Jackknife empirical likelihood for the error variance in linear models

Variance estimation is a fundamental yet important problem in statistical modelling. In this paper, we propose jackknife empirical likelihood (JEL) methods for the error variance in a linear regression model. We prove that the JEL ratio converges to the standard chi-squared distribution. The asymptotic chi-squared properties for the adjusted JEL and extended JEL estimators are also established. Extensive simulation studies to compare the new JEL methods with the standard method in terms of coverage probability and interval length are conducted, and the simulation results show that our proposed JEL methods perform better than the standard method. We also illustrate the proposed methods using two real data sets.     

Jackknife empirical likelihood-based inferences for Lorenz curve with kernel smoothing

The Lorenz curve describes the wealth proportion for an income-ordered population. In this paper, we introduce a kernel smoothing estimator for the Lorenz curve and propose a smoothed jackknife empirical likelihood method for constructing confidence intervals of Lorenz ordinates. Extensive simulation studies are conducted to evaluate finite sample performances of the proposed methods. A real dataset of Georgia professor’s income is used to illustrate the proposed methods.     

Statistical Inference for a Relative Risk Measure

ABSTRACT

For monitoring systemic risk from regulators’ point of view, this article proposes a relative risk measure, which is sensitive to the market comovement. The asymptotic normality of a nonparametric estimator and its smoothed version is established when the observations are independent. To effectively construct an interval without complicated asymptotic variance estimation, a jackknife empirical likelihood inference procedure based on the smoothed nonparametric estimation is provided with a Wilks type of result in case of independent observations. When data follow from AR-GARCH models, the relative risk measure with respect to the errors becomes useful and so we propose a corresponding nonparametric estimator. A simulation study and real-life data analysis show that the proposed relative risk measure is useful in monitoring systemic risk.     

A smoothed Q‐learning algorithm for estimating optimal dynamic treatment regimes

In this paper, we propose a smoothed Q‐learning algorithm for estimating optimal dynamic treatment regimes. In contrast to the Q‐learning algorithm in which nonregular inference is involved, we show that, under assumptions adopted in this paper, the proposed smoothed Q‐learning estimator is asymptotically normally distributed even when the Q‐learning estimator is not and its asymptotic variance can be consistently estimated. As a result, inference based on the smoothed Q‐learning estimator is standard. We derive the optimal smoothing parameter and propose a data‐driven method for estimating it. The finite sample properties of the smoothed Q‐learning estimator are studied and compared with several existing estimators including the Q‐learning estimator via an extensive simulation study. We illustrate the new method by analyzing data from the Clinical Antipsychotic Trials of Intervention Effectiveness–Alzheimer's Disease (CATIE‐AD) study.     

Smoothed estimates for nonlinear time series models with irregular data

Time series data observed at unequal time intervals (irregular data) occur quite often and this usually poses problems in its analysis. A recursive form of the exponentially smoothed estimated is here proposed for a nonlinear model with irregularly observed data and its asymptotic properties are discussed An alternative smoother to that of Wright (1985) is also derived. Numerical comparison is made between the resulting estimates and other smoothed estimates.     

Hazard function estimation from homogeneous right censored data with missing censoring indicators

The kernel smoothed Nelson–Aalen estimator has been well investigated, but is unsuitable when some of the censoring indicators are missing. A representation introduced by Dikta, however, facilitates hazard estimation when there are missing censoring indicators. In this article, we investigate (i) a kernel smoothed semiparametric hazard estimator and (ii) a kernel smoothed “pre-smoothed” Nelson–Aalen estimator. We derive the asymptotic normality of the proposed estimators and compare their asymptotic variances.     

Smoothed empirical likelihood for quantile regression models with response data missing at random

This paper studies smoothed quantile linear regression models with response data missing at random. Three smoothed quantile empirical likelihood ratios are proposed first and shown to be asymptotically Chi-squared. Then, the confidence intervals for the regression coefficients are constructed without the estimation of the asymptotic covariance. Furthermore, a class of estimators for the regression parameter is presented to derive its asymptotic distribution. Simulation studies are conducted to assess the finite sample performance. Finally, a real-world data set is analyzed to illustrated the effectiveness of the proposed methods.     

Smoothed alternatives of the two-sample median and Wilcoxon's rank sum tests

We discuss smoothed rank statistics for testing the location shift parameter of the two-sample problem. They are based on discrete test statistics – the median and Wilcoxon's rank sum tests. For the one-sample problem, Maesono et al. [Smoothed nonparametric tests and their properties. arXiv preprint. 2016; ArXiv:1610.02145] reported that some nonparametric discrete tests have a problem with their p-values because of their discreteness. The p-values of Wilcoxon's test are frequently smaller than those of the median test in the tail area. This leads to an arbitrary choice of the median and Wilcoxon's rank sum tests. To overcome this problem, we propose smoothed versions of those tests. The smoothed tests inherit the good properties of the original tests and are asymptotically equivalent to them. We study the significance probabilities and local asymptotic powers of the proposed tests.     

Orthogonal series density estimation for complex surveys

We propose an orthogonal series density estimator for complex surveys, where samples are neither independent nor identically distributed. The proposed estimator is proved to be design-unbiased and asymptotically design-consistent. The asymptotic normality is proved under both design and combined spaces. Two data driven estimators are proposed based on the proposed oracle estimator. We show the efficiency of the proposed estimators in simulation studies. A real survey data example is provided for an illustration.     

Goodness-of-fit test under length-biased sampling

In this article, we study a goodness-of-fit (GOF) test in the presence of length-biased sampling. For this purpose, we introduce a smoothed estimator of distribution function (d.f.) and we investigate its asymptotic behaviors, such as uniform consistency and asymptotic normality. Based on this estimator, we define a one-sample Kolmogorov type of GOF test for length-biased data. We conduct Monte Carlo simulations to evaluate the performance of the proposed test statistic and compare it with the one-sample Kolmogorov type of GOF test obtained by the non smoothed estimator of d.f.     

Consistent Nonparametric Tests for Lorenz Dominance

This article proposes consistent nonparametric methods for testing the null hypothesis of Lorenz dominance. The methods are based on a class of statistical functionals defined over the difference between the Lorenz curves for two samples of welfare-related variables. We present two specific test statistics belonging to the general class and derive their asymptotic properties. As the limiting distributions of the test statistics are nonstandard, we propose and justify bootstrap methods of inference. We provide methods appropriate for case where the two samples are independent as well as the case where the two samples represent different measures of welfare for one set of individuals. The small sample performance of the two tests is examined and compared in the context of a Monte Carlo study and an empirical analysis of income and consumption inequality.     

Statistical inference for multiple change-point models

In this article, we propose a new technique for constructing confidence intervals for the mean of a noisy sequence with multiple change-points. We use the weighted bootstrap to generalize the bootstrap aggregating or bagging estimator. A standard deviation formula for the bagging estimator is introduced, based on which smoothed confidence intervals are constructed. To further improve the performance of the smoothed interval for weak signals, we suggest a strategy of adaptively choosing between the percentile intervals and the smoothed intervals. A new intensity plot is proposed to visualize the pattern of the change-points. We also propose a new change-point estimator based on the intensity plot, which has superior performance in comparison with the state-of-the-art segmentation methods. The finite sample performance of the confidence intervals and the change-point estimator are evaluated through Monte Carlo studies and illustrated with a real data example.     

Theoretical grounding for estimation in conditional independence multivariate finite mixture models

For the nonparametric estimation of multivariate finite mixture models with the conditional independence assumption, we propose a new formulation of the objective function in terms of penalised smoothed Kullback–Leibler distance. The nonlinearly smoothed majorisation-minimisation (NSMM) algorithm is derived from this perspective. An elegant representation of the NSMM algorithm is obtained using a novel projection-multiplication operator, a more precise monotonicity property of the algorithm is discovered, and the existence of a solution to the main optimisation problem is proved for the first time.     

Smoothed empirical likelihood analysis of partially linear quantile regression models with missing response variables

In this paper, we consider the estimation and inference of the parameters and the nonparametric part in partially linear quantile regression models with responses that are missing at random. First, we extend the normal approximation (NA)-based methods of Sun (2005) to the missing data case. However, the asymptotic covariance matrices of NA-based methods are difficult to estimate, which complicates inference. To overcome this problem, alternatively, we propose the smoothed empirical likelihood (SEL)-based methods. We define SEL statistics for the parameters and the nonparametric part and demonstrate that the limiting distributions of the statistics are Chi-squared distributions. Accordingly, confidence regions can be obtained without the estimation of the asymptotic covariance matrices. Monte Carlo simulations are conducted to evaluate the performance of the proposed method. Finally, the NA- and SEL-based methods are applied to real data.     

Goodness‐of‐fit Test for Directional Data

In this paper, we study the problem of testing the hypothesis on whether the density f of a random variable on a sphere belongs to a given parametric class of densities. We propose two test statistics based on the L² and L¹ distances between a non‐parametric density estimator adapted to circular data and a smoothed version of the specified density. The asymptotic distribution of the L² test statistic is provided under the null hypothesis and contiguous alternatives. We also consider a bootstrap method to approximate the distribution of both test statistics. Through a simulation study, we explore the moderate sample performance of the proposed tests under the null hypothesis and under different alternatives. Finally, the procedure is illustrated by analysing a real data set based on wind direction measurements.     

On Testing Exponentiality against NBAFR Alternatives

In this article, we propose an interesting approach for testing exponentiality against NBAFR alternatives. A measure of deviation from exponentiality has been derived on the basis of an inequality which we have proved. A test statistic has been constructed using density estimators and its asymptotic normality established. The consistency of the said test is also proved.     

Robust Smoothed Rank Estimation Methods for Accelerated Failure Time Model Allowing Clusters

Smoothed Gehan rank estimation methods are widely used in accelerated failure time (AFT) models with/without clusters. However, most methods are sensitive to outliers in the covariates. In order to solve this problem, we propose robust approaches based on the smoothed Gehan rank estimation methods for the AFT model, allowing for clusters by employing two different weight functions. Simulation studies show that the proposed methods outperform existing smoothed rank estimation methods regarding their biases and standard deviations when there are outliers in the covariates. The proposed methods are also applied to a real dataset from the “Major cardiovascular interventions” study.     

Smoothing for discrete-valued time series

We deal with smoothed estimators for conditional probability functions of discrete-valued time series { Y_t } under two different settings. When the conditional distribution of Y_t given its lagged values falls in a parametric family and depends on exogenous random variables, a smoothed maximum (partial) likelihood estimator for the unknown parameter is proposed. While there is no prior information on the distribution, various nonparametric estimation methods have been compared and the adjusted Nadaraya–Watson estimator stands out as it shares the advantages of both Nadaraya–Watson and local linear regression estimators. The asymptotic normality of the estimators proposed has been established in the manner of sparse asymptotics, which shows that the smoothed methods proposed outperform their conventional, unsmoothed, parametric counterparts under very mild conditions. Simulation results lend further support to this assertion. Finally, the new method is illustrated via a real data set concerning the relationship between the number of daily hospital admissions and the levels of pollutants in Hong Kong in 1994–1995. An ad hoc model selection procedure based on a local Akaike information criterion is proposed to select the significant pollutant indices.     

Integrated-Quantile-Based Estimation for First-Price Auction Models

This article considers nonparametric estimation of first-price auction models under the monotonicity restriction on the bidding strategy. Based on an integrated-quantile representation of the first-order condition, we propose a tuning-parameter-free estimator for the valuation quantile function. We establish its cube-root-n consistency and asymptotic distribution under weaker smoothness assumptions than those typically assumed in the empirical literature. If the latter are true, we also provide a trimming-free smoothed estimator and show that it is asymptotically normal and achieves the optimal rate of Guerre, Perrigne, and Vuong (2000). We illustrate our method using Monte Carlo simulations and an empirical study of the California highway procurement auctions. Supplementary materials for this article are available online.