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Moment Convergence In Conditional Limit Theorems
, 2000
"... . Consider a sum P N 1 Y i of random variables conditioned on a given value of the sum P N 1 X i of some other variables, where X i and Y i are dependent but the pairs (X i ; Y i ) form an i.i.d. sequence. We prove, for a triangular array (X ni ; Y ni ) of such pairs satisfying certain condi ..."
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. Consider a sum P N 1 Y i of random variables conditioned on a given value of the sum P N 1 X i of some other variables, where X i and Y i are dependent but the pairs (X i ; Y i ) form an i.i.d. sequence. We prove, for a triangular array (X ni ; Y ni ) of such pairs satisfying certain conditions, both convergence of the distribution of the conditioned sum (after suitable normalization) to a normal distribution, and convergence of its moments. The results are motivated by an application to hashing with linear probing; we give also some other applications to occupancy problems, random forests, and branching processes. 1. Introduction Many random variables arising in different areas of probability theory, combinatorics and statistics turn out to have the same distribution as a sum of independent random variables conditioned on a specific value of another such sum. More precisely, we are concerned with variables with the distribution of P N 1 Y i conditioned on P N 1 X...
Monotonicity, asymptotic normality and vertex degrees in random graphs
 Bernoulli
, 2007
"... Abstract. We exploit a result by Nerman [23] which shows that conditional limit theorems hold when a certain monotonicity condition is satisfied. Our main result is an application to vertex degrees in random graphs where we obtain asymptotic normality for the number of vertices with a given degree i ..."
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Abstract. We exploit a result by Nerman [23] which shows that conditional limit theorems hold when a certain monotonicity condition is satisfied. Our main result is an application to vertex degrees in random graphs where we obtain asymptotic normality for the number of vertices with a given degree in the random graph G(n, m) with a fixed number of edges from the corresponding result for the random graph G(n, p) with independent edges. We give also some simple applications to random allocations and to spacings. Finally, inspired by these results but logically independent from them, we investigate whether a onesided version of the Cramér–Wold theorem holds. We show that such a version holds under a weak supplementary condition, but not without it. 1.
Model Checking for Incomplete High Dimensional Categorical Data
, 1999
"... OF THE DISSERTATION Model Checking for Incomplete High Dimensional Categorical Data by MingYi Hu Doctor of Philosophy in Statistics University of California, Los Angeles, 1999 Professor Thomas R. Belin, Cochair Professor Robert I. Jennrich, Cochair Categorical data are often arranged in ..."
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OF THE DISSERTATION Model Checking for Incomplete High Dimensional Categorical Data by MingYi Hu Doctor of Philosophy in Statistics University of California, Los Angeles, 1999 Professor Thomas R. Belin, Cochair Professor Robert I. Jennrich, Cochair Categorical data are often arranged in a contingency table and summarized by a loglinear model. A standard approach for comparing two competing models is to calculate twice the discrepancy between maximized loglikelihoods, which follows a 2 distribution asymptotically. But when data are sparse, the 2 approximation may be questionable. xii As an alternative to a largesample approximation to the reference distribution, we implement the framework introduced by Rubin (1984) for finding the posterior predictive check (PPC) distribution. The PPC distribution represents the conditional probability of a future value of a test statistic based on the information given by observed data along with model specifications, which can se...
Submitted to the Bernoulli The Loglinear GroupLasso Estimator and Its Asymptotic Properties
"... We define the grouplasso estimator for the natural parameters of the exponential families of distributions representing hierarchical loglinear models under multinomial sampling scheme. Such estimator arises as the solution of a convex penalized likelihood optimization problem based on the groupla ..."
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We define the grouplasso estimator for the natural parameters of the exponential families of distributions representing hierarchical loglinear models under multinomial sampling scheme. Such estimator arises as the solution of a convex penalized likelihood optimization problem based on the grouplasso penalty. We illustrate how it is possible to construct an estimator of the underlying loglinear model using the blocks of nonzero coefficients recovered by the grouplasso procedure. We investigate the asymptotic properties of the grouplasso estimator as a model selection method in a doubleasymptotic framework, in which both the sample size and the model complexity grow simultaneously. We provide conditions guaranteeing that the grouplasso estimator is model selection consistent, in the sense that, with overwhelming probability as the sample size increases, it correctly identifies all the sets of nonzero interactions among the variables. Provided the sequences of true underlying models is sparse enough, recovery is possible even if the number of cells grows larger than the sample size. Finally, we derive some central limit type of results for the loglinear grouplasso estimator.