Jacobi and Laguerre polynomial approximations for the distributions of statistics useful in testing for outliers in exponential and gamma samples

Recently, Sanjel and Balakrishnan [A Laguerre Polynomial Approximation for a goodness-of-fit test for exponential distribution based on progressively censored data, J. Stat. Comput. Simul. 78 (2008), pp. 503–513] proposed the use of Laguerre orthogonal polynomials for a goodness-of-fit test for the exponential distribution based on progressively censored data. In this paper, we use Jacobi and Laguerre orthogonal polynomials in order to obtain density approximants for some test statistics useful in testing for outliers in gamma and exponential samples. We first obtain the exact moments of the statistics and then the density approximants, based on these moments, are expressed in terms of Jacobi and Laguerre polynomials. A comparative study is carried out of the critical values obtained by using the proposed methods to the corresponding results given by Barnett and Lewis [Outliers in Statistical Data, 3rd ed., John Wiley & Sons, New York, 1993]. This reveals that the proposed techniques provide very accurate approximations to the distributions. Finally, we present some numerical examples to illustrate the proposed approximations. Monte Carlo simulations suggest that the proposed approximate densities are very accurate.