Estimation of parameters of a mixture distribution using quantile functions

Suppose a sample of size n is drawn from a mixture distribution F ( x ) = δ F 1 ( x ) + ( 1 − δ ) F 2 ( x ) where component distribution functions F 1 ( x ) and F 2 ( x ) are such that F 1 ( x ) is stochastically smaller than F 2 ( x ) . Out of these n observations, the first r 1 smallest and last r 2 largest observations could have, respectively, come from distributions F 1 ( x ) and F 2 ( x ) . It is proposed to estimate parameters of F(x) using least-squares type criterion given in terms of quantile functions of F 1 ( x ) , F 2 ( x ) and a quantile function obtained using a convex combination of quantile functions of F 1 ( x ) and F 2 ( x ) . Extensive numerical results that compare the mean squared errors of these newly proposed estimators of different parameters of a mixture of exponential distributions with those of standard estimators, such as maximum likelihood and moment estimators etc., obtained using individual components F 1 ( x ) = 1 − exp ( − λ 1 x ) and F 2 ( x ) = 1 − exp ( − λ 2 x ) of the given mixture distribution are given. It is to be highlighted that these newly proposed estimators outperform the standard ones for various values of r 1 and r 2 and δ. Similar results are obtained by comparing generalized variance of a bivariate vector made up of new estimators of two exponential distributions with that of a bivariate vector formed using existing set of estimators for two cases, namely, (i) δ is known and (ii) δ is unknown and they are also found to be encouraging.