Large Sample Properties Research Articles

GIVE statistic for goodness of fit in instrumental variables models with application to COVID data.

Since COVID-19 outbreak, scientists have been interested to know whether there is any impact of the Bacillus Calmette–Guerin (BCG) vaccine against COVID-19 mortality or not. It becomes more relevant as a large population in the world may have latent tuberculosis infection (LTBI), for which a person may not have active tuberculosis but persistent immune responses stimulated by Mycobacterium tuberculosis antigens, and that means, both LTBI and BCG generate immunity against COVID-19. In order to understand the relationship between LTBI and COVID-19 mortality, this article proposes a measure of goodness of fit, viz., Goodness of Instrumental Variable Estimates (GIVE) statistic, of a model obtained by Instrumental Variables estimation. The GIVE statistic helps in finding the appropriate choice of instruments, which provides a better fitted model. In the course of study, the large sample properties of the GIVE statistic are investigated. As indicated before, the COVID-19 data is analysed using the GIVE statistic, and moreover, simulation studies are also conducted to show the usefulness of the GIVE statistic along with analysis of well-known Card data.

Approximate least squares estimators of a two-dimensional chirp model

In this paper, we propose a method based on maximizing a periodogram-type function for parameter estimation of a two-dimensional (2-D) mono-component chirp signal model. The obtained estimators are called approximate least squares estimators (ALSEs). We also put forward a sequential algorithm for parameter estimation of a more general version of the model with multiple number of components. The main focus of the paper has been on developing the large-sample properties of the proposed estimators. Derived theoretical results show that the proposed estimators are strongly consistent and asymptotically normally distributed. Moreover, it is shown that the derived asymptotic distribution is identical to that of the traditional least squares estimators (LSEs). The numerical performance of the ALSEs and sequential ALSEs is demonstrated through extensive simulation studies. The results are positioned parallel to that obtained using the least squares method and 2-D multilag High Order Ambiguity function (2D-ml-HAF) for a comparative study. These simulations make use of the true values as the initial guesses in the optimization algorithm as using a fine grid search for this purpose is practically infeasible. We also provide a practical alternative for modeling of real world signals in an another set of simulation experiments. This involves a combination of two methods- 2D-ml-HAF estimators and genetic algorithm (GA). The results show that the proposed combination is a reasonable scheme to find initial guesses and provides satisfactory performance for varying signal-to-noise ratio (SNR).

Analysis of parent-of-origin effects for secondary phenotypes using case-control mother-child pair data.

The detection of parent-of-origin effects (POEs) has become a research focus in genetic association studies since POEs play an important role in explaining the heritability of many complex human disorders. Genetic association studies are commonly conducted based on case-control designs. Case-control genetic association studies often collect additional information on secondary phenotypes other than the case-control status. Various statistical methods have been proposed to analyze the secondary phenotypes, but no methods are specifically tailored for identifying POEs of offspring genes on the secondary phenotypes. The parental origin information may not be determined unambiguously using the genotypes of the test locus for some families, and ignoring such families would lose considerable information. In this article, we focus on case-control mother-child pair design that has been widely used for studying human early life growth and development, and propose a robust and efficient retrospective likelihood method to detect POEs for the secondary phenotypes using multilocus genotypes. The proposed method fully utilizes the information from multilocus genotypes, Hardy-Weinberg equilibrium (HWE), Mendelian inheritance law, and conditional independence between child genotype and maternal covariate given maternal genotype. Large sample properties, including consistency and asymptotic normality, are established for our proposed statistical method. The finite sample performance of our method are demonstrated through extensive simulation studies and application to the Danish National Birth Cohort data.

Estimating multivariate density and its derivatives for mixed measurement error data

In this paper, we propose a nonparametric mixed kernel estimator for a multivariate density function and its derivatives when the data are contaminated with different sources of measurement errors. The proposed estimator is a mixture of the classical and the deconvolution kernels, accounting for the error-free and error-prone variables, respectively. Large sample properties of the proposed nonparametric estimator, including the order of the mean squares error, the consistency, and the asymptotic normality, are thoroughly investigated. The optimal convergence rates among all nonparametric estimators for different measurement error structures are derived, and it is shown that the proposed mixed kernel estimators achieve the optimal convergence rate. A simulation study is conducted to evaluate the finite sample performance of the proposed estimators.

Functional group bridge for simultaneous regression and support estimation.

This paper is motivated by studying differential brain activities to multiple experimental condition presentations in intracranial electroencephalography (iEEG) experiments. Contrasting effects of experimental conditions are often zero in most regions and nonzero in some local regions, yielding locally sparse functions. Such studies are essentially a function-on-scalar regression problem, with interest being focused not only on estimating nonparametric functions but also on recovering the function supports. We propose a weighted group bridge approach for simultaneous function estimation and support recovery in function-on-scalar mixed effect models, while accounting for heterogeneity present in functional data. We use B-splines to transform sparsity of functions to its sparse vector counterpart of increasing dimension, and propose a fast nonconvex optimization algorithm using nested alternative direction method of multipliers (ADMM) for estimation. Large sample properties are established. In particular, we show that the estimated coefficient functions are rate optimal in the minimax sense under the L2 norm and resemble a phase transition phenomenon. For support estimation, we derive a convergence rate under the norm that leads to a selection consistency property under δ-sparsity, and obtain a result under strict sparsity using a simple sufficient regularity condition. An adjusted extended Bayesian information criterion is proposed for parameter tuning. The developed method is illustrated through simulations and an application to a novel iEEG data set to study multisensory integration.

Asymptotic properties of Bayesian inference in linear regression with a structural break

This paper studies large sample properties of a Bayesian approach to inference about slope parameters γ in linear regression models with a structural break. In contrast to the conventional approach to inference about γ that does not take into account the uncertainty of the unknown break date, the Bayesian approach that we consider incorporates such uncertainty. Our main theoretical contribution is a Bernstein–von Mises type theorem (Bayesian asymptotic normality) for γ under a wide class of priors, which essentially indicates an asymptotic equivalence between the conventional frequentist and Bayesian inference. Consequently, a frequentist researcher could look at credible intervals of γ to check robustness with respect to the uncertainty of the break date. Simulation studies show that the conventional confidence intervals of γ tend to undercover in finite samples whereas the credible intervals offer more reasonable coverages in general. As the sample size increases, the two methods coincide, as predicted from our theoretical conclusion. Using data from Paye and Timmermann (2006) on stock return prediction, we illustrate that the traditional confidence intervals on γ might underrepresent the true sampling uncertainty.

GDUS-Modified Topp-Leone Distribution: A New Distribution with Increasing, Decreasing, and Bathtub Hazard Functions

In this paper, we propose an extension to the Topp-Leone distribution, as introduced by [20] using the Generalized-DUS transformation given by [8]. The Topp-Leone distribution is defined on interval (0,1) and has a characteristic J-shaped frequency curve. The newly extended version of Topp-Leone distribution accommodates a variety of shapes of hazard rate functions making it a versatile distribution. We have also derived explicit expressions for some properties like ordinary moments, conditional moments, distribution of order statistics, quantiles, mean deviation, and entropy. Further, we have also discussed results on identifiability, stress-strength reliability, and stochastic ordering that are concerned with two independent random variables. For inference regarding the unknown parameters of the distribution, we derive the equations which give their maximum likelihood estimators. We also present the asymptotic confidence intervals of the unknown parameters of the distribution, based on large sample property, using the Fisher information matrix. To facilitate further studies, a step-by-step algorithm is presented to produce a random sample from the distribution. Further, extensive simulation experiments are done to study the long-term behavior of the maximum likelihood estimators of the parameters through their mean squared error and mean absolute bias on the basis of large number of samples. The consistency of the MLEs is empirically proved. Lastly, the application of the proposed distribution is shown by fitting a real-life dataset over some existing distributions in the same range.

Extension of Relative-Risk Power Estimator under Dependent Random Censored Data

In this paper, the considered problem consists in estimation of conditional survival function by right random censoring model in the presence of a covariate. We propose a new estimator of conditional survival function which is extension of relative-risk power estimator of independent censoring and study its large sample properties. We present result of asymptotic normality with the same limiting Gaussian process as for copula-graphic estimator.

Nonparametric and semiparametric estimation with sequentially truncated survival data.

In observational cohort studies with complex sampling schemes, truncation arises when the time to event of interest is observed only when it falls below or exceeds another random time, that is, the truncation time. In more complex settings, observation may require a particular ordering of event times; we refer to this as sequential truncation. Estimators of the event time distribution have been developed for simple left-truncated or right-truncated data. However, these estimators may be inconsistent under sequential truncation. We propose nonparametric and semiparametric maximum likelihood estimators for the distribution of the event time of interest in the presence of sequential truncation, under two truncation models. We show the equivalence of an inverse probability weighted estimator and a product limit estimator under one of these models. We study the large sample properties of the proposed estimators and derive their asymptotic variance estimators. We evaluate the proposed methods through simulation studies and apply the methods to an Alzheimer's disease study. We have developed an R package, seqTrun, for implementation of ourmethod.

Log-Rank-Type Tests for Equality of Distributions in High-Dimensional Spaces

Motivated by applications in high-dimensional settings, we propose a novel approach for testing the equality of two or more populations by constructing a class of intensity centered score processes. The resulting tests are analogous in spirit to the well-known class of weighted log-rank statistics that are widely used in survival analysis. The test statistics are nonparametric, computationally simple, and applicable to high-dimensional data. We establish the usual large sample properties by showing that the underlying log-rank score process converges weakly to a Gaussian random field with zero mean under the null hypothesis and with a drift under the contiguous alternatives. For the Kolmogorov–Smirnov-type and the Cramer-von Mises-type statistics, we also establish the consistency result for any fixed alternative. Cutoff points for the test statistics are obtained by permutations or a simulation-based resampling method. The new approach is applied to a study of brain activation measured by functional magnetic resonance imaging when performing two linguistic tasks and also to a prostate cancer DNA microarray dataset. Supplementary materials for this article are available online.

Regression modeling of restricted mean survival time for left-truncated right-censored data.

The restricted mean survival time (RMST) is a clinically meaningful summary measure in studies with survival outcomes. Statistical methods have been developed for regression analysis of RMST to investigate impacts of covariates on RMST, which is a useful alternative to the Cox regression analysis. However, existing methods for regression modeling of RMST are not applicable to left-truncated right-censored data that arise frequently in prevalent cohort studies, for which the sampling bias due to left truncation and informative censoring induced by the prevalent sampling scheme must be properly addressed. The pseudo-observation (PO) approach has been used in regression modeling of RMST for right-censored data and competing-risks data. For left-truncated right-censored data, we propose to directly model RMST as a function of baseline covariates based on POs under general censoring mechanisms. We adjust for the potential covariate-dependent censoring or dependent censoring by the inverse probability of censoring weighting method. We establish large sample properties of the proposed estimators and assess their finite sample performances by simulation studies under various scenarios. We apply the proposed methods to a prevalent cohort of women diagnosed with stage IV breast cancer identified from surveillance, epidemiology, and end results-medicare linked database.

Extrapolation estimation in parametric regression models with measurement error

For the general parametric regression models with covariates contaminated with normal measurement errors, an alternative method to the traditional simulation extrapolation algorithm is proposed to estimate the unknown parameters in the regression function. By applying the conditional expectation directly to the target function, the proposed algorithm successfully removes the simulation step, by generating an estimation equation either for immediate use or for extrapolating, thus providing a possibility of reducing the computational time or the Monte Carlo simulation error. Large sample properties of the resulting estimator, including the consistency and the asymptotic normality, are thoroughly discussed. Potential wide applications of the proposed estimation procedure are illustrated by examples, simulation studies, as well as a real data analysis.

On the estimation of hazard rate in mixed populations with its application

Mixture of survival and hazard functions have been widely applied to the analysis of data from heterogeneous populations. From the Bayesian point of view, two different predictive mixture hazard rates can be considered: prior and posterior predictive hazard rates. In this paper, we consider a heterogeneous population consists of two sub-populations with different hazard rates where each one follows a Cox proportional hazards model. A prior predictive mixture hazard model is proposed to estimate the hazard rate of the population through the assessment of some potential regression covariates. Under right-censoring, the estimating equations based on martingale are developed to estimate the parameters of the assumed mixture model. The large sample properties of the proposed estimators are established. The finite sample behavior of the resulting estimators is evaluated through simulation studies, and the approach is also applied to a kidney cancer data set collected from a clinical trial.

An Intensity Mapping Constraint on the CO-galaxy Cross-power Spectrum at Redshift ∼3

Abstract The abundance of cold molecular gas plays a crucial role in models of galaxy evolution. While deep spectroscopic surveys of CO emission lines have been a primary tool for measuring this abundance, the difficulty of these observations has motivated alternative approaches to studying molecular gas content. One technique, line intensity mapping, seeks to constrain the average molecular gas properties of large samples of individually undetectable galaxies through the CO brightness power spectrum. Here we present constraints on the cross-power spectrum between CO intensity maps and optical galaxy catalogs. This cross-measurement allows us to check for systematic problems in CO intensity mapping data, and validate the data analysis used for the auto-power spectrum measurement of the CO Power Spectrum Survey. We place a 2σ upper limit on the band-averaged CO-galaxy cross-power of P × < 540 μK h−3 Mpc3. Our measurement favors a nonzero 〈T CO〉 at around 90% confidence and gives an upper limit on the mean molecular gas density at z ∼ 2.6 of 7.7 × 108 M ⊙ Mpc−3. We forecast the expected cross-power spectrum by applying a number of literature prescriptions for the CO luminosity–halo mass relation to a suite of mock light cones. Under the most optimistic forecasts, the cross-spectrum could be detected with only moderate extensions of the data used here, while more conservative models could be detected with a factor of 10 increase in sensitivity. Ongoing CO intensity mapping experiments will target fields allowing for extensive cross-correlation analysis and should reach the sensitivity required to detect the cross-spectrum signal.

Inference for the tail conditional allocation: large sample properties, insurance risk assessment, and compound sums of concomitants

Inference for the tail conditional allocation: large sample properties, insurance risk assessment, and compound sums of concomitants

Asymptotic properties of Bayesian inference in linear regression with a structural break

This paper studies large sample properties of a Bayesian approach to inference about slope parameters $\gamma$ in linear regression models with a structural break. In contrast to the conventional approach to inference about $\gamma$ that does not take into account the uncertainty of the unknown break location $\tau$, the Bayesian approach that we consider incorporates such uncertainty. Our main theoretical contribution is a Bernstein-von Mises type theorem (Bayesian asymptotic normality) for $\gamma$ under a wide class of priors, which essentially indicates an asymptotic equivalence between the conventional frequentist and Bayesian inference. Consequently, a frequentist researcher could look at credible intervals of $\gamma$ to check robustness with respect to the uncertainty of $\tau$. Simulation studies show that the conventional confidence intervals of $\gamma$ tend to undercover in finite samples whereas the credible intervals offer more reasonable coverages in general. As the sample size increases, the two methods coincide, as predicted from our theoretical conclusion. Using data from Paye and Timmermann (2006) on stock return prediction, we illustrate that the traditional confidence intervals on $\gamma$ might underrepresent the true sampling uncertainty.

Semiparametric Single-Index Estimation for Average Treatment Effects

We propose a semiparametric method to estimate the average treatment effect under the assumption of unconfoundedness given observational data. Our estimation method alleviates misspecification issues of the propensity score function by estimating the single-index link function involved through Hermite polynomials. Our approach is computationally tractable and allows for moderately large dimension covariates. We provide the large sample properties of the estimator and show its validity. Also, the average treatment effect estimator achieves the parametric rate and asymptotic normality. Our extensive Monte Carlo study shows that the proposed estimator is valid in finite samples. We also provide an empirical analysis on the effect of maternal smoking on babies' birth weight and the effect of job training program on future earnings.

Estimation and group variable selection for additive partial linear models withwavelets and splines

In this paper we study sparse high dimensional additive partial linear models with nonparametric additive components of heterogeneous smoothness. We review several existing algorithms that have been developed for this problem in the recent literature, highlighting the connections between them, and present some computationally efficient algorithms for fitting such models. To achieve optimal rates in large sample situations we use hybrid P-splines and block wavelet penalisation techniques combined with adaptive (group) LASSO-like procedures for selecting the additive components in the nonparametric part of the models. Hence, the component selection and estimation in the nonparametric part may be viewed as a functional version of estimation and grouped variable selection. This allows to take advantage of several oracle results which yield asymptotic optimality of estimators in high-dimensional but sparse additive models. Numerical implementations of our procedures for proximal like algorithms are discussed. Large sample properties of the estimates and of the model selection are presented and the results are illustrated with simulated examples and a real data analysis. Keywords: Additive models, Backfitting, Penalisation, Proximal algorithms, Squared group-LASSO, Splines, Wavelets

Improved Modified Ratio Estimation of Population Mean Using Information on Size of the Sample

In sample surveys, auxiliary information is used for estimation to improve the efficiency of estimators. Increased precision can be obtained when the variable under study is highly correlated with auxiliary information. In this study, the sample size has been used as information for improved estimation of population mean of the main variable under study. A new modified generalized ratio type estimator of population mean has been proposed and the efficiency was examined using Murthy (1967) and Mukhopadhyay (2009) dataset. The large sample properties, the bias and the mean squared error of the newly proposed modified ratio estimator were obtained up to first order of approximation. The optimum value of the characterizing scalar which minimizes the mean squared error was obtained and the minimum value of the mean squared error of the proposed modified ratio estimator for this optimum value was also obtained. A theoretical comparison of the proposed modified ratio estimators was made with the other existing related estimators of population mean using auxiliary information. The conditions under which the proposed modified ratio estimators perform better than the other existing estimators of population mean are given. A numerical study was also carried out to see the performances of the proposed modified ratio estimators and some existing related ratio estimators of population mean and verify the conditions under which the proposed modified ratio estimators are better than some other existing related ratio estimators considered. It was shown that the proposed modified ratio estimators perform better than some existing related ratio estimators as they are having lower mean squared errors.
 Keywords: Ratio Estimator, Sample size, Bias, Mean Squared Error, Efficiency

Censored quantile regression based on multiply robust propensity scores.

Censored quantile regression has elicited extensive research interest in recent years. One class of methods is based on an informative subset of a sample, selected via the propensity score. Propensity score can either be estimated using parametric methods, which poses the risk of misspecification or obtained using nonparametric approaches, which suffer from "curse of dimensionality." In this study, we propose a new estimation method based on multiply robust propensity score for censored quantile regression. This method only requires one of the multiple candidate models for propensity score to be correctly specified, and thus, it provides a certain level of resistance to the misspecification of parametric models. Large sample properties, such as the consistency and asymptotic normality of the proposed estimator, are thoroughly investigated. Extensive simulation studies are conducted to assess the performance of the proposed estimator. The proposed method is also applied to a study on human immunodeficiency viruses.