Local Central Limit Theorems , the High Order Correlations of Rejective Sampling , and Applications to Conditional Logistic Likelihood

Local Central Limit Theorems, the High Order Correlations of Rejective Sampling, and Applications to Conditional Logistic Likelihood Asymptotics

LOCAL CENTRAL LIMIT THEOREMS, THE HIGH-ORDER CORRELATIONS OF REJECTIVE SAMPLING AND LOGISTIC LIKELIHOOD ASYMPTOTICS BY RICHARD ARRATIA, LARRY GOLDSTEIN1 AND BRYAN LANGHOLZ1 University of Southern California

Let I1, . . . , In be independent but not necessarily identically distributed Bernoulli random variables, and let Xn = ∑nj=1 Ij . For ν in a bounded region, a local central limit theorem expansion of P(Xn = EXn + ν) is developed to any given degree. By conditioning, this expansion provides information on the high-order correlation structure of dependent, weighted sampling schemes of a populatio...

Local Central Limit Theorems , the High - Order Correlations of Rejective Sampling and Logistic Likelihood

Let I1, . . . , In be independent but not necessarily identically distributed Bernoulli random variables, and let Xn = ∑n j=1 Ij . For ν in a bounded region, a local central limit theorem expansion of P(Xn = EXn + ν) is developed to any given degree. By conditioning, this expansion provides information on the high-order correlation structure of dependent, weighted sampling schemes of a populati...

The Local Limit Theorem: A Historical Perspective

The local limit theorem describes how the density of a sum of random variables follows the normal curve. However the local limit theorem is often seen as a curiosity of no particular importance when compared with the central limit theorem. Nevertheless the local limit theorem came first and is in fact associated with the foundation of probability theory by Blaise Pascal and Pierre de Fer...

Second-order asymptotic theory for calibration estimators in sampling and missing-data problems

Consider three different but related problems with auxiliary information: infinite population sampling or Monte Carlo with control variates, missing response with explanatory variables, and Poisson and rejective sampling with auxiliary variables. We demonstrate unified regression and likelihood estimators and study their second-order properties. The likelihood estimators are second-order unbias...