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Register a new userPreliminary testing derivatives of a linear unified estimator in the logistic regression model
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Added by
Yasin Asar
Publication date
2017
fields
Mathematical Statistics
and research's language is
English
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Recently, the well known Liu estimator (Liu, 1993) is attracted researchers attention in regression parameter estimation for an ill conditioned linear model. It is also argued that imposing sub-space hypothesis restriction on parameters improves estimation by shrinking toward non-sample information. Chang (2015) proposed the almost unbiased Liu estimator (AULE) in the binary logistic regression. In this article, some improved unbiased Liu type estimators, namely, restricted AULE, preliminary test AULE, Stein-type shrinkage AULE and its positive part for estimating the regression parameters in the binary logistic regression model are proposed based on the work Chang (2015). The performances of the newly defined estimators are analysed through some numerical results. A real data example is also provided to support the findings.
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The Chebyshev or $ell_{infty}$ estimator is an unconventional alternative to the ordinary least squares in solving linear regressions. It is defined as the minimizer of the $ell_{infty}$ objective function begin{align*} hat{boldsymbol{beta}} := argmin_{boldsymbol{beta}} |boldsymbol{Y} - mathbf{X}boldsymbol{beta}|_{infty}. end{align*} The asymptotic distribution of the Chebyshev estimator under fixed number of covariates were recently studied (Knight, 2020), yet finite sample guarantees and generalizations to high-dimensional settings remain open. In this paper, we develop non-asymptotic upper bounds on the estimation error $|hat{boldsymbol{beta}}-boldsymbol{beta}^*|_2$ for a Chebyshev estimator $hat{boldsymbol{beta}}$, in a regression setting with uniformly distributed noise $varepsilon_isim U([-a,a])$ where $a$ is either known or unknown. With relatively mild assumptions on the (random) design matrix $mathbf{X}$, we can bound the error rate by $frac{C_p}{n}$ with high probability, for some constant $C_p$ depending on the dimension $p$ and the law of the design. Furthermore, we illustrate that there exist designs for which the Chebyshev estimator is (nearly) minimax optimal. In addition we show that Chebyshevs LASSO has advantages over the regular LASSO in high dimensional situations, provided that the noise is uniform. Specifically, we argue that it achieves a much faster rate of estimation under certain assumptions on the growth rate of the sparsity level and the ambient dimension with respect to the sample size.
We propose a censored quantile regression estimator motivated by unbiased estimating equations. Under the usual conditional independence assumption of the survival time and the censoring time given the covariates, we show that the proposed estimator is consistent and asymptotically normal. We develop an efficient computational algorithm which uses existing quantile regression code. As a result, bootstrap-type inference can be efficiently implemented. We illustrate the finite-sample performance of the proposed method by simulation studies and analysis of a survival data set.
We reexamine the classical linear regression model when the model is subject to two types of uncertainty: (i) some of covariates are either missing or completely inaccessible, and (ii) the variance of the measurement error is undetermined and changing according to a mechanism unknown to the statistician. By following the recent theory of sublinear expectation, we propose to characterize such mean and variance uncertainty in the response variable by two specific nonlinear random variables, which encompass an infinite family of probability distributions for the response variable in the sense of (linear) classical probability theory. The approach enables a family of estimators under various loss functions for the regression parameter and the parameters related to model uncertainty. The consistency of the estimators is established under mild conditions on the data generation process. Three applications are introduced to assess the quality of the approach including a forecasting model for the S&P Index.
This short note is to point the reader to notice that the proof of high dimensional asymptotic normality of MLE estimator for logistic regression under the regime $p_n=o(n)$ given in paper: Maximum likelihood estimation in logistic regression models with a diverging number of covariates. Electronic Journal of Statistics, 6, 1838-1846. is wrong.
In the last decade, the secondary use of large data from health systems, such as electronic health records, has demonstrated great promise in advancing biomedical discoveries and improving clinical decision making. However, there is an increasing concern about biases in association studies caused by misclassification in the binary outcomes derived from electronic health records. We revisit the classical logistic regression model with misclassified outcomes. Despite that local identification conditions in some related settings have been previously established, the global identification of such models remains largely unknown and is an important question yet to be answered. We derive necessary and sufficient conditions for global identifiability of logistic regression models with misclassified outcomes, using a novel approach termed as the submodel analysis, and a technique adapted from the Picard-Lindel{o}f existence theorem in ordinary differential equations. In particular, our results are applicable to logistic models with discrete covariates, which is a common situation in biomedical studies, The conditions are easy to verify in practice. In addition to model identifiability, we propose a hypothesis testing procedure for regression coefficients in the misclassified logistic regression model when the model is not identifiable under the null.
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