Approximations for saddlepoint densities of M-estimators

grantor: University of TorontoWe examine the implications of using estimated cumulants in saddlepoint density function approximations. Much of the current literature on empirical saddlepoint methodology is concerned with the relative error of the density estimate. Our interest is not in the order of the relative error, since large cumulants may have an effect which outweighs the effect of the sample size. Steepest descents methodology can be used to express many density functions as asymptotic series in powers of 'n'-1/2 For many M-estimators, and other statistics, each term in the saddlepoint series can be expressed as series in powers of 'n'-1/2 . Thus, the leading terms in the overall series forms a density approximation. The saddlepoint density series is a function of cumulants. An estimated density can be determined by estimating the cumulants. Cumulants estimation can take many forms, in particular empirical estimation, or estimation with zero. Thus there are many possible density estimates. We show how steepest descents methodology can often be used to create a series, proportional to the saddlepoint approximation 'fSP'(' x'), for the difference between the true and estimated density. When the series exists, and the estimated cumulants are asymptotically Gaussian, f&d4;x -fx is asymptotically Gaussian. This limiting distribution is not constant over all 'x'. We show how the expansion of nf&d4; x-fx can be used to assess various cumulant estimators. In particular, the series can be used to determine if a cumulant should be empirically estimated or estimated with zero. Conclusions are not independent of 'x'. However, if all higher order cumulants are known, the leading term in Ef&d4;x; k1&ldots;kj-1,k &d4;j-f x2 is smaller than the leading term in Ef&d4;x; k1&ldots;kj-1,0 -fx 2 if the coefficient of variation for the empirical cumulant is less than one. This conclusion is independent of 'x'0. We show there is often a point in the empirical cumulant sequence after which estimates should be omitted from an empirical saddlepoint density. The point depends on the empirical cumulant coefficient of variation sequence. Examples show the for M-estimators cumulant estimation should rarely exceed the fourth cumulant, often the Gaussian approximation is sufficient for smooth densities and small sample sizes.Ph.D