The efficient hybrid bootstrap estimator, exploiting unbiasedness of Monte Carlo and efficiency of symbolic computation

grantor: University of TorontoThe bootstrap is a computational method of statistical inference. The efficient calculation of bootstrap estimates using both of symbolic computation and Monte Carlo estimation is discussed. The method is applied to M-estimators and smooth functions of these. These statistics can be approximated as power series in 'n' -1/2. These series expansions permit the symbolic calculation of bootstrap expectations. The finite approximation of the given M-estimator is used as a control function, and the bootstrap estimate associated with it is calculated analytically using symbolic computation. The remainder is estimated using the standard bootstrap Monte Carlo method. Typically, the variance of this hybrid estimator is much smaller than that of the conventional bootstrap Monte Carlo estimator. The method is applied to the bootstrap estimation of moments, cumulants, and tail areas. In particular, the analytic component of the tail area is computed using a saddlepoint approximation. Symbolic computation is the main tool for the computation of the series approximation of the statistic to any specific order and for the analytic bootstrap computation associated with it. This method generalizes the use of linear control functions in bootstrap variance estimation by Davison, Hinkley, and Schechtman [1986] and Efron [1990].Ph.D