11,803 research outputs found
New Flexible Regression Models Generated by Gamma Random Variables with Censored Data
We propose and study a new log-gamma Weibull regression model. We obtain explicit expressions for the raw and incomplete moments, quantile and generating functions and mean deviations of the log-gamma Weibull distribution. We demonstrate that the new regression model can be applied to censored data since it represents a parametric family of models which includes as sub-models several widely-known regression models and therefore can be used more effectively in the analysis of survival data. We obtain the maximum likelihood estimates of the model parameters by considering censored data and evaluate local influence on the estimates of the parameters by taking different perturbation schemes. Some global-influence measurements are also investigated. Further, for different parameter settings, sample sizes and censoring percentages, various simulations are performed. In addition, the empirical distribution of some modified residuals are displayed and compared with the standard normal distribution. These studies suggest that the residual analysis usually performed in normal linear regression models can be extended to a modified deviance residual in the proposed regression model applied to censored data. We demonstrate that our extended regression model is very useful to the analysis of real data and may give more realistic fits than other special regression models
Moving Beyond Sub-Gaussianity in High-Dimensional Statistics: Applications in Covariance Estimation and Linear Regression
Concentration inequalities form an essential toolkit in the study of high
dimensional (HD) statistical methods. Most of the relevant statistics
literature in this regard is based on sub-Gaussian or sub-exponential tail
assumptions. In this paper, we first bring together various probabilistic
inequalities for sums of independent random variables under much weaker
exponential type (namely sub-Weibull) tail assumptions. These results extract a
part sub-Gaussian tail behavior in finite samples, matching the asymptotics
governed by the central limit theorem, and are compactly represented in terms
of a new Orlicz quasi-norm - the Generalized Bernstein-Orlicz norm - that
typifies such tail behaviors.
We illustrate the usefulness of these inequalities through the analysis of
four fundamental problems in HD statistics. In the first two problems, we study
the rate of convergence of the sample covariance matrix in terms of the maximum
elementwise norm and the maximum k-sub-matrix operator norm which are key
quantities of interest in bootstrap, HD covariance matrix estimation and HD
inference. The third example concerns the restricted eigenvalue condition,
required in HD linear regression, which we verify for all sub-Weibull random
vectors through a unified analysis, and also prove a more general result
related to restricted strong convexity in the process. In the final example, we
consider the Lasso estimator for linear regression and establish its rate of
convergence under much weaker than usual tail assumptions (on the errors as
well as the covariates), while also allowing for misspecified models and both
fixed and random design. To our knowledge, these are the first such results for
Lasso obtained in this generality. The common feature in all our results over
all the examples is that the convergence rates under most exponential tails
match the usual ones under sub-Gaussian assumptions.Comment: 64 pages; Revised version (discussions added and some results
modified in Section 4, minor changes made throughout
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