Approximate MLEs for the location and scale parameters of the skew logistic distribution

Abstract

Azzalini (Scand J Stat 12:171–178, 1985) provided a methodology to introduce skewness in a normal distribution. Using the same method of Azzalini (1985), the skew logistic distribution can be easily obtained by introducing skewness to the logistic distribution. For the skew logistic distribution, the likelihood equations do not provide explicit solutions for the location and scale parameters. We present a simple method of deriving explicit estimators by approximating the likelihood equations appropriately. We examine numerically the bias and variance of these estimators and show that these estimators are as efficient as the maximum likelihood estimators (MLEs). The coverage probabilities of the pivotal quantities (for location and scale parameters) based on asymptotic normality are shown to be unsatisfactory, especially when the effective sample size is small. To improve the coverage probabilities and for constructing confidence intervals, we suggest the use of simulated percentage points. Finally, we present a numerical example to illustrate the methods of inference developed here.