In this paper we consider an extension of the beta regression model proposed by Ferrari and Cribari-Neto (2004). We extend their model in two different ways, first, we let the regression structure be nonlinear, second, we allow a regression structure for the precision parameter, moreover, this regression structure may also be nonlinear. Generally, the beta regression is useful to situations where the response is restricted to the standard unit interval and the regression structure involves regressors and unknown parameters. We derive general formulae for second-order biases of the maximum likelihood estimators and use them to define bias-corrected estimators. Our formulae generalizes the results obtained by Ospina et al. (2006), and are easily implemented by means of supplementary weighted linear regressions. We also compare these bias-corrected estimators with three different estimators which are also bias-free to the second-order, one analytical and the other two based on bootstrap methods. These estimators are compared by simulation. We present an empirical application.
The replacement of indicator functions by integrated beta kernels in the definition of the empirical stable tail dependence function is shown to produce a smoothed version of the latter estimator with the same asymptotic distribution but superior finite-sample performance. The link of the new estimator with the empirical beta copula enables a simple but effective resampling scheme.
In this paper, we consider the beta prime regression model recently proposed by \cite{bour18}, which is tailored to situations where the response is continuous and restricted to the positive real line with skewed and long tails and the regression structure involves regressors and unknown parameters. We consider two different strategies of bias correction of the maximum-likelihood estimators for the parameters that index the model. In particular, we discuss bias-corrected estimators for the mean and the dispersion parameters of the model. Furthermore, as an alternative to the two analytically b…
Motivated by the case fatality rate (CFR) of COVID-19, in this paper, we develop a fully parametric quantile regression model based on the generalized three-parameter beta (GB3) distribution. Beta regression models are primarily used to model rates and proportions. However, these models are usually specified in terms of a conditional mean. Therefore, they may be inadequate if the observed response variable follows an asymmetrical distribution, such as CFR data. In addition, beta regression models do not consider the effect of the covariates across the spectrum of the dependent variable, which …
We consider the issue of performing accurate small-sample testing inference in beta regression models, which are useful for modeling continuous variates that assume values in $(0,1)$, such as rates and proportions. We derive the Bartlett correction to the likelihood ratio test statistic and also consider a bootstrap Bartlett correction. Using Monte Carlo simulations we compare the finite sample performances of the two corrected tests to that of the standard likelihood ratio test and also to its variant that employs Skovgaard's adjustment; the latter is already available in the literature. The …
The inflated beta regression model aims to enable the modeling of responses in the intervals $(0,1]$, $[0,1)$ or $[0,1]$. In this model, hypothesis testing is often performed based on the likelihood ratio statistic. The critical values are obtained from asymptotic approximations, which may lead to distortions of size in small samples. In this sense, this paper proposes the bootstrap Bartlett correction to the statistic of likelihood ratio in the inflated beta regression model. The proposed adjustment only requires a simple Monte Carlo simulation. Through extensive Monte Carlo simulations the f…
In this paper, we are interested in the study of beta kernel estimators from an asymptotic minimax point of view. It is well known that beta kernel estimators are, on the contrary of classical kernel estimators, "free of boundary effect" and thus are very useful in practice. The goal of this paper is to prove that there is a price to pay: for very regular functions or for certain losses, these estimators are not minimax. Nevertheless they are minimax for classical regularities such as regularity of order two or less than two, supposed commonly in the practice and for some classical losses.
In order to overcome multicollinearity, we propose a stochastic restricted Liu-type max- imum likelihood estimator by incorporating Liu-type maximum likelihood estimator (Inan and Erdo- gan, 2013) to the logistic regression model when the linear restrictions are stochastic. We also discuss the properties of the new estimator. Moreover, we give a method to choose the biasing parameter in the new estimator. Finally, a simulation study is given to show the performance of the new estimator.
In this work we introduce the class of beta autoregressive fractionally integrated moving average models for continuous random variables taking values in the continuous unit interval $(0,1)$. The proposed model accommodates a set of regressors and a long-range dependent time series structure. We derive the partial likelihood estimator for the parameters of the proposed model, obtain the associated score vector and Fisher information matrix. We also prove the consistency and asymptotic normality of the estimator under mild conditions. Hypotheses testing, diagnostic tools and forecasting are als…
We consider a linear regression model with regression parameter beta =(beta_1, ..., beta_p) and independent and identically N(0, sigma^2)distributed errors. Suppose that the parameter of interest is theta = a^T beta where a is a specified vector. Define the parameter tau = c^T beta - t where the vector c and the number t are specified and a and c are linearly independent. Also suppose that we have uncertain prior information that tau = 0. Kabaila and Giri (2009c) present a new frequentist 1-alpha confidence interval for theta that utilizes this prior information. This interval has expected len…
We show that the generalized method of moments (GMM) estimation problem in instrumental variable quantile regression (IVQR) models can be equivalently formulated as a mixed integer quadratic programming problem. This enables exact computation of the GMM estimators for the IVQR models. We illustrate the usefulness of our algorithm via Monte Carlo experiments and an application to demand for fish.
In linear regression, the least squares (LS) estimator has certain optimality properties if the errors are normally distributed. This assumption is often violated in practice, partly caused by data outliers. Robust estimators can cope with this situation and thus they are widely used in practice. One example of robust estimators for regression are adaptive modified maximum likelihood (AMML) estimators (Donmez, 2010). However, they are not robust to $x$ outliers, so-called leverage points. In this study, we propose a new regression estimator by employing an appropriate weighting scheme in the A…
This paper introduces a new biased estimator for the negative binomial regression model that is a generalization of Liu-type estimator proposed for the linear model in [12]. Since the variance of the maximum likelihood estimator (MLE) is inflated when there is multicollinearity between the explanatory variables, a new biased estimator is proposed to solve the problem and decrease the variance of MLE in order to make stable inferences. Moreover, we obtain some theoretical comparisons between the new estimator and some others via matrix mean squared error (MMSE) criterion. Furthermore, a Monte C…
A broad class of smooth, possibly data-adaptive nonparametric copula estimators that contains empirical Bernstein copulas introduced by Sancetta and Satchell (and thus the empirical beta copula proposed by Segers, Sibuya and Tsukahara) is studied. Within this class, a subclass of estimators that depend on a scalar parameter determining the amount of marginal smoothing and a functional parameter controlling the shape of the smoothing region is specifically considered. Empirical investigations of the influence of these parameters suggest to focus on two particular data-adaptive smooth copula est…
It is well-known that kernel regression estimators do not produce a constant estimator variance over a domain. To correct this problem, Nishida and Kanazawa (2015) proposed a variance-stabilizing (VS) local variable bandwidth for Local Linear (LL) regression estimator. In contrast, Choi and Hall (1998) proposed the skewing (SK) methods for a univariate LL estimator and constructed a convex combination of one LL estimator and two SK estimators that are symmetrically placed on both sides of the LL estimator (the convex combination (CC) estimator) to eliminate higher-order terms in its asymptotic…
We propose a new class of estimators of the multivariate response linear regression coefficient matrix that exploits the assumption that the response and predictors have a joint multivariate Normal distribution. This allows us to indirectly estimate the regression coefficient matrix through shrinkage estimation of the parameters of the inverse regression, or the conditional distribution of the predictors given the responses. We establish a convergence rate bound for estimators in our class and we study two examples. The first example estimator exploits an assumption that the inverse regression…
In this paper, we propose an empirical likelihood-based weighted estimator of regression parameter in quantile regression model with nonignorable missing covariates. The proposed estimator is computationally simple and achieves semiparametric efficiency if the probability of missingness on the fully observed variables is correctly specified. The efficiency gain of the proposed estimator over the complete-case-analysis estimator is quantified theoretically and illustrated via simulation and a real data application.
In this paper we propose a family of robust estimates for isotonic regression: isotonic M-estimators. We show that their asymptotic distribution is, up to an scalar factor, the same as that of Brunk's classical isotonic estimator. We also derive the influence function and the breakdown point of these estimates. Finally we perform a Monte Carlo study that shows that the proposed family includes estimators that are simultaneously highly efficient under gaussian errors and highly robust when the error distribution has heavy tails.
In this article, we propose the Sample Information Optimal Estimator (SIOE) and the Stochastic Restricted Optimal Estimator (SROE) for misspecified linear regression model when multicollinearity exists among explanatory variables. Further, we obtain the superiority conditions of proposed estimators over some other existing estimators in the Mean Square Error Matrix (MSEM) criterion in a standard form which can apply to all estimators considered in this study. Finally, a real world example and a Monte Carlo simulation study are presented for the proposed estimators to illustrate the theoretical…
Estimating linear regression using least squares and reporting robust standard errors is very common in financial economics, and indeed, much of the social sciences and elsewhere. For thick tailed predictors under heteroskedasticity this recipe for inference performs poorly, sometimes dramatically so. Here, we develop an alternative approach which delivers an unbiased, consistent and asymptotically normal estimator so long as the means of the outcome and predictors are finite. The new method has standard errors under heteroskedasticity which are easy to reliably estimate and tests which are cl…
This paper studies robust regression for data on Riemannian manifolds. Geodesic regression is the generalization of linear regression to a setting with a manifold-valued dependent variable and one or more real-valued independent variables. The existing work on geodesic regression uses the sum-of-squared errors to find the solution, but as in the classical Euclidean case, the least-squares method is highly sensitive to outliers. In this paper, we use M-type estimators, including the $L_1$, Huber and Tukey biweight estimators, to perform robust geodesic regression, and describe how to calculate …
We establish asymptotic normality for estimators of the additive regression components under random censorship. To build our estimators, we couple the marginal integration method (Newey (1994)) with an initial Inverse Probability of Censoring Weighted estimator of the multivariate censored regression function introduced by Carbonez et al. (1995) and Kohler et al. (2002). Asymptotic confidence bands are derived from our result.
This paper discusses minimum distance estimation method in the linear regression model with dependent errors which are strongly mixing. The regression parameters are estimated through the minimum distance estimation method, and asymptotic distributional properties of the estimators are discussed. A simulation study compares the performance of the minimum distance estimator with other well celebrated estimator. This simulation study shows the superiority of the minimum distance estimator over another estimator. KoulMde (R package) which was used for the simulation study is available online. See…
We study the isotonic regression estimator over a general countable pre-ordered set. We obtain the limiting distribution of the estimator and study its properties. It is proved that, under some general assumptions, the limiting distribution of the isotonized estimator is given by the concatenation of the separate isotonic regressions of the certain subvectors of an unrestrecred estimator's asymptotic distribution. Also, we show that the isotonization preserves the rate of convergence of the underlying estimator. We apply these results to the problems of estimation of a bimonotone regression fu…
We study the least square estimator, in the framework of simple linear regression, when the deviance term $\varepsilon$ with respect to the linear model is modeled by a uniform distribution. In particular, we give the law of this estimator, and prove some convergence properties.
This paper concerns the estimation of the regression function at a given point in nonparametric heteroscedastic models with Gaussian noise or with noise having unknown distribution. In the two cases an asymptotically efficient kernel estimator is constructed for the minimax absolute error risk.
This article studies a trimmed version of the Nadaraya-Watson estimator to estimate the unknown non-parametric regression function. The characterization of the estimator through minimization problem is established, and its pointwise asymptotic distribution is also derived. The robustness property of the proposed estimator is also studied through breakdown point. Besides, the asymptotic efficiency study along with an extensive simulation study shows that this estimator performs well for various cases. The practicability of the estimator is shown for three benchmark real data as well.
Consider a linear regression model. Fan and Li (2001) describe the smoothly clipped absolute deviation (SCAD) point estimator of the regression parameter vector. To gain insight into the properties of this estimator, they consider an orthonormal design matrix and focus on the estimation of a specified component of this vector. They show that the SCAD point estimator has three attractive properties. We answer the question: To what extent can an interval estimator, centred on the SCAD estimator, have similar attractive properties?
Consider a random vector (X, T), where X is d-dimensional and T is one-dimensional. We suppose that the random variable T is subject to random right censoring and satisfies the $α$-mixing property. The aim of this paper is to study the behavior of the kernel estimator of the relative error regression and to establish its uniform almost sure consistency with rate. Furthermore, we have highlighted the covariance term which measures the dependency. The simulation study shows that the proposed estimator performs well for a finite sample size in different cases.
We study asymptotic behavior of one-step $M$-estimators based on samples from arrays of not necessarily identically distributed random variables and representing explicit approximations to the corresponding consistent $M$-estimators. These estimators generalize Fisher's one-step approximations to consistent maximum likelihood estimators. As a consequence, we consider some nonlinear regression problems where the procedure mentioned allow us to construct explicit asymptotically optimal estimators. We also consider the problem of constructing initial estimators which are needed for one-step estim…
Penalized regression estimators are a popular tool for the analysis of sparse and high-dimensional data sets. However, penalized regression estimators defined using an unbounded loss function can be very sensitive to the presence of outlying observations, especially high leverage outliers. Moreover, it can be particularly challenging to detect outliers in high-dimensional data sets. Thus, robust estimators for sparse and high-dimensional linear regression models are in need. In this paper, we study the robust and asymptotic properties of MM-Bridge and adaptive MM-Bridge estimators: $\ell_q$-pe…
It is known that when the multicollinearity exists in the logistic regression model, variance of maximum likelihood estimator is unstable. As a remedy, in the context of biased shrinkage ridge estimation, Chang (2015) introduced an almost unbiased Liu estimator in the logistic regression model. Making use of his approach, when some prior knowledge in the form of linear restrictions are also available, we introduce a restricted almost unbiased Liu estimator in the logistic regression model. Statistical properties of this newly defined estimator are derived and some comparison result are also pr…
An Efficient Minimax Optimal Estimator For Multivariate Convex Regression
Electrical Engineering and Computer Science, Massachusetts Institute of Technology（麻省理工学院电气工程与计算机科学系）；Department of Mathematics, Tel Aviv University（特拉维夫大学数学系）
We study asymptotic behavior of one-step weighted $M$-estimators based on samples from arrays of not necessarily identically distributed random variables and representing explicit approximations to the corresponding consistent weighted $M$-estimators. Sufficient conditions are presented for asymptotic normality of the one-step weighted $M$-estimators under consideration. As a consequence, we consider some well-known nonlinear regression models where the procedure mentioned allow us to construct explicit asymptotically optimal estimators.
In this paper, we prove large deviations principle for the Nadaraya-Watson estimator and for the semi-recursive kernel estimator of the regression in the multidimensional case. Under suitable conditions, we show that the rate function is a good rate function. We thus generalize the results already obtained in the unidimensional case for the Nadaraya-Watson estimator. Moreover, we give a moderate deviations principle for these two estimators. It turns out that the rate function obtained in the moderate deviations principle for the semi-recursive estimator is larger than the one obtained for the…
Nonparametric density and regression estimators commonly depend on a bandwidth. The asymptotic properties of these estimators have been widely studied when bandwidths are nonstochastic. In practice, however, in order to improve finite sample performance of these estimators, bandwidths are selected by data driven methods, such as cross-validation or plug-in procedures. As a result nonparametric estimators are usually constructed using stochastic bandwidths. In this paper we establish the asymptotic equivalence in probability of local polynomial regression estimators under stochastic and nonstoc…
Quantile regression predicts the $τ$-quantile of the conditional distribution of a response variable given the explanatory variable for $τ\in(0,1)$. The aim of this paper is to establish the asymptotic distribution of the quantile estimator obtained by penalized spline method. A simulation and an exploration of real data are performed to validate our results.
A continuous-time nonlinear regression model with Lévy-driven linear noise process is considered. Sufficient conditions of consistency and asymptotic normality of the Whittle estimator for the parameter of the noise spectral density are obtained in the paper.
The main Theorem of Jain et al.[Jain, K., Singh, S., and Sharma, S. (2011), Re- stricted estimation in multivariate measurement error regression model; JMVA, 102, 2, 264-280] is established in its full generality. Namely, we derive the joint asymp- totic normality of the unrestricted estimator (UE) and the restricted estimators of the matrix of the regression coefficients. The derived result holds under the hypothesized restriction as well as under the sequence of alternative restrictions. In addition, we establish Asymptotic Distributional Risk for the estimators and compare their relative pe…
We propose a class of weighted least squares estimators for the tail index of a distribution function with a regularly varying upper tail. Our approach is based on the method developed by \cite{Holan2010} for the Parzen tail index. Asymptotic normality of the estimators is proved. Through a simulation study, these and earlier estimators are compared in the Pareto and Hall models using the mean squared error as criterion. The results show that the weighted least squares estimator is better than the other estimators investigated.
In this article, basing on NQD samples, we investigate the fixed design nonparametric regression model, where the errors are pairwise NQD random errors, with fixed design points, and an unknown function. Nonparametric weighted estimator will be introduced and its consistency is studied. As special case, the consistency result for weighted kernel estimators of the model is obtained. This extends the earlier work on independent random and dependent random errors to NQD case.
The multivariate errors-in-variables regression model is applicable when both dependent and independent variables in a multivariate regression are subject to measurement errors. In such a scenario it is long established that the traditional least squares approach to estimating the model parameters is biased and inconsistent. The generalized least squares, ordinary least squares and maximum likelihood estimators (under the assumption of Gaussian errors) were derived in the seminal paper of Gleser (1981). However, the ordinary least squares and maximum likelihood estimators for the mean vectors …
Under a partially linear models we study a family of robust estimates for the regression parameter and the regression function when some of the predictor variables take values on a Riemannian manifold. We obtain the consistency and the asymptotic normality of the proposed estimators. Also, we consider a robust cross validation procedure to select the smoothing parameter. Simulations and application to real data show the performance of our proposal under small samples and contamination.
In this article, we consider an imputation method to handle missing response values based on semiparametric quantile regression estimation. In the proposed method, the missing response values are generated using the estimated conditional quantile regression function at given values of covariates. We adopt the generalized method of moments for estimation of parameters defined through a general estimation equation. We demonstrate that the proposed estimator, which combines both semiparametric quantile regression imputation and generalized method of moments, has competitive edge against some of t…
