Testing symmetry based on empirical likelihood

In this paper, we propose a general kth correlation coefficient between the density function and distribution function of a continuous variable as a measure of symmetry and asymmetry. We first propose a root-n moment-based estimator of the kth correlation coefficient and present its asymptotic results. Next, we consider statistical inference of the kth correlation coefficient by using the empirical likelihood (EL) method. The EL statistic is shown to be asymptotically a standard chi-squared distribution. Last, we propose a residual-based estimator of the kth correlation coefficient for a parametric regression model to test whether the density function of the true model error is symmetric or not. We present the asymptotic results of the residual-based kth correlation coefficient estimator and also construct its EL-based confidence intervals. Simulation studies are conducted to examine the performance of the proposed estimators, and we also use our proposed estimators to analyze the air quality dataset.     

Recursive kernel estimate of the conditional quantile for functional ergodic data

ABSTRACT

In this article, we study the recursive kernel estimator of the conditional quantile of a scalar response variable Y given a random variable (rv) X taking values in a semi-metric space. Two estimators are considered. While the first one is given by inverting the double-kernel estimate of the conditional distribution function, the second estimator is obtained by using the robust approach. We establish the almost complete consistency of these estimates when the observations are sampled from a functional ergodic process. Finally, a simulation study is carried out to illustrate the finite sample performance of these estimators.     

A Pointwise Estimator for the k-Fold Convolution of a Distribution Function

ABSTRACT

This article is concerned with some parametric and nonparametric estimators for the k-fold convolution of a distribution function. An alternative estimator is proposed and its unbiasedness, asymptotic unbiasedness, and consistency properties are investigated. The asymptotic normality of this estimator is established. Some applications of the estimator are given in renewal processes. Finally, the computational procedures are described and the relative performance of these estimators for small sample sizes is investigated by a simulation study.     

Asymptotic normality of locally modelled regression estimator for functional data

We focus on the nonparametric regression of a scalar response on a functional explanatory variable. As an alternative to the well-known Nadaraya-Watson estimator for regression function in this framework, the locally modelled regression estimator performs very well [cf. [Barrientos-Marin, J., Ferraty, F., and Vieu, P. (2010), ‘Locally Modelled Regression and Functional Data’, Journal of Nonparametric Statistics, 22, 617–632]. In this paper, the asymptotic properties of locally modelled regression estimator for functional data are considered. The mean-squared convergence as well as asymptotic normality for the estimator are established. We also adapt the empirical likelihood method to construct the point-wise confidence intervals for the regression function and derive the Wilk's phenomenon for the empirical likelihood inference. Furthermore, a simulation study is presented to illustrate our theoretical results.     

HAZARD RATE ESTIMATION FOR CENSORED DATA BY WAVELET METHODS

ABSTRACT

We study the estimation of a hazard rate function based on censored data by non-linear wavelet method. We provide an asymptotic formula for the mean integrated squared error (MISE) of nonlinear wavelet-based hazard rate estimators under randomly censored data. We show this MISE formula, when the underlying hazard rate function and censoring distribution function are only piecewise smooth, has the same expansion as analogous kernel estimators, a feature not available for the kernel estimators. In addition, we establish an asymptotic normality of the nonlinear wavelet estimator.     

A study on the least squares estimator of multivariate isotonic regression function

Consider the problem of pointwise estimation of f in a multivariate isotonic regression model Z=f(X₁,…,X_d)+ϵ, where Z is the response variable, f is an unknown nonparametric regression function, which is isotonic with respect to each component, and ϵ is the error term. In this article, we investigate the behavior of the least squares estimator of f. We generalize the greatest convex minorant characterization of isotonic regression estimator for the multivariate case and use it to establish the asymptotic distribution of properly normalized version of the estimator. Moreover, we test whether the multivariate isotonic regression function at a fixed point is larger (or smaller) than a specified value or not based on this estimator, and the consistency of the test is established. The practicability of the estimator and the test are shown on simulated and real data as well.     

Power periodic threshold GARCH model: Structure and estimation

Abstract

In this paper, we introduce and study the Power Periodic Threshold GARCH Model (PPTGARCH). We give the necessary and sufficient conditions for the existence of the unique strictly periodically stationary solution of the model and the necessary and sufficient conditions for the existence of moments. A sufficient condition for the periodic geometric ergodicity and β – mixing property using the uniform countable additivity condition is given. We prove the consistency and asymptotic normality of the Quasi-Maximum Likelihood estimator (QMLE) of the parameters. Simulation studies to illustrate consistency and asymptotic normality of the estimators for different underlying error distributions are presented.     

Similarity-based model for ordered categorical data

In a large variety of applications, the data for a variable we wish to explain are ordered and categorical. In this paper, we present a new similarity-based model for the scenario and investigate its properties. We establish that the process is ψ-mixing and strictly stationary and derive the explicit form of the autocorrelation function in some special cases. Consistency and asymptotic normality of the maximum likelihood estimator of the model’s parameters are proven. A simulation study supports our findings. The results are applied to the Netflix data set, comprised of a survey on users’ grading of movies.     

Principal varying coefficient estimator for high-dimensional models

ABSTRACT

We consider principal varying coefficient models in the high-dimensional setting, combined with variable selection, to reduce the effective number of parameters in semiparametric modelling. The estimation is based on B-splines approach. For the unpenalized estimator, we establish non-asymptotic bounds of the estimator and then establish the (asymptotic) local oracle property of the penalized estimator, as well as non-asymptotic error bounds. Monte Carlo studies reveal the favourable performance of the estimator and an application on a real dataset is presented.     

Generation of pseudo random numbers and estimation under Cox models with time-dependent covariates

ABSTRACT

We study the method for generating pseudo random numbers under various special cases of the Cox model with time-dependent covariates when the baseline hazard function may not be constant and the random variable may equal infinity with a positive probability. During our simulation studies in computing the partial likelihood estimates, in between 3% and 20% of the time with a moderate sample size, it happens that the partial likelihood estimate of the regression coefficient is ∞ for the data from the Cox model. We propose a semi-parametric estimator as a modification for such a case. We present simulation results on the asymptotic properties of the semi-parametric estimator.     

Point estimation after early stopping in a repeated measures trial

When the individual measurements are statistically independent, the maximum likelihood estimator calculated at the end of a sequential procedure overestimates the underlying effect. There are many clinical trials in which we are interested in comparing changes in responses between two treatment groups sequentially. Lee and DeMets (1991, JASA 86, 757–762) proposed a group sequential method for comparing rates of change when a response variable is measured for eaeh patient at successive follow-up visits. They assumed that the response follows the linear mixed effects model and derived the asymptotic joint distribution of the sequentially computed statistics. In this article, we consider the maximum likelihood estimator (MLE), the median unbiased estimator (MUE) and the midpoint of a 100(1-α)% confidence interval as point estimators for the rate of change in the linear mixed effects model, and investigate their properties by Monte Carlo simulation.     

The limiting distribution of the least-squares estimator in nearly integrated seasonal models

We consider the least-squares estimator of the autoregressive parameter in a nearly integrated seasonal model. Building on the study by Chan (1989), who obtained the limiting distribution, we derive a closed-form expression for the appropriate limiting joint moment generating function. We use this function to tabulate percentage points of the asymptotic distribution for various seasonal periods via numerical integration. The results are extended by deriving a stochastic asymptotic expansion to order O_p(T^-l), whose percentage points are also obtained by numerically integrating the appropriate limiting joint moment generating function. The adequacy of the approximation to the finite-sample distribution is discussed.     

Bayes and Empirical Bayes Inference in Changepoint Problems

ABSTRACT

In the case of the random design nonparametric regression, the double smoothing technique is applied to estimate the multivariate regression function. The proposed estimator has desirable properties in both the finite sample and the asymptotic cases. In the finite sample case, it has bounded conditional (and unconditional) bias and variance. On the other hand, in the asymptotic case, it has the same mean square error as the local linear estimator in Fan (Design-Adaptive Nonparametric Regression. Journal of the American Statistical Association 1992, 87, 998–1004; Local Linear Regression Smoothers and Their Minimax Efficiencies. Annals of Statistics 1993, 21, 196–216). Simulation studies demonstrate that the proposed estimator is better than the local linear estimator, because it has a smaller sample mean integrated square error and gives smoother estimates.     

Variable selection in the high-dimensional continuous generalized linear model with current status data

In survival studies, current status data are frequently encountered when some individuals in a study are not successively observed. This paper considers the problem of simultaneous variable selection and parameter estimation in the high-dimensional continuous generalized linear model with current status data. We apply the penalized likelihood procedure with the smoothly clipped absolute deviation penalty to select significant variables and estimate the corresponding regression coefficients. With a proper choice of tuning parameters, the resulting estimator is shown to be a root n/p_n-consistent estimator under some mild conditions. In addition, we show that the resulting estimator has the same asymptotic distribution as the estimator obtained when the true model is known. The finite sample behavior of the proposed estimator is evaluated through simulation studies and a real example.     

Survival Estimation with Missing Censoring Times for the Generalized Koziol–Green Model

ABSTRACT

When analyzing time-to-event data, there are various situations in which right censoring times for unfailed units are missing. In that context, by taking a supplementary sample of a convenient percentage of unfailed units, we propose a semi-parametric method for estimating a survival function under the natural extension of the Koziol–Green model to double random censoring. Some large sample properties of this estimator are derived. We prove uniform strong consistency and asymptotic weak convergence to a Gaussian process. A simulation study is also presented in order to analyze the behavior of the proposed estimator.     

Statistical Analysis of Non Linear Least Squares Estimation for Harmonic Signals in Multiplicative and Additive Noise

Abstract

This article concerns the stochastically constrained linear model under a biased assumption. We propose a quasi-stochastically constrained least squares estimator. Furthermore, we provide the expectation of this estimator, demonstrate its consistency and asymptotic normality. In the end of the article, the simulation study of the new estimator shows that it is superior to the least squares estimator, ridge estimator, and the linear constrained estimators under certain conditions by comparing the mean squared errors of these estimators.     

Additive Transformation Models for Multivariate Interval-Censored Data

ABSTRACT

We present a new estimator of extreme quantiles dedicated to Weibull tail distributions. This estimate is based on a consistent estimator of the Weibull tail coefficient. This parameter is defined as the regular variation coefficient of the inverse cumulative hazard function. We give conditions in order to obtain the weak consistency and the asymptotic distribution of the extreme quantiles estimator. Its asymptotic as well as its finite sample performances are compared to classical ones.     

On the relevance of weaker instruments

ABSTRACT

We study the asymptotic properties of the standard GMM estimator when additional moment restrictions, weaker than the original ones, are available. We provide conditions under which these additional weaker restrictions improve the efficiency of the GMM estimator. To detect “spurious” identification that may come from invalid moments, we rely on the Hansen J-test that assesses the compatibility between existing restrictions and additional ones. Our simulations reveal that the J-test has good power properties and that its power increases with the weakness of the additional restrictions. Our theoretical characterization of the J-test provides some intuition for why that is.     

Estimation and Simulation for the M-Wright Function

In this article, a structural form of an M-Wright distributed random variable is derived. The mixture representation then led to a random number generation algorithm. A formal parameter estimation procedure is also proposed. This procedure is needed to make the M-Wright function usable in practice. The asymptotic normality of the estimator is established as well. The estimator and the random number generation algorithm are then tested using synthetic data.     

Inference on proportional hazard rate model parameter under Type-I progressively hybrid censoring scheme

ABSTRACT

In this paper, under Type-I progressive hybrid censoring sample, we obtain maximum likelihood estimator of unknown parameter when the parent distribution belongs to proportional hazard rate family. We derive the conditional probability density function of the maximum likelihood estimator using moment-generating function technique. The exact confidence interval is obtained and compared by conducting a Monte Carlo simulation study for burr Type XII distribution. Finally, we obtain the Bayes and posterior regret gamma minimax estimates of the parameter under a precautionary loss function with precautionary index k = 2 and compare their behavior via a Monte Carlo simulation study.