In the fields of clinical trials, biomedical surveys, marketing, banking, with dichotomous response variable, the logistic regression is considered as an alternative convenient approach to linear regression. In this paper, we develop a novel bootstrap technique based on perturbation resampling method for approximating the distribution of the maximum likelihood estimator (MLE) of the regression parameter vector. We establish second order correctness of the proposed bootstrap method after proper studentization and smoothing. It is shown that inferences drawn based on the proposed bootstrap method are more accurate compared to that based on asymptotic normality. The main challenge in establishing second order correctness remains in the fact that the response variable being binary, the resulting MLE has a lattice structure. We show the direct bootstrapping approach fails even after studentization. We adopt smoothing technique developed in Lahiri (1993) to ensure that the smoothed studentized version of the MLE has a density. Similar smoothing strategy is employed to the bootstrap version also to achieve second order correct approximation.
Consider the multiple linear regression model $y_{i} = \boldsymbol{x}'_{i} \boldsymbolβ + ε_{i}$, where $ε_i$'s are independent and identically distributed random variables, $\mathbf{x}_i$'s are known design vectors and $\boldsymbolβ$ is the $p \times 1$ vector of parameters. An effective way of approximating the distribution of the M-estimator $\boldsymbol{\barβ}_n$, after proper centering and scaling, is the Perturbation Bootstrap Method. In this current work, second order results of this non-naive bootstrap method have been investigated. Second order correctness is important for reducing th…
The logistic regression analysis proposed by Schouten et al. (Stat Med. 1993;12:1733-1745) has been a standard method in current statistical analysis of case-cohort studies, and it enables effective estimation of risk ratio from selected subsamples. Schouten et al. (1993) also proposed the standard error estimate of the risk ratio estimator can be calculated by the robust variance estimator. In this article, however, we show that the robust variance estimator does not account for the duplications of case and subcohort samples and generally has certain bias, i.e., inaccurate confidence interval…
Logistic regression with unknown sizes has many important applications in biological and medical sciences. All models about this problem in the literature are parametric ones. A semiparametric regression model is proposed. This model incorporates overdispersion due to the variation of sizes, and allows general dose-response relations. An Expectation Conditional Maximization algorithm is provided to maximize the log likelihood. The bootstrap method can be used to construct confidence intervals for regression coefficients. Simulation is performed to study the behavior of the proposed model. Two …
Binomial data with unknown sizes often appear in biological and medical sciences and are usually overdispersed. All previous methods used parametric models and only considered overdispersion due to the variation of sizes. The proposed semiparametric model considers overdispersion due to the variation of sizes and that of probabilities. By doing this, it can include variations caused by observations, missing covariates, and random measurement errors in covariates. An Expectation Conditional Maximization algorithm is provided to stabilize the loglikelihood optimization. Selecting the number of s…
Accurate statistical inference in logistic regression models remains a critical challenge when the ratio between the number of parameters and sample size is not negligible. This is because approximations based on either classical asymptotic theory or bootstrap calculations are grossly off the mark. This paper introduces a resized bootstrap method to infer model parameters in arbitrary dimensions. As in the parametric bootstrap, we resample observations from a distribution, which depends on an estimated regression coefficient sequence. The novelty is that this estimate is actually far from the …
This paper considers a new bootstrap procedure to estimate the distribution of high-dimensional $\ell_p$-statistics, i.e. the $\ell_p$-norms of the sum of $n$ independent $d$-dimensional random vectors with $d \gg n$ and $p \in [1, \infty]$. We provide a non-asymptotic characterization of the sampling distribution of $\ell_p$-statistics based on Gaussian approximation and show that the bootstrap procedure is consistent in the Kolmogorov-Smirnov distance under mild conditions on the covariance structure of the data. As an application of the general theory we propose a bootstrap hypothesis test …
For a variant of the algorithm in [Pit19] (arXiv:1903.10816) to compute the approximate density or distribution function of a linear mixture of independent random variables known by a finite sample, it is presented a proof of the functional correctness, i.e. the convergence of the computed distribution function towards the true distribution function (given the observations) as the algorithm resolution is increased to infinity. The algorithm (like its predecessor version) bears elements which are closely related to early known methods for numerical inversion of the characteristic function of a …
A novel yet simple extension of the symmetric logistic distribution is proposed by introducing a skewness parameter. It is shown how the three parameters of the ensuing skew logistic distribution may be estimated using maximum likelihood. The skew logistic distribution is then extended to the skew bi-logistic distribution to allow the modelling of multiple waves in epidemic time series data. The proposed skew-logistic model is validated on COVID-19 data from the UK, and is evaluated for goodness-of-fit against the logistic and normal distributions using the recently formulated empirical surviv…
Standard statistical methods that do not take proper account of the complexity of survey design can lead to erroneous inferences when applied to survey data due to unequal selection probabilities, clustering, and other design features. In particular, the actual type I error rates of tests of hypotheses using standard methods can be much bigger than the nominal significance level. Methods that take account of survey design features in testing hypotheses have been proposed, including Wald tests and quasi-score tests that involve the estimated covariance matrices of parameter estimates. In this p…
In this work, we provide a refinement of the selective CLT result of Tian and Taylor (2015), which allows for selective inference in non-parametric settings by adjusting for the asymptotic Gaussian limit for selection. Under some regularity assumptions on the density of the randomization, including heavier tails than Gaussian satisfied by e.g. logistic distribution, we prove the selective CLT holds without any assumptions on the underlying parameter, allowing for rare selection events. We also show a selective CLT result for Gaussian randomization, though the quantitative results are qualitati…
Separation in logistic regression is a common problem causing failure of the iterative estimation process when finding maximum likelihood estimates. Firth's correction (FC) was proposed as a solution, providing estimates also in presence of separation. In this paper we evaluate whether ridge regression (RR) could be considered instead, specifically, if it could reduce the mean squared error (MSE) of coefficient estimates in comparison to FC. In RR the tuning parameter determining the penalty strength is usually obtained by minimizing some measure of the out-of-sample prediction error or inform…
Methods based on partial least squares (PLS) regression, which has recently gained much attention in the analysis of high-dimensional genomic datasets, have been developed since the early 2000s for performing variable selection. Most of these techniques rely on tuning parameters that are often determined by cross-validation (CV) based methods, which raises important stability issues. To overcome this, we have developed a new dynamic bootstrapbased method for significant predictor selection, suitable for both PLS regression and its incorporation into generalized linear models (GPLS). It relies …
The bootstrap, based on resampling, has, for several decades, been a widely used method for computing confidence intervals for applications where no exact method is available and when sample sizes are not large enough to be able to rely on easy-to-compute large-sample approximate methods, such a Wald (normal-approximation) confidence intervals. Simulation based bootstrap intervals have been proven useful in that their actual coverage probabilities are close to the nominal confidence level in small samples. Small samples analytical approximations such as the Wald method, however, tend to have c…
Differentially Private Bootstrap: New Privacy Analysis and Inference Strategies
Department of Statistics Purdue University West Lafayette, IN 47906, USA（统计学系 帕克森大学）；Department of Statistics University of California, Los Angeles Los Angeles, CA 90095, USA（统计学系 加州大学洛杉矶分校）；Department of Statistics Purdue University West Lafayette, IN 47906, USA and Department of Statistics University of Pittsburgh Pittsburgh, PA 15260, USA（统计学系 帕克森大学 and 统计学系 肯塔基大学）
In this paper, we extend to generalized linear models (including logistic and other binary regression models, Poisson regression and gamma regression models) the robust model selection methodology developed by Mueller and Welsh (2005; JASA) for linear regression models. As in Mueller and Welsh (2005), we combine a robust penalized measure of fit to the sample with a robust measure of out of sample predictive ability which is estimated using a post-stratified m-out-of-n bootstrap. A key idea is that the method can be used to compare different estimators (robust and nonrobust) as well as differe…
A regression method for proportional, or fractional, data with mixed effects is outlined, designed for analysis of datasets in which the outcomes have substantial weight at the bounds. In such cases a normal approximation is particularly unsuitable as it can result in incorrect inference. To resolve this problem, we employ a logistic regression model and then apply a bootstrap method to correct conservative confidence intervals. This paper outlines the theory of the method, and demonstrates its utility using simulated data. Working code for the R platform is provided through the package glmmbo…
Biomediq A/S（Biomediq公司）；Cerebriu A/S（Cerebriu公司）；University of Copenhagen（哥本哈根大学）；University College London（伦敦大学学院）；King’s College London（伦敦国王学院）
Quantitative characterization of disease progression using longitudinal data can provide long-term predictions for the pathological stages of individuals. This work studies the robust modeling of Alzheimer's disease progression using parametric methods. The proposed method linearly maps the individual's age to a disease progression score (DPS) and jointly fits constrained generalized logistic functions to the longitudinal dynamics of biomarkers as functions of the DPS using M-estimation. Robustness of the estimates is quantified using bootstrapping via Monte Carlo resampling, and the estimated…
Let $\{X_i\}_{i=-\infty}^{\infty}$ be a sequence of random vectors and $Y_{in}=f_{in}(\mathcal{X}_{i,\ell})$ be zero mean block-variables where $\mathcal{X}_{i,\ell}=(X_i,...,X_{i+\ell-1}),i\geq 1$, are overlapping blocks of length $\ell$ and where $f_{in}$ are Borel measurable functions. This paper establishes valid joint asymptotic expansions of general orders for the joint distribution of the sums $\sum_{i=1}^nX_i$ and $\sum_{i=1}^nY_{in}$ under weak dependence conditions on the sequence $\{X_i\}_{i=-\infty}^{\infty}$ when the block length $\ell$ grows to infinity. In contrast to the classi…
Considering the increasing size of available data, the need for statistical methods that control the finite sample bias is growing. This is mainly due to the frequent settings where the number of variables is large and allowed to increase with the sample size bringing standard inferential procedures to incur significant loss in terms of performance. Moreover, the complexity of statistical models is also increasing thereby entailing important computational challenges in constructing new estimators or in implementing classical ones. A trade-off between numerical complexity and statistical proper…
Zou [J. Amer. Statist. Assoc. 101 (2006) 1418-1429] proposed the Adaptive LASSO (ALASSO) method for simultaneous variable selection and estimation of the regression parameters, and established its oracle property. In this paper, we investigate the rate of convergence of the ALASSO estimator to the oracle distribution when the dimension of the regression parameters may grow to infinity with the sample size. It is shown that the rate critically depends on the choices of the penalty parameter and the initial estimator, among other factors, and that confidence intervals (CIs) based on the oracle l…
In this paper, we derive higher order Edgeworth expansions for the finite sample distributions of the subsampling-based t-statistic and the Wald statistic in the Gaussian location model under the so-called fixed-smoothing paradigm. In particular, we show that the error of asymptotic approximation is at the order of the reciprocal of the sample size and obtain explicit forms for the leading error terms in the expansions. The results are used to justify the second-order correctness of a new bootstrap method, the Gaussian dependent bootstrap, in the context of Gaussian location model.
