Asymptotic Variance-covariance Matrix Research Articles

Pivotal-based inference for a Pareto distribution under the adaptive progressive Type-II censoring scheme

<abstract><p>This paper proposes an inference approach based on a pivotal quantity under the adaptive progressive Type-II censoring scheme. To exemplify the proposed methodology, an extensively employed distribution, a Pareto distribution, is utilized. This distribution has limitations in estimating confidence intervals for unknown parameters from classical methods such as the maximum likelihood and bootstrap methods. For example, in the maximum likelihood method, the asymptotic variance-covariance matrix does not always exist. In addition, both classical methods can yield confidence intervals that do not satisfy nominal levels when a sample size is not large enough. Our approach resolves these limitations by allowing us to construct exact intervals for unknown parameters with computational simplicity. Aside from this, the proposed approach leads to closed-form estimators with properties such as unbiasedness and consistency. To verify the validity of the proposed methodology, two approaches, a Monte Carlo simulation and a real-world data analysis, are conducted. The simulation testifies to the superior performance of the proposed methodology as compared to the maximum likelihood method, and the real-world data analysis examines the applicability and scalability of the proposed methodology.</p></abstract>

Simpler standard errors for two-stage optimization estimators revisited

“Simpler standard errors for two-stage optimization estimators” (Terza, 2016a, Stata Journal 16: 368–385) offers an analytic simplification of the daunting textbook formulations of the asymptotic variance–covariance matrix of a class of two-stage optimization estimators. Here I revisit that simplification and show that it applies to a much broader class of estimators than was originally considered. I also offer a correction that further enhances the generality of this asymptotic variance–covariance matrix formulation. These points are illustrated via a real-data application.

Analysis of the Stress–Strength Model Using Uniform Truncated Negative Binomial Distribution under Progressive Type-II Censoring

In this study, we introduce a novel estimation technique for assessing the reliability parameter R=P(Y<X) of the uniform truncated negative binomial distribution (UTNBD) in the context of stress–strength analysis. We base our inferences on the assumption that both the strength (X) and stress (Y) random variables follow a UTNBD with identical first shape and scale parameters. In the presence of a progressive type-II censoring scheme, we employ maximum likelihood, two parametric bootstrap methods, and Bayesian estimation approaches to derive the estimators. Due to the complexity introduced by censoring, the estimators are not available in explicit forms and are instead obtained through numerical approximation techniques. Furthermore, we compute the highest posterior density credible intervals and determine the asymptotic variance-covariance matrix. To assess the performance of our proposed estimators, we conduct a Monte Carlo simulation study and provide a comparative analysis. Finally, we illustrate the practical applicability of our study with an engineering application.

Approximating Partial Likelihood Estimators via Optimal Subsampling

With the growing availability of large-scale biomedical data, it is often time-consuming or infeasible to directly perform traditional statistical analysis with relatively limited computing resources at hand. We propose a fast subsampling method to effectively approximate the full data maximum partial likelihood estimator in Cox’s model, which largely reduces the computational burden when analyzing massive survival data. We establish consistency and asymptotic normality of a general subsample-based estimator. The optimal subsampling probabilities with explicit expressions are determined via minimizing the trace of the asymptotic variance-covariance matrix for a linearly transformed parameter estimator. We propose a two-step subsampling algorithm for practical implementation, which has a significant reduction in computing time compared to the full data method. The asymptotic properties of the resulting two-step subsample-based estimator is also established. Extensive numerical experiments and a real-world example are provided to assess our subsampling strategy. Supplemental materials for this article are available online.

Fitting additive risk models using auxiliary information.

There has been a growing interest in incorporating auxiliary summary information from external studies into the analysis of internal individual-level data. In this paper, we propose an adaptive estimation procedure for an additive risk model to integrate auxiliary subgroup survival information via a penalized method of moments technique. Our approach can accommodate information from heterogeneous data. Parameters to quantify the magnitude of potential incomparability between internal data and external auxiliary information are introduced in our framework while nonzero components of these parameters suggest a violation of the homogeneity assumption. We further develop an efficient computational algorithm to solve the numerical optimization problem by profiling out the nuisance parameters. In an asymptotic sense, our method can be as efficient as if all the incomparable auxiliary information is accurately acknowledged and has been automatically excluded from consideration. The asymptotic normality of the proposed estimator of the regression coefficients is established, with an explicit formula for the asymptotic variance-covariance matrix that can be consistently estimated from the data. Simulation studies show that the proposed method yields a substantial gain in statistical efficiency over the conventional method using the internal data only, and reduces estimation biases when the given auxiliary survival information is incomparable. We illustrate the proposed method with a lung cancer survival study.

Asymptotic Standard Errors of IRT Linking Coefficients for the Bifactor Extension of the 3PL Model

The bifactor extension of the three-parameter logistic (B3PL) model has been used in applications of multidimensional item response theory (IRT) such as test equating and vertical scaling. Developing a common multidimensional IRT scale (that is, multidimensional coordinate system) is critical in those applications. Three common-item scale linking methods, the direct least squares (DLS), mean/least squares (MLS), and item response function (IRF) methods, for the B3PL model have been found to be effective in developing a common multidimensional IRT ability scale between two test forms to be linked. In this paper, the asymptotic standard errors (SEs) of IRT linking coefficients estimated by the DLS, MLS, and IRF methods are derived assuming that the B3PL model holds and the asymptotic variance-covariance matrix of item parameter estimates from separate calibrations is available. The delta method is used for the derivations. Computer simulations which investigate the accuracy of the derivations under various conditions are given, showing that the derivations are reasonably accurate when sufficiently large samples are used and that in general the SEs of the IRF method are smaller than those of the DLS and MLS methods. The simulation results also suggest that the SEs of linking coefficient estimates are, approximately, inversely proportional to the square root of the sample size when two test forms are administered to the same number of examinees.

Finite-sample performance of the robust variance estimator in the presence of missing data

Theoretically, the maximum likelihood estimator has the sandwich-type asymptotic variance-covariance matrix under model misspecification. Its empirical estimator, that is called the robust variance estimator, is consistent. Thus, the estimator is asymptotically valid even under model misspecification. In practice, the robust variance estimator is used for computation of standard errors in longitudinal data analysis. Recently, Golden et al. (2019 Econometrics, 7, 1-27) showed that the maximum likelihood estimator retains a sandwich-type asymptotic variance-covariance matrix in the presence of missing data even when the missing-data mechanism is missing not at random. Although they revealed the asymptotic validity of the robust variance estimator in the simultaneous presence of both model misspecification and missing data, its finite-sample performance did not be investigated. In this article, we evaluated the finite-sample performance via simulation studies and clarify its small-sample problems. In addition, we illustrated the robust variance estimator using longitudinal CD4 count data from a randomized double-blind study.

Inferences for Stress-Strength Reliability Model in the Presence of Partially Accelerated Life Test to Its Strength Variable.

We focus on estimating the stress-strength reliability model when the strength variable is subjected to the step-stress partially accelerated life test. Based on the assumption that both stress and strength random variables follow Weibull distribution with a common first shape parameter, the inferences for this reliability system are constructed. The maximum likelihood, two parametric bootstraps, and Bayes estimates are obtained. Moreover, approximate confidence intervals, asymptotic variance-covariance matrix, and highest posterior density credible intervals are derived. A simulation study and application to real-life data are conducted to compare the proposed estimation methods developed here and also check the accuracy of the results.

Fluctuation Results for Multi-species Sherrington-Kirkpatrick Model in the Replica Symmetric Regime

We study the Replica Symmetric region of general multi-species Sherrington-Kirkpatrick (MSK) Model and answer some of the questions raised in Ann. Probab. 43 (2015), no. 6, 3494–3513, where the author proved the Parisi formula under positive-definite assumption on the disorder covariance matrix $$\Delta ^2$$ . First, we prove exponential overlap concentration at high temperature for both indefinite and positive-definite $$\Delta ^2$$ MSK model. We also prove a central limit theorem for the free energy using overlap concentration. Furthermore, in the zero external field case, we use a quadratic coupling argument to prove overlap concentration up to $$\beta _c$$ , which is expected to be the critical inverse temperature. The argument holds for both positive-definite and emphindefinite $$\Delta ^2$$ , and $$\beta _c$$ has the same expression in two different cases. Second, we develop a species-wise cavity approach to study the overlap fluctuation, and the asymptotic variance-covariance matrix of overlap is obtained as the solution to a matrix-valued linear system. The asymptotic variance also suggests the de Almeida–Thouless (AT) line condition from the Replica Symmetry (RS) side. Our species-wise cavity approach does not require the positive-definiteness of $$\Delta ^2$$ . However, it seems that the AT line conditions in positive-definite and indefinite cases are different. Finally, in the case of positive-definite $$\Delta ^2$$ , we prove that above the AT line, the MSK model is in Replica Symmetry Breaking phase under some natural assumption. This generalizes the results of J. Stat. Phys. 174 (2019), no. 2, 333–350, from 2-species to general species.

Two-phase sample selection strategies for design and analysis in post-genome-wide association fine-mapping studies.

Post‐GWAS analysis, in many cases, focuses on fine‐mapping targeted genetic regions discovered at GWAS‐stage; that is, the aim is to pinpoint potential causal variants and susceptibility genes for complex traits and disease outcomes using next‐generation sequencing (NGS) technologies. Large‐scale GWAS cohorts are necessary to identify target regions given the typically modest genetic effect sizes. In this context, two‐phase sampling design and analysis is a cost‐reduction technique that utilizes data collected during phase 1 GWAS to select an informative subsample for phase 2 sequencing. The main goal is to make inference for genetic variants measured via NGS by efficiently combining data from phases 1 and 2. We propose two approaches for selecting a phase 2 design under a budget constraint. The first method identifies sampling fractions that select a phase 2 design yielding an asymptotic variance covariance matrix with certain optimal characteristics, for example, smallest trace, via Lagrange multipliers (LM). The second relies on a genetic algorithm (GA) with a defined fitness function to identify exactly a phase 2 subsample. We perform comprehensive simulation studies to evaluate the empirical properties of the proposed designs for a genetic association study of a quantitative trait. We compare our methods against two ranked designs: residual‐dependent sampling and a recently identified optimal design. Our findings demonstrate that the proposed designs, GA in particular, can render competitive power in combined phase 1 and 2 analysis compared with alternative designs while preserving type 1 error control. These results are especially evident under the more practical scenario where design values need to be defined a priori and are subject to misspecification. We illustrate the proposed methods in a study of triglyceride levels in the North Finland Birth Cohort of 1966. R code to reproduce our results is available at github.com/egosv/TwoPhase_postGWAS.

Inferences for Weibull Fréchet Distribution Using a Bayesian and Non-Bayesian Methods on Gastric Cancer Survival Times.

In this article, based on progressively type-II censored schemes, the maximum likelihood, Bayes, and two parametric bootstrap methods are used for estimating the unknown parameters of the Weibull Fréchet distribution and some lifetime indices as reliability and hazard rate functions. Moreover, approximate confidence intervals and asymptotic variance-covariance matrix have been obtained. Markov chain Monte Carlo technique based on Gibbs sampler within Metropolis–Hasting algorithm is used to generate samples from the posterior density functions. Furthermore, Bayesian estimate is computed under both balanced square error loss and balanced linear exponential loss functions. Simulation results have been implemented to obtain the accuracy of the estimators. Finally, application on the survival times in years of a group of patients given chemotherapy and radiation treatment is presented for illustrating all the inferential procedures developed here.

Quantile Regression with Generated Regressors

This paper studies estimation and inference for linear quantile regression models with generated regressors. We suggest a practical two-step estimation procedure, where the generated regressors are computed in the first step. The asymptotic properties of the two-step estimator, namely, consistency and asymptotic normality are established. We show that the asymptotic variance-covariance matrix needs to be adjusted to account for the first-step estimation error. We propose a general estimator for the asymptotic variance-covariance, establish its consistency, and develop testing procedures for linear hypotheses in these models. Monte Carlo simulations to evaluate the finite-sample performance of the estimation and inference procedures are provided. Finally, we apply the proposed methods to study Engel curves for various commodities using data from the UK Family Expenditure Survey. We document strong heterogeneity in the estimated Engel curves along the conditional distribution of the budget share of each commodity. The empirical application also emphasizes that correctly estimating confidence intervals for the estimated Engel curves by the proposed estimator is of importance for inference.

On a new generalized lindley distribution: Properties, estimation and applications.

In this study, an extension of the generalized Lindley distribution using the Marshall-Olkin method and its own sub-models is presented. This new model for modelling survival and lifetime data is flexible. Several statistical properties and characterizations of the subject distribution along with its reliability analysis are presented. Statistical inference for the new family such as the Maximum likelihood estimators and the asymptotic variance covariance matrix of the unknown parameters are discussed. A simulation study is considered to compare the efficiency of the different estimators based on mean square error criterion. Finally, a real data set is analyzed to show the flexibility of our proposed model compared with the fit attained by some other competitive distributions.

Optimal subsampling for quantile regression in big data

Summary We investigate optimal subsampling for quantile regression. We derive the asymptotic distribution of a general subsampling estimator and then derive two versions of optimal subsampling probabilities. One version minimizes the trace of the asymptotic variance-covariance matrix for a linearly transformed parameter estimator and the other minimizes that of the original parameter estimator. The former does not depend on the densities of the responses given covariates and is easy to implement. Algorithms based on optimal subsampling probabilities are proposed and asymptotic distributions, and the asymptotic optimality of the resulting estimators are established. Furthermore, we propose an iterative subsampling procedure based on the optimal subsampling probabilities in the linearly transformed parameter estimation which has great scalability to utilize available computational resources. In addition, this procedure yields standard errors for parameter estimators without estimating the densities of the responses given the covariates. We provide numerical examples based on both simulated and real data to illustrate the proposed method.

On the consistency of the logistic quasi-MLE under conditional symmetry

For estimating the parameters of a linear conditional mean, I show that the quasi-maximum likelihood estimator (QMLE) obtained under the nominal assumption that the error term is independent of the explanatory variables with a logistic distribution is consistent provided the conditional distribution of the error term is symmetric. No other restrictions are required for Fisher consistency; in particular, the error and covariates need not be independent, and so general heteroskedasticity of unknown form is allowed. Importantly, the influence function of the logistic quasi-log likelihood is bounded, making it more resilient to outliers than ordinary least squares. Inference using the logistic QMLE is straightforward using a robust asymptotic variance–covariance matrix estimator.

Efficient and direct estimation of the variance–covariance matrix in EM algorithm with interpolation method

The expectation–maximization (EM) algorithm is a seminal method to calculate the maximum likelihood estimators (MLEs) for incomplete data. However, one drawback of this algorithm is that the asymptotic variance–covariance matrix of the MLE is not automatically produced. Although there are several methods proposed to resolve this drawback, limitations exist for these methods. In this paper, we propose an innovative interpolation procedure to directly estimate the asymptotic variance–covariance matrix of the MLE obtained by the EM algorithm. Specifically we make use of the cubic spline interpolation to approximate the first-order and the second-order derivative functions in the Jacobian and Hessian matrices from the EM algorithm. It does not require iterative procedures as in other previously proposed numerical methods, so it is computationally efficient and direct. We derive the truncation error bounds of the functions theoretically and show that the truncation error diminishes to zero as the mesh size approaches zero. The optimal mesh size is derived as well by minimizing the global error. The accuracy and the complexity of the novel method is compared with those of the well-known SEM method. Two numerical examples and a real data are used to illustrate the accuracy and stability of this novel method.

THE BETA TRANSMUTED POWER DISTRIBUTION: PROPERTIES AND APPLICATION

In this this paper, we define and study a new generalization of the Power distribution and the quadratic rank transmutation map (QRTM) in order to generate a flexible family of probability distribution taking Power distribution as the base distribution. The new distribution is called the beta transmuted Power (BTP) distribution. Some properties of the distribution such as moments, quantiles, mean deviation and order statistics are derived. The method of maximum likelihood is proposed to estimate the model parameters. The asymptotic confidence intervals for the parameters are also obtained based on asymptotic variance-covariance matrix. A simulation study is conducted to study the performance of the estimators. The importance and flexibility of the new model is proved empirically using a real data set.

Statistical Inference for Gompertz Distribution based on Progressive Type-II Censored Data with Binomial Removals

In this paper, the problem of estimation of parameters for a two-parameterGompertz distribution is considered based on a progressively type-II censored sample with binomial removals. Together with the unknown parameters, the removal probability is also estimated. The maximum likelihood estimators of the parameters and the asymptotic variance-covariance matrix of the estimates are obtained. Bayes estimators are also obtained using different loss functions such as squared error, LINEX and general entropy. A simulation study is performed for comparison between various estimators developed in this paper. A real data set is also used for illustration.

The quantile-based flattened logistic distribution: Some properties and applications

In this paper, the quantile-based flattened logistic distribution has been studied. Some classical and quantile-based properties of the distribution have been obtained. Closed form expressions of L-moments, L-moment ratios and expectation of order statistics of the distribution have been obtained. A quantile-based analysis concerning the method of matching L-moments estimation is employed to estimate the parameters of the proposed model. We further derive the asymptotic variance–covariance matrix of the matching L-Moments estimators of the proposed model. Finally, we apply the proposed model to simulated as well as two real life datasets and compare the fit with the logistic distribution.

Inferences for Weibull-Exponential Distribution Based on Progressive Type-II Censoring Under Step-Stress Partially Accelerated Life Test Model

The object of this paper is to obtain point and interval estimations for the parameters of the Weibull-exponential distribution (WED) based on step-stress partially accelerated life tests model with progressive type-II censoring (PRO-II-C). Maximum likelihood (ML) and Bayes estimation method are used to estimate the unknown parameters of WED and the acceleration factor. Moreover, the approximate confidence intervals and asymptotic variance–covariance matrix have been acquired. Markov Chain Monte Carlo (MCMC) technique is applied to estimate the unknown parameters of WED and the acceleration factor. The Metropolis–Hastings algorithm is a MCMC method which used to generate samples from the posterior density functions. An example is applied for different estimation methods. Finally, a Monte Carlo simulation study is carried out to investigate the precision of Bayes estimates with MLEs and to compare the performance of different corresponding CIs.