## An Asymptotically Efficient Stepwise Estimator for Exponential Families with Applications to Certain Multivariate Normal Distributions |

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Appendix assumed Assumption asymptotic distribution asymptotic properties asymptotic variance asymptotically efficient asymptotically normal estimator Borel measurable Chapter closed form computation conditions of Theorem considered consistent asymptotically consistent root continuous and finite convergence Corollary 2.5 covariance matrix Cramer-Rao lower bound cubic equation defined density diagonal matrix differentiable distributed as N(0 Doctor of Philosophy efficient and asymptotically EMLE entries esti exists exponential family function Hence i=l i=l identically distributed independently and identically iteration likelihood equation Madansky mator multidimensional multivariate normal distribution nonsymmetric notation Note Observe obtain Oh rr open interval orthogonal matrix parameter space positive definite problem r x r random variable real numbers real root regularity conditions result Robbins-Monro procedure Sacks satisfies shown situation solution stepwise estimation procedure stochastic approximation procedure stopping rules sufficient statistics symmetric derivative symmetric matrix Theorem 3-8 Theorem J>.k Truncation Procedure unbiased estimator uniparameter verify x e a,b