Robust pairwise multiple comparisons under short-tailed symmetric distributions

In one-way ANOVA, most of the pairwise multiple comparison procedures depend on normality assumption of errors. In practice, errors have non-normal distributions so frequently. Therefore, it is very important to develop robust estimators of location and the associated variance under non-normality. In this paper, we consider the estimation of one-way ANOVA model parameters to make pairwise multiple comparisons under short-tailed symmetric (STS) distribution. The classical least squares method is neither efficient nor robust and maximum likelihood estimation technique is problematic in this situation. Modified maximum likelihood (MML) estimation technique gives the opportunity to estimate model parameters in closed forms under non-normal distributions. Hence, the use of MML estimators in the test statistic is proposed for pairwise multiple comparisons under STS distribution. The efficiency and power comparisons of the test statistic based on sample mean, trimmed mean, wave and MML estimators are given and the robustness of the test obtained using these estimators under plausible alternatives and inlier model are examined. It is demonstrated that the test statistic based on MML estimators is efficient and robust and the corresponding test is more powerful and having smallest Type I error.     

A robust estimation method for the linear regression model parameters with correlated error terms and outliers

Independence of error terms in a linear regression model, often not established. So a linear regression model with correlated error terms appears in many applications. According to the earlier studies, this kind of error terms, basically can affect the robustness of the linear regression model analysis. It is also shown that the robustness of the parameters estimators of a linear regression model can stay using the M-estimator. But considering that, it acquires this feature as the result of establishment of its efficiency. Whereas, it has been shown that the minimum Matusita distance estimators, has both features robustness and efficiency at the same time. On the other hand, because the Cochrane and Orcutt adjusted least squares estimators are not affected by the dependence of the error terms, so they are efficient estimators. Here we are using of a non-parametric kernel density estimation method, to give a new method of obtaining the minimum Matusita distance estimators for the linear regression model with correlated error terms in the presence of outliers. Also, simulation and real data study both are done for the introduced estimation method. In each case, the proposed method represents lower biases and mean squared errors than the other two methods.KEYWORDS: Robust estimation method, minimum Matusita distance estimation method, non-parametric kernel density estimation method, correlated error terms, outliers     

Estimation of the Mean and Standard Deviation of the Logistic Distribution Based on Multiply Type-II Censored Samples

In this paper, we discuss the problem of estimating the mean and standard deviation of a logistic population based on multiply Type-II censored samples. First, we discuss the best linear unbiased estimation and the maximum likelihood estimation methods. Next, by appropriately approximating the likelihood equations we derive approximate maximum likelihood estimators for the two parameters and show that these estimators are quite useful as they do not need the construction of any special tables (as required for the best linear unbiased estimators) and are explicit estimators (unlike the maximum likelihood estimators which need to be determined by numerical methods). We show that these estimators are also quite efficient, and derive the asymptotic variances and covariance of the estimators. Finally, we present an example to illustrate the methods of estimation discussed in this paper.     

A weighted linear quantile regression

In this article, we introduce a new weighted quantile regression method. Traditionally, the estimation of the parameters involved in quantile regression is obtained by minimizing a loss function based on absolute distances with weights independent of explanatory variables. Specifically, we study a new estimation method using a weighted loss function with the weights associated with explanatory variables so that the performance of the resulting estimation can be improved. In full generality, we derive the asymptotic distribution of the weighted quantile regression estimators for any uniformly bounded positive weight function independent of the response. Two practical weighting schemes are proposed, each for a certain type of data. Monte Carlo simulations are carried out for comparing our proposed methods with the classical approaches. We also demonstrate the proposed methods using two real-life data sets from the literature. Both our simulation study and the results from these examples show that our proposed method outperforms the classical approaches when the relative efficiency is measured by the mean-squared errors of the estimators.     

Non-existence of an optimum estimator in a class of ratio estimators

A number of estimators formulated in the field of the ratio method of estimation has been presented. A class of estimators encompassing these estimators is constructed. It is noted that an optimum estimator does not exist uniformly in this class. The “Optimum” so obtained reduces to the usual regression estimator.     

Stabilizing the Performance of Kurtosis Estimator of Multivariate Data

The estimation of the kurtosis parameter of the underlying distribution plays a central role in many statistical applications. The central theme of the article is to improve the estimation of the kurtosis parameter using a priori information. More specifically, we consider the problem of estimating kurtosis parameter of a multivariate population when some prior information regarding the the parameter is available. The rationale is that the sample estimator of the kurtosis parameter has a large estimation error. In this situation we consider shrinkage and pretest estimation methodologies and reappraise their statistical properties. The estimation based on these strategies yield relatively smaller estimation error in comparison with the sample estimator in the candidate subspace. A large sample theory of the suggested estimators are developed and compared. The results demonstrate that suggested estimators outperform the estimator based on the sample data only in the candidate subspace. In an effort to appreciate the relative behavior of the estimators in a finite sample scenario, a Monte-carlo simulation study is planned and performed. The result of simulation study strongly corroborates the asymptotic result. To illustrate the application of the estimators, some example are showcased based on recently published data.     

An Investigation of Quantile Function Estimators Relative to Quantile Confidence Interval Coverage

In this article, we investigate the limitations of traditional quantile function estimators and introduce a new class of quantile function estimators, namely, the semi-parametric tail-extrapolated quantile estimators, which has excellent performance for estimating the extreme tails with finite sample sizes. The smoothed bootstrap and direct density estimation via the characteristic function methods are developed for the estimation of confidence intervals. Through a comprehensive simulation study to compare the confidence interval estimations of various quantile estimators, we discuss the preferred quantile estimator in conjunction with the confidence interval estimation method to use under different circumstances. Data examples are given to illustrate the superiority of the semi-parametric tail-extrapolated quantile estimators. The new class of quantile estimators is obtained by slight modification of traditional quantile estimators, and therefore, should be specifically appealing to researchers in estimating the extreme tails.     

Small area estimation strategies for large population surveys: a comparison of design and model-based methods

Small area estimation (SAE) concerns with how to reliably estimate population quantities of interest when some areas or domains have very limited samples. This is an important issue in large population surveys, because the geographical areas or groups with only small samples or even no samples are often of interest to researchers and policy-makers. For example, large population health surveys, such as Behavioural Risk Factor Surveillance System and Ohio Mecaid Assessment Survey (OMAS), are regularly conducted for monitoring insurance coverage and healthcare utilization. Classic approaches usually provide accurate estimators at the state level or large geographical region level, but they fail to provide reliable estimators for many rural counties where the samples are sparse. Moreover, a systematic evaluation of the performances of the SAE methods in real-world setting is lacking in the literature. In this paper, we propose a Bayesian hierarchical model with constraints on the parameter space and show that it provides superior estimators for county-level adult uninsured rates in Ohio based on the 2012 OMAS data. Furthermore, we perform extensive simulation studies to compare our methods with a collection of common SAE strategies, including direct estimators, synthetic estimators, composite estimators, and Datta GS, Ghosh M, Steorts R, Maples J.'s [Bayesian benchmarking with applications to small area estimation. Test 2011;20(3):574–588] Bayesian hierarchical model-based estimators. To set a fair basis for comparison, we generate our simulation data with characteristics mimicking the real OMAS data, so that neither model-based nor design-based strategies use the true model specification. The estimators based on our proposed model are shown to outperform other estimators for small areas in both simulation study and real data analysis.     

Inference for the extreme value distribution under progressive Type-II censoring

The extreme value distribution has been extensively used to model natural phenomena such as rainfall and floods, and also in modeling lifetimes and material strengths. Maximum likelihood estimation (MLE) for the parameters of the extreme value distribution leads to likelihood equations that have to be solved numerically, even when the complete sample is available. In this paper, we discuss point and interval estimation based on progressively Type-II censored samples. Through an approximation in the likelihood equations, we obtain explicit estimators which are approximations to the MLEs. Using these approximate estimators as starting values, we obtain the MLEs using an iterative method and examine numerically their bias and mean squared error. The approximate estimators compare quite favorably to the MLEs in terms of both bias and efficiency. Results of the simulation study, however, show that the probability coverages of the pivotal quantities (for location and scale parameters) based on asymptotic normality are unsatisfactory for both these estimators and particularly so when the effective sample size is small. We, therefore, suggest the use of unconditional simulated percentage points of these pivotal quantities for the construction of confidence intervals. The results are presented for a wide range of sample sizes and different progressive censoring schemes. We conclude with an illustrative example.     

Simultaneous estimation of restricted location parameters based on permutation and sign-change

The problem of simultaneously estimating location parameters is addressed, where the vector of location parameters belongs to a polyhedral cone including simple order, tree order and positive orthant restrictions and so forth. This paper proposes modified estimators based on orthogonal transformations such as sign-change and permutation and proves that, in a multivariate location family, the modified estimators are minimax under quadratic loss. Shrinkage minimax estimators improving on the modified estimators are obtained for a restricted mean vector of spherically symmetric distribution. An application of sign-change transformation is also given in estimation of a bounded normal mean.     

An Adaptive Two-stage Estimation Method for Additive Models

Abstract.  In this paper, a two-stage estimation method for non-parametric additive models is investigated. Differing from Horowitz and Mammen's two-stage estimation, our first-stage estimators are designed not only for dimension reduction but also as initial approximations to all of the additive components. The second-stage estimators are obtained by using one-dimensional non-parametric techniques to refine the first-stage ones. From this procedure, we can reveal a relationship between the regression function spaces and convergence rate, and then provide estimators that are optimal in the sense that, better than the usual one-dimensional mean-squared error (MSE) of the order n ^−4/5 , the MSE of the order n ^{− 1} can be achieved when the underlying models are actually parametric. This shows that our estimation procedure is adaptive in a certain sense. Also it is proved that the bandwidth that is selected by cross-validation depends only on one-dimensional kernel estimation and maintains the asymptotic optimality. Simulation studies show that the new estimators of the regression function and all components outperform the existing estimators, and their behaviours are often similar to that of the oracle estimator.     

Estimating the fundamental frequency of a periodic function

In this paper we consider the problem of estimation of the fundamental frequency of a periodic function, which has several applications in Speech Signal Processing. The problem was originally proposed by Hannan (1974) and later on Quinn and Thomson (1991) provided an estimation procedure of the unknown parameters. It is observed that the estimation procedure of Quinn and Thomson (1991) is quite involved numerically. In this paper we propose to use two simple estimators and it is observed that their performance are quite satisfactory. Asymptotic properties of the proposed estimators are obtained. The large sample properties of the estimators are compared theoretically. We present some simulation results to compare their small sample performance. One speech data is analyzed using this particular model.     

Ridge regression from principal component point of view

Ridge regression is often discussed as an estimation procedure for producing estimators which are biased but with a smaller mean squared error than the usual least square estimators. In this paper we show that this procedure can also be used to reflect the nature of dependency among a set of highly collinear regressor variables. In particular, we prove that, when data are severely multicollinear, the ridge estimators can be made very close to the principal component estimators. Examples are given to illustrate the point.     

相依函数型数据的局部回归估计的渐近正态性

函数型数据研究近年来为越来越多的学者所重视,其在天文,医药,经济现象,生态环境及工业制造等诸多方面均有重要应用.非参数统计是统计研究的一个重要方面,其中核函数估计和局部多项式方法是这一类研究中重要常用方法.函数型数据的非参数方法中以核函数估计方法较为常见,且其收敛速度与极限分布无论在独立情形还是相依情形都有理论结果.而局部多项式的研究在函数型数据背景下较为少见,原因在于将局部多项式方法推广到函数型数据背景一直是一个难题. Marin, Ferraty, Vieu [Journal of Nonparametric Statistics, 22 (5) (2010), pp.617-632] 提出了非参函数型模型的局部回归估计. 这种估计可以看作是局部多项式估计在函数型数据背景下的一个推广.这种方法提出后,许多学者进一步研究了这种方法,考察了这种方法的收敛速度和极限分布,并将这种方法应用到不同的模型中以适应实际需求.但是,前人的研究都要求数据具有独立同分布的性质.然而许多实际数据并不符合这一假设.本文研究了在相依函数型数据情形下局部回归估计的渐近正态性.由于估计方法有差异,核函数估计的研究方法无法直接推广到局部回归估计,而相依性结构也给研究带来了一些挑战,我们采用Bernstein分块方法将相依性问题转化为渐近独立的问题,从而得到了估计的渐近正态性.此外我们还采用数据模拟的方法进一步验证了渐近正态的结果.     

On characterizations of pitman closeness of some shrinkage estimators

For the multivariate normal mean (vector) estimation problem, some characterizations of the Pitman closest property of a general class of shrinkage (or Stein-rule) estimators (including the so called positive-rule versions) are studied. Further, for the same model when the parameter is restricted to a positively homogeneous cone, Pitman closeness of restricted shrinkage maximum likelihood estimators is established.     

Semiparametric estimation of the single-index varying-coefficient model

In this paper, we consider the choice of pilot estimators for the single-index varying-coefficient model, which may result in radically different estimators, and develop the method for estimating the unknown parameter in this model. To estimate the unknown parameters efficiently, we use the outer product of gradient method to find the consistent initial estimators for interest parameters, and then adopt the refined estimation method to improve the efficiency, which is similar to the refined minimum average variance estimation method. An algorithm is proposed to estimate the model directly. Asymptotic properties for the proposed estimation procedure have been established. The bandwidth selection problem is also considered. Simulation studies are carried out to assess the finite sample performance of the proposed estimators, and efficiency comparisons between the estimation methods are made.     

Nonparametric density estimation from censored data

Nonparametric estimation of the probability density function f° of a lifetime distribution based on arbitrarily right-censor-ed observations from f° has been studied extensively in recent years. In this paper the density estimators from censored data that have been obtained to date are outlined. Histogram, kernel-type, maximum likelihood, series-type, and Bayesian nonparametric estimators are included. Since estimation of the hazard rate function can be considered as giving a density estimate, all known results concerning nonparametric hazard rate estimation from censored samples are also briefly mentioned.     

Non-stationary spatiotemporal analysis of karst water levels

Summary.  We consider non-stationary spatiotemporal modelling in an investigation into karst water levels in western Hungary. A strong feature of the data set is the extraction of large amounts of water from mines, which caused the water levels to reduce until about 1990 when the mining ceased, and then the levels increased quickly. We discuss some traditional hydrogeological models which might be considered to be appropriate for this situation, and various alternative stochastic models. In particular, a separable space–time covariance model is proposed which is then deformed in time to account for the non-stationary nature of the lagged correlations between sites. Suitable covariance functions are investigated and then the models are fitted by using weighted least squares and cross-validation. Forecasting and prediction are carried out by using spatiotemporal kriging. We assess the performance of the method with one-step-ahead forecasting and make comparisons with naïve estimators. We also consider spatiotemporal prediction at a set of new sites. The new model performs favourably compared with the deterministic model and the naïve estimators, and the deformation by time shifting is worthwhile.     

On empirical Bayes estimation of variance components in random effects model

In this paper, Bayes estimators of variance components are derived for the one-way random effects model, and empirical Bayes (EB) estimators are constructed by the kernel estimation method of a multivariate density and its mixed partial derivatives. It is shown that the EB estimators are asymptotically optimal and convergence rates are established. Finally, an example concerning the main results is given.     

Quantile regression estimation of partially linear additive models

In this paper, we consider the estimation of partially linear additive quantile regression models where the conditional quantile function comprises a linear parametric component and a nonparametric additive component. We propose a two-step estimation approach: in the first step, we approximate the conditional quantile function using a series estimation method. In the second step, the nonparametric additive component is recovered using either a local polynomial estimator or a weighted Nadaraya–Watson estimator. Both consistency and asymptotic normality of the proposed estimators are established. Particularly, we show that the first-stage estimator for the finite-dimensional parameters attains the semiparametric efficiency bound under homoskedasticity, and that the second-stage estimators for the nonparametric additive component have an oracle efficiency property. Monte Carlo experiments are conducted to assess the finite sample performance of the proposed estimators. An application to a real data set is also illustrated.