Asymptotic inference for high-dimensional data

In this paper, we study inference for high-dimensional data characterized by small sample sizes relative to the dimension of the data. In particular, we provide an infinite-dimensional framework to study statistical models that involve situations in which (i) the number of parameters increase with the sample size (that is, allowed to be random) and (ii) there is a possibility of missing data. Under a variety of tail conditions on the components of the data, we provide precise conditions for the joint consistency of the estimators of the mean. In the process, we clarify and improve some of the recent consistency results that appeared in the literature. An important aspect of the work presented is the development of asymptotic normality results for these models. As a consequence, we construct different test statistics for one-sample and two-sample problems concerning the mean vector and obtain their asymptotic distributions as a corollary of the infinite-dimensional results. Finally, we use these theoretical results to develop an asymptotically justifiable methodology for data analyses. Simulation results presented here describe situations where the methodology can be successfully applied. They also evaluate its robustness under a variety of conditions, some of which are substantially different from the technical conditions. Comparisons to other methods used in the literature are provided. Analyses of real-life data is also included.Comment: Published in at http://dx.doi.org/10.1214/09-AOS718 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Corrections and acknowledgment for ``Local limit theory and large deviations for supercritical branching processes''

Corrections and acknowledgment for ``Local limit theory and large deviations for supercritical branching processes'' [math.PR/0407059]Comment: Published at http://dx.doi.org/10.1214/105051606000000574 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Local limit theory and large deviations for supercritical Branching processes

In this paper we study several aspects of the growth of a supercritical Galton-Watson process {Z_n:n\ge1}, and bring out some criticality phenomena determined by the Schroder constant. We develop the local limit theory of Z_n, that is, the behavior of P(Z_n=v_n) as v_n\nearrow \infty, and use this to study conditional large deviations of {Y_{Z_n}:n\ge1}, where Y_n satisfies an LDP, particularly of {Z_n^{-1}Z_{n+1}:n\ge1} conditioned on Z_n\ge v_n

Large deviation results for branching processes in fixed and random environments

This thesis considers three different aspects of large deviations for branching processes. First, we study the deviation between the empirical mean and the true mean. Second, we investigate the large deviation behavior exhibited by the tail of the random variable W occurring in multi-type branching processes. Finally, we discuss the large deviations as they apply to a branching random walk in stationary ergodic environments

Effect of diethylstilbesterol and prolactin on the induction of follicle stimulating hormone receptors in immature and cycling rats

Induction of follicle stimulating hormone receptor in the granulosa cells of intact immature rat ovary by diethylstilbesterol, an estrogen, has been studied. A single injection of 4 mg of diethylstilbesterol produced 72 h later a 3-fold increase in follicle stimulating hormone receptor concentration as monitored by [125I]-oFSH binding to isolated cells. The newly induced receptors were kinetically indistinguishable from the preexisting ones, as determined by Lineweaver-Burk plot of the binding data. The induced receptors were functional as evidenced by increased ability of the granulosa cells to incorporate [3H]-leucine into cellular proteins. Neutralization of endogenous follicle stimulating hormone and luteinizing hormone by administering specific antisera had no effect on the ability of diethylstilbesterol to induce follicle stimulating hormone receptors, whereas blockade of endogenous prolactin secretion by ergobromocryptin administration significantly inhibited (~ 30 %) the response to diethylstilbesterol; this inhibition could be completely relieved by ovine prolactin treatment. However, ovine prolactin at the dose tried did not by itself enhance follicle stimulating hormone receptor level. Administration of ergobromocryptin to adult cycling rats at noon of proestrus brought about as measured on diestrusII, (a) a reduction of both follicle stimulating hormone (~ 30 %) and luteinizing hormone (~ 45 %) receptor concentration in granulosa cells, (b) a drastic reduction in the ovarian tissue estradiol with no change in tissue progesterone and (c) reduction in the ability of isolated granulosa cells to convert testosterone to estradiol in response to follicle stimulating hormone. Ergobromocryptin treatment affected only prolactin and not follicle stimulating hormone or luteinizing hormone surges on the proestrus evening. Treatment of rats with ergobromocryptin at proestrus noon followed by an injection of ovine prolactin (1 mg) at 1700 h of the same day completely reversed the ergobromocryptin induced reduction in ovarian tissue estradiol as well as the aromatase activity of the granulosa cells on diestrus II, thus suggesting a role for proestrus prolactin surge in the follicular maturation process

Functional limit laws for the intensity measure of point processes and applications

Motivated by applications to the study of depth functions for tree-indexed random variables generated by point processes, we describe functional limit theorems for the intensity measure of point processes. Specifically, we establish uniform laws of large numbers and uniform central limit theorems over a class of bounded measurable functions for estimates of the intensity measure. Using these results, we derive the uniform asymptotic properties of half-space depth and, as corollaries, obtain the asymptotic behavior of medians and other quantiles of the standardized intensity measure. Additionally, we obtain uniform concentration upper bound for the estimator of half-space depth. As a consequence of our results, we also derive uniform consistency and uniform asymptotic normality of Lotka-Nagaev and Harris-type estimators for the Laplace transform of the point processes in a branching random walk.Comment: 38 page