Convergence Rates for Probabilities of Moderate Deviations for Multidimensionally Indexed Random Variables

Let {X,Xn¯;n¯∈Z+d} be a sequence of i.i.d. real-valued random variables, and Sn¯=∑k¯≤n¯Xk¯, n¯∈Z+d. Convergence rates of moderate deviations are derived; that is, the rates of convergence to zero of certain tail probabilities of the partial sums are determined. For example, we obtain equivalent conditions for the convergence of the series ∑n¯b(n¯)ψ2(a(n¯))P{|Sn¯|≥a(n¯)ϕ(a(n¯))}, where a(n¯)=n11/α1⋯nd1/αd, b(n¯)=n1β1⋯ndβd, ϕ and ψ are taken from a broad class of functions. These results generalize and improve some results of Li et al. (1992) and some previous work of Gut (1980)

Asymptotics for non-parametric likelihood estimation with doubly censored multivariate failure times

AbstractThis paper considers non-parametric estimation of a multivariate failure time distribution function when only doubly censored data are available, which occurs in many situations such as epidemiological studies. In these situations, each of multivariate failure times of interest is defined as the elapsed time between an initial event and a subsequent event and the observations on both events can suffer censoring. As a consequence, the estimation of multivariate distribution is much more complicated than that for multivariate right- or interval-censored failure time data both theoretically and practically. For the problem, although several procedures have been proposed, they are only ad-hoc approaches as the asymptotic properties of the resulting estimates are basically unknown. We investigate both the consistency and the convergence rate of a commonly used non-parametric estimate and show that as the dimension of multivariate failure time increases or the number of censoring intervals of multivariate failure time decreases, the convergence rate for non-parametric estimate decreases, and is slower than that with multivariate singly right-censored or interval-censored data

Goodness-of-fit, score test, zero-inflation and over-dispersion in generalized linear models.

In this thesis we develop goodness of fit tests of the generalized linear model with non-canonical links for data that are extensive but sparse. We derive approximations to the first three moments of the deviance statistic. A supplementary estimating equation is proposed from which the modified deviance statistic is obtained. Applications of the modified deviance statistic to binomial and Poisson data are shown. A simulation study is conducted to compare the behavior, in terms of size and power, of the modified deviance statistic and the modified Pearson statistic developed earlier by Farrington (1996). Three sets of data with different degrees, of sparseness and different link functions are analyzed. The simulation results and examples indicate that both the modified Pearson statistic and the modified deviance statistic perform well in terms of holding nominal levels. However, the modified deviance statistic shows much better power properties for the range of parameters investigated under the alternative hypothesis. Theses results also answer a question posed by Farrington (1996) and extend results of McCullagh (1986) for Poisson log-linear models. In some instances a score or a C(alpha) statistic performs well. In this thesis we also develop a score test statistic to assess goodness of fit of the generalized linear model for data that are extensive but sparse. The performance of this statistic is then compared with the modified Pearson statistic. Results of simulation show that both the modified score test statistic developed in our paper and the modified Pearson statistic developed by Farrington (1996) maintain nominal levels. However the modified score test has some edge over the modified Pearson statistic in terms of power. In practice, sometimes, discrete data contain excess zeros that can not be explained by a simple model. In this thesis we develop score tests for testing zero-inflation in generalized linear models. These score tests are then applied to binomial models and Poisson models and their performances are evaluated. A limited simulation study shows that the score tests reasonably maintain the nominal levels. The power of the tests for detecting zero-inflation increases very slowly for Poisson mean mu or binomial parameter p. For large values of mu and p power increases very fast and approaches 1.0 even for moderate zero-inflation. A discrete generalized linear model (Poisson or binomial) may fall to fit a set of data having a lot of zeros either because of zero-inflation only, because of over-dispersion only, or because there is zero-inflation as well as over-dispersion in the data. In this thesis we obtain score tests (i) for zero-inflation in presence of over-dispersion, (ii) for over-dispersion in presence of zero-inflation, and (iii) simultaneously for testing for zero-inflation and over-dispersion. For Poisson and binomial data these score tests are compared with those obtained from the zero-inflated negative binomial model and the zero-inflated beta-binomial model. Some simulations are performed for Poisson data to study type I error properties of the tests. In general the score tests developed here hold nominal levels reasonably well. The data sets are analyzed to illustrate model section procedure by the score tests. (Abstract shortened by UMI.)Dept. of Mathematics and Statistics. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis2001 .D46. Source: Dissertation Abstracts International, Volume: 62-10, Section: B, page: 4616. Adviser: S. R. Paul. Thesis (Ph.D.)--University of Windsor (Canada), 2001

On the bounded laws of iterated logarithm in Banach space

In the present paper, by using the inequality due to Talagrand's isoperimetric method, several versions of the bounded law of iterated logarithm for a sequence of independent Banach space valued random variables are developed and the upper limits for the non-random constant are given

On the self-normalized bounded laws of iterated logarithm in Banach space

For a sequence of independent symmetric Banach space valued random variables {Xn,n[greater-or-equal, slanted]1}, we obtain the self-normalized law of iterated logarithm and give the upper bound for the non-random constant.Banach space Bounded law of iterated logarithm Rademacher series Self-normalizer Symmetric random variables

Self-normalized Wittmann's laws of iterated logarithm in Banach space

For a sequence of independent symmetric Banach space valued random variables {Xn,n[greater-or-equal, slanted]1}, we obtain the self-normalized Wittmann's law of iterated logarithm (LIL) and give the upper bound for the non-random constant.Banach space Law of iterated logarithm Rademacher series Self-normalizer Symmetric random variables

The complete convergence of subsequence for sums of independent B-valued random variables

In the present paper, the sufficient and necessary conditions that are derived for the sequence of B-valued i.i.d. random variables {Xi}, the strictly increasing subsequence of positive integers {nk} and the positive monotone sequence of real numbers {an} with an[short up arrow][infinity].Banach space Complete convergence Entropy estimate Isoperimetric methods Rademacher series Subsequence

Quantile Regression Approach for Analyzing Similarity of Gene Expressions under Multiple Biological Conditions

Temporal gene expression data contain ample information to characterize gene function and are now widely used in bio-medical research. A dense temporal gene expression usually shows various patterns in expression levels under different biological conditions. The existing literature investigates the gene trajectory using the mean function. However, temporal gene expression curves usually show a strong degree of heterogeneity under multiple conditions. As a result, rates of change for gene expressions may be different in non-central locations and a mean function model may not capture the non-central location of the gene expression distribution. Further, the mean regression model depends on the normality assumptions of the error terms of the model, which may be impractical when analyzing gene expression data. In this research, a linear quantile mixed model is used to find the trajectory of gene expression data. This method enables the changes in gene expression over time to be studied by estimating a family of quantile functions. A statistical test is proposed to test the similarity between two different gene expressions based on estimated parameters using a quantile model. Then, the performance of the proposed test statistic is examined using extensive simulation studies. Simulation studies demonstrate the good statistical performance of this proposed test statistic and show that this method is robust against normal error assumptions. As an illustration, the proposed method is applied to analyze a dataset of 18 genes in P. aeruginosa, expressed in 24 biological conditions. Furthermore, a minimum Mahalanobis distance is used to find the clustering tree for gene expressions

Asymptotics for non-parametric likelihood estimation with doubly censored multivariate failure times

This paper considers non-parametric estimation of a multivariate failure time distribution function when only doubly censored data are available, which occurs in many situations such as epidemiological studies. In these situations, each of multivariate failure times of interest is defined as the elapsed time between an initial event and a subsequent event and the observations on both events can suffer censoring. As a consequence, the estimation of multivariate distribution is much more complicated than that for multivariate right- or interval-censored failure time data both theoretically and practically. For the problem, although several procedures have been proposed, they are only ad-hoc approaches as the asymptotic properties of the resulting estimates are basically unknown. We investigate both the consistency and the convergence rate of a commonly used non-parametric estimate and show that as the dimension of multivariate failure time increases or the number of censoring intervals of multivariate failure time decreases, the convergence rate for non-parametric estimate decreases, and is slower than that with multivariate singly right-censored or interval-censored data.Multivariate doubly interval-censored Non-parametric maximum likelihood estimation Strong consistency Convergence rate