Large Sample Properties Research Articles

On negative cumulative extropy with applications

Recently, an alternative measure of uncertainty termed by extropy was proposed by Lad et al. Accordingly, a cumulative residual extropy is also introduced by replacing the survival function in place of the density function in the extropy. In this paper, we define negative cumulative extropy (NCEX) measure of information and study several properties of this concept. We then elaborate the NCEX of coherent system’s lifetime when lifetimes of components are dependent and identically distributed. We also consider the conditional negative cumulative extropy and discuss some its properties. Finally, we propose the estimator of this measure using empirical approach and study it as a risk measure and large sample properties of this estimator.

A weighting method for simultaneous adjustment for confounding and joint exposure-outcome misclassifications.

Joint misclassification of exposure and outcome variables can lead to considerable bias in epidemiological studies of causal exposure-outcome effects. In this paper, we present a new maximum likelihood based estimator for marginal causal effects that simultaneously adjusts for confounding and several forms of joint misclassification of the exposure and outcome variables. The proposed method relies on validation data for the construction of weights that account for both sources of bias. The weighting estimator, which is an extension of the outcome misclassification weighting estimator proposed by Gravel and Platt (Weighted estimation for confounded binary outcomes subject to misclassification. Stat Med 2018; 37: 425–436), is applied to reinfarction data. Simulation studies were carried out to study its finite sample properties and compare it with methods that do not account for confounding or misclassification. The new estimator showed favourable large sample properties in the simulations. Further research is needed to study the sensitivity of the proposed method and that of alternatives to violations of their assumptions. The implementation of the estimator is facilitated by a new R function (ipwm) in an existing R package (mecor).

Two classes of dynamic binomial integer-valued ARCH models

This paper introduces two classes of binomial integer-valued ARCH models with dynamic survival probabilities, each of which is controlled by a stochastic recurrence equation. Stationarity and ergodicity of the process are established, and stochastic properties are given. Conditional least squares and conditional maximum likelihood estimators for the parameters of interest are considered, and their large-sample properties are established. The performances of these estimators are compared via simulation studies. Finally, we demonstrate the usefulness of the proposed models by analyzing real datasets.

Reweighting estimators for the transformation models with length-biased sampling data and missing covariates

Length-biased sampling data are commonly observed in cross-sectional surveys and epidemiological cohort studies. Due to study design or accident, some components of the covariate vector are often missing. This article considers the statistical inference for the transformation models with length-biased sampling data and missing covariates. The reweighting estimating procedures are proposed for the unknown regression parameters when the selection probability is known, estimated non parametrically, or estimated parametrically. The large sample properties of the resulting estimators are studied. Simulation studies are presented to demonstrate the utility and efficiency of the proposed methods.

Finite-sample generalized confidence distributions and sign-based robust estimators in median regressions with heterogeneous dependent errors

We study the problem of estimating the parameters of a linear median regression without any assumption on the shape of the error distribution – including no condition on the existence of moments – allowing for heterogeneity (or heteroskedasticity) of unknown form, noncontinuous distributions, and very general serial dependence (linear and nonlinear). This is done through a reverse inference approach, based on a distribution-free sign-based testing theory, from which confidence sets and point estimators are subsequently generated. We propose point estimators, which have a natural association with confidence distributions. These estimators are based on maximizing test p-values and inherit robustness properties from the generating distribution-free tests. Both finite-sample and large-sample properties of the proposed estimators are established under weak regularity conditions. We show that they are median-unbiased (under symmetry and estimator unicity) and possess equivariance properties. Consistency and asymptotic normality are established without any moment existence assumption on the errors. A Monte Carlo study of bias and RMSE shows sign-based estimators perform better than LAD-type estimators in various heteroskedastic settings. We illustrate the use of sign-based estimators on an example of β-convergence of output levels across U.S. states.

The Role of Parallel Trends in Event Study Settings: An Application to Environmental Economics

Difference-in-differences (DID) research designs usually rely on variation of treatment timing such that, after making an appropriate parallel trends assumption, one can identify, estimate, and make inference about causal effects. In practice, however, different DID procedures rely on different parallel trends assumptions (PTAs), and recover different causal parameters. In this paper, we focus on staggered DID (also referred as event studies) and discuss the role played by the PTA in terms of identification and estimation of causal parameters. We document a “robustness” versus “efficiency” trade-off in terms of the strength of the underlying PTA and argue that practitioners should be explicit about these trade-offs whenever using DID procedures. We propose new DID estimators that reflect these trade-offs and derive their large sample properties. We illustrate the practical relevance of these results by assessing whether the transition from federal to state management of the Clean Water Act affects compliance rates.

Eigengalaxies: describing galaxy morphology using principal components in image space

ABSTRACTWe demonstrate how galaxy morphologies can be represented by weighted sums of ‘eigengalaxies’ and how eigengalaxies can be used in a probabilistic framework to enable principled and simplified approaches in a variety of applications. Eigengalaxies can be derived from a Principal Component Analysis (PCA) of sets of single- or multiband images. They encode the image space equivalent of basis vectors that can be combined to describe the structural properties of large samples of galaxies in a massively reduced manner. As an illustration, we show how a sample of 10243 galaxies in the Hubble Space Telescope CANDELS survey can be represented by just 12 eigengalaxies. We show in some detail how this image space may be derived and tested. We also describe a probabilistic extension to PCA (PPCA) which enables the eigengalaxy framework to assign probabilities to galaxies. We present four practical applications of the probabilistic eigengalaxy framework that are particularly relevant for the next generation of large imaging surveys: we (i) show how low likelihood galaxies make for natural candidates for outlier detection; (ii) demonstrate how missing data can be predicted; (iii) show how a similarity search can be performed on exemplars; (iv) demonstrate how unsupervised clustering of objects can be implemented.

Efficient estimation of human immunodeficiency virus incidence rate using a pooled cross-sectional cohort study design.

Development of methods to accurately estimate human immunodeficiency virus (HIV) incidence rate remains a challenge. Ideally, one would follow a random sample of HIV-negative individuals under a longitudinal study design and identify incident cases as they arise. Such designs can be prohibitively resource intensive and therefore alternative designs may be preferable. We propose such a simple, less resource-intensive study design and develop a weighted log likelihood approach which simultaneously accounts for selection bias and outcome misclassification error. The design is based on a cross-sectional survey which queries individuals' time since last HIV-negative test, validates their test results with formal documentation whenever possible, and tests all persons who do not have documentation of being HIV-positive. To gain efficiency, we update the weighted log likelihood function with potentially misclassified self-reports from individuals who could not produce documentation of a prior HIV-negative test and investigate large sample properties of validated sub-sample only versus pooled sample estimators through extensive Monte Carlo simulations. We illustrate our method by estimating incidence rate for individuals who tested HIV-negative within 1.5 and 5 years prior to Botswana Combination Prevention Project enrolment. This article establishes that accurate estimates of HIV incidence rate can be obtained from individuals' history of testing in a cross-sectional cohort study design by appropriately accounting for selection bias and misclassification error. Moreover, this approach is notably less resource-intensive compared to longitudinal and laboratory-based methods.

Causal inference from observational studies with clustered interference, with application to a cholera vaccine study

Understanding the population-level effects of vaccines has important public health policy implications. Inferring vaccine effects from an observational study is challenging because participants are not randomized to vaccine (i.e., treatment). Observational studies of infectious diseases present the additional challenge that vaccinating one participant may affect another participant’s outcome, that is, there may be interference. In this paper recent approaches to defining vaccine effects in the presence of interference are considered, and new causal estimands designed specifically for use with observational studies are proposed. Previously defined estimands target counterfactual scenarios in which individuals independently choose to be vaccinated with equal probability. However, in settings where there is interference between individuals within clusters, it may be unlikely that treatment selection is independent between individuals in the same cluster. The proposed causal estimands instead describe counterfactual scenarios which allow for within-cluster dependence in the individual treatment selections. These estimands may be more relevant for policy-makers or public health officials who desire to quantify the effect of increasing the proportion of vaccinated individuals in a population. Inverse probability-weighted estimators for these estimands are proposed. The large-sample properties of the estimators are derived, and a simulation study demonstrating the finite-sample performance of the estimators is presented. The proposed methods are illustrated by analyzing data from a study of cholera vaccination in over 100,000 individuals in Bangladesh.

Semiparametric estimation with spatially correlated recurrent events

AbstractThis article pertains to the analysis of recurrent event data in the presence of spatial correlation. Consider units located at n possibly spatially correlated geographical areas described by their longitude and latitude and monitored for the occurrence of an event that can recur. We propose a new class of semiparametric models for recurrent events that simultaneously account for risk factors and correlation among the spatial locations, and that subsumes the current methods. Since the parameters involved in the models are not directly estimable because of the high dimension of the likelihood, we use composite likelihood approach for estimation. The approach leads to estimates with population interpretation where their large sample properties are obtained under a reasonable set of regularity conditions. Simulation studies suggest that the resulting estimators have a very good finite sampling properties. The methods are illustrated using spatial data on recurrent esophageal cancer in the northern region of France and recurrent wildfire data in the province of Alberta, Canada.

Partial Dynamic Dimension Reduction for Conditional Mean in Regression

In many regression analysis, the authors are interested in regression mean of response variate given predictors, not its the conditional distribution. This paper is concerned with dimension reduction of predictors in sense of mean function of response conditioning on predictors. The authors introduce the notion of partial dynamic central mean dimension reduction subspace, different from central mean dimension reduction subspace, it has varying subspace in the domain of predictors, and its structural dimensionality may not be the same point by point. The authors study the property of partial dynamic central mean dimension reduction subspace, and develop estimated methods called dynamic ordinary least squares and dynamic principal Hessian directions, which are extension of ordinary least squares and principal Hessian directions based on central mean dimension reduction subspace. The kernel estimate methods for dynamic ordinary least squares and dynamic Principal Hessian Directions are employed, and large sample properties of estimators are given under the regular conditions. Simulations and real data analysis demonstrate that they are effective.

Extended cumulative entropy based on kth lower record values for the coherent systems lifetime

Kayal (Probab. Eng. Inf. Sci. 30(4):640–662, 2016) first proposed generalized cumulative entropy based on lower record values. Motivated by Kayal (Probab. Eng. Inf. Sci. 30(4):640–662, 2016), recently, Tahmasebi and Eskandarzadeh (Stat. Probab. Lett. 126:164–172, 2017) proposed an extended cumulative entropy (ECE) based on kth lower record values. In this paper, we obtain some properties of ECE. We study this measure of information for the coherent systems lifetime with identically distributed components. We define the conditional ECE for the system lifetime and discuss some properties of it. We also use this idea to propose a measure of extended cumulative past inaccuracy. Finally, we propose the estimators of these measures using empirical approach. In addition, we study large sample properties of these estimators.

Estimation for a Class of Semiparametric Pareto Mixture Densities

We study the estimation of a class of semiparametric mixture models, where the models have a symmetric nonparametric component and a parametric component of Pareto distribution with unknown parameters. We establish an estimation procedure by minimizing a criterion function after dealing with the jump point. We study the large sample properties of the proposed estimator, and prove consistency and asymptotic normality of the parameter estimation. For the nonparametric component, bias and variance are derived, and a rule-of-thumb bandwidth selection method is given. Simulation studies demonstrate good performance of the proposed methodology.

Large sample properties of the regression depth induced median

Notions of depth in regression have been introduced and studied in the literature. Regression depth (RD) of Rousseeuw and Hubert (1999), the most famous one, is a direct extension of Tukey location depth (Tukey, 1975) to regression.Like its location counterpart, the most remarkable advantage of the notion of depth in regression is to directly introduce the maximum (or deepest) regression depth estimator (aka depth induced median) for regression parameters in a multi-dimensional setting. Classical questions for the regression depth induced median include (i) is it a consistent estimator (or rather under what sufficient conditions, it is consistent)? and (ii) is there any limiting distribution?Bai and He (1999) (BH99) pioneered an attempt to answer these questions. Under some stringent conditions on (i) the design points, (ii) the conditional distributions of y given xi, and (iii) the error distributions, BH99 proved the strong consistency of the depth induced median. Under another set of conditions, BH99 showed the existence of the limiting distribution of the estimator.This article establishes the strong consistency of the depth induced median without any of the stringent conditions in BH99, and proves the existence of the limiting distribution of the estimator by sufficient conditions and an approach different from BH99.

Nonparametric regression estimate with Berkson Laplace measurement error

In this paper, a nonparametric estimator for the regression function is constructed when the covariates are contaminated with the multivariate Laplace measurement error. The proposed estimator is based upon a simple relationship between the regression function and the conditional expectation of the regression function given the proxy data, as well as the second derivative of this expectation. Large sample properties of the proposed estimator, including the consistency and asymptotic normality, are established. The theoretical optimal bandwidth based on asymptotic integrated mean squared error is derived, and a data-driven bandwidth selector is recommended. Finite sample performance of the proposed estimator is evaluated by a simulation study.

Optimal subsampling for large-scale quantile regression

To deal with massive data sets, subsampling is known as an effective method which can significantly reduce computational costs in estimating model parameters. In this article, an efficient subsampling method is developed for large-scale quantile regression via Poisson sampling framework, which can solve the memory constraint problem imposed by big data. Under some mild conditions, large sample properties for the estimator involving the weak and strong consistencies, and asymptotic normality are established. Furthermore, the optimal subsampling probabilities are derived according to the A-optimality criterion. It is shown that the estimator based on the optimal subsampling asymptotically achieves a smaller variance than that by the uniform random subsampling. The proposed method is illustrated and evaluated through numerical analyses on both simulated and real data sets.

Some New Copula Based Distribution-free Tests of Independence among Several Random Variables

Over the last couple of decades, several copula based methods have been proposed in the literature to test for the independence among several random variables. But these existing tests are not invariant under monotone transformations of the variables, and they often perform poorly if the dependence among the variables is highly non-monotone in nature. In this article, we propose a copula based measure of dependency and use it to construct some new distribution-free tests of independence. The proposed measure and the resulting tests, all are invariant under permutations and monotone transformations of the variables. Our dependency measure involves a kernel function, and we use the Gaussian kernel for that purpose. We adopt a multi-scale approach, where we look at the results obtained for several choices of the bandwidth parameter associated with the Gaussian kernel and aggregate them judiciously. Large sample properties of the dependency measure and the resulting tests are derived under appropriate regularity conditions. Several simulated and real data sets are analyzed to compare the performance of the proposed tests with some popular tests available in the literature.

Estimating sparse networks with hubs

Graphical modelling techniques based on sparse estimation have been applied to infer complex networks in many fields, including biology and medicine, engineering, finance and social sciences. One structural feature of some of these networks that poses a challenge for statistical inference is the presence of a small number of strongly interconnected nodes, which are called hubs. For example, in microbiome research hubs or microbial taxa play a significant role in maintaining stability of the microbial community structure. In this paper, we investigate the problem of estimating sparse networks in which there are a few highly connected hub nodes. Methods based on L1-regularization have been widely used for performing sparse estimation in the graphical modelling context. However, while these methods encourage sparsity, they do not take into account structural information of the network. We introduce a new method for estimating networks with hubs that exploits the ability of (inverse) covariance estimation methods to include structural information about the underlying network. Our method is a weighted lasso approach with novel row/column sum weights, which we refer to as the hubs weighted graphical lasso. A practical advantage of the new method is that it leads to an optimization problem that is solved using the efficient graphical lasso algorithm that is already implemented in the R package glasso (Friedman et al., 2019). We establish large sample properties of the method when the number of parameters diverges with the sample size. We then show via simulations that the method outperforms competing methods and illustrate its performance with an application to microbiome data.

Nonsmooth backfitting for the excess risk additive regression model with two survival time scales

Summary We consider an extension of Aalen’s additive regression model that allows covariates to have effects that vary on two different time scales. The two time scales considered are equal up to a constant for each individual and vary across individuals, such as follow-up time and age in medical studies or calendar time and age in longitudinal studies. The model was introduced in Scheike (2001), where it was solved using smoothing techniques. We present a new backfitting algorithm for estimating the structured model without having to use smoothing. Estimators of the cumulative regression functions on the two time scales are suggested by solving local estimating equations jointly on the two time scales. We provide large-sample properties and simultaneous confidence bands. The model is applied to data on myocardial infarction, providing a separation of the two effects stemming from time since diagnosis and age.

Bounds on Phase, Frequency, and Timing Synchronization in Fully Digital Receivers With 1-bit Quantization and Oversampling

Digital receivers based on 1-bit quantization and oversampling w.r.t. the transmit signal bandwidth promise lower energy consumption. However, since 1-bit quantization is a highly non-linear operation, standard receiver algorithms cannot be applied. Thus, we derive performance bounds for phase, timing, and frequency estimation in order to gain a deeper insight into the impact of 1-bit quantization and oversampling. We identify uniform phase and sample dithering as crucial to combat the effect of the non-linearity introduced by 1-bit quantization. Since oversampling results in noise correlation, a closed form of the likelihood function is not available. Thus, we study a system model with white noise by adapting the receive filter bandwidth to the sampling rate. Considering the aforementioned dithering, we obtain very tight closed form lower bounds on the Cramér-Rao lower bound (CRLB) in the large sample regime. We show that with uniform phase and sample dithering, all large sample properties of the CRLB of the unquantized receiver are preserved under 1-bit quantization, except for an signal-to-noise ratio (SNR) dependent performance loss that can be decreased by oversampling. Numerical computations show that the properties of the CRLB for white noise still hold for colored noise except that the performance loss due to 1-bit quantization is reduced.