in these scenarios.
Laheetharan et al. (Sat,) studied this question.