A new estimate of skewness with mean-squared error smaller than that of the sample skewness

Recently, a new procedure for distribution fitting, based on matching of the first two moments, partial and complete, was introduced (Shore, 1995). When the sampling skewness of the fitted distribution is compared to the sample skewness, and both are regarded as estimates of the skewness of the underlying distribution, the mean-squared-error of the former is appreciably lower than that of the latter. In this paper we present some simulation results to support this claim and demonstrate its magnitude. An alternative two-moment distributional fitting procedure, based on a new family of four-parameter distributions, is also introduced and studied. Since three-moment distribution fitting is very common practice in simulation studies, these results may have important implications for the current state-of-the-art of simulation     

Goodness‐of‐fit testing based on a weighted bootstrap: A fast large‐sample alternative to the parametric bootstrap

The process comparing the empirical cumulative distribution function of the sample with a parametric estimate of the cumulative distribution function is known as the empirical process with estimated parameters and has been extensively employed in the literature for goodness‐of‐fit testing. The simplest way to carry out such goodness‐of‐fit tests, especially in a multivariate setting, is to use a parametric bootstrap. Although very easy to implement, the parametric bootstrap can become very computationally expensive as the sample size, the number of parameters, or the dimension of the data increase. An alternative resampling technique based on a fast weighted bootstrap is proposed in this paper, and is studied both theoretically and empirically. The outcome of this work is a generic and computationally efficient multiplier goodness‐of‐fit procedure that can be used as a large‐sample alternative to the parametric bootstrap. In order to approximately determine how large the sample size needs to be for the parametric and weighted bootstraps to have roughly equivalent powers, extensive Monte Carlo experiments are carried out in dimension one, two and three, and for models containing up to nine parameters. The computational gains resulting from the use of the proposed multiplier goodness‐of‐fit procedure are illustrated on trivariate financial data. A by‐product of this work is a fast large‐sample goodness‐of‐fit procedure for the bivariate and trivariate t distribution whose degrees of freedom are fixed. The Canadian Journal of Statistics 40: 480–500; 2012 © 2012 Statistical Society of Canada