Bivariate Birnbaum–Saunders distribution and associated inference

Univariate Birnbaum–Saunders distribution has been used quite effectively to model positively skewed data, especially lifetime data and crack growth data. In this paper, we introduce bivariate Birnbaum–Saunders distribution which is an absolutely continuous distribution whose marginals are univariate Birnbaum–Saunders distributions. Different properties of this bivariate Birnbaum–Saunders distribution are then discussed. This new family has five unknown parameters and it is shown that the maximum likelihood estimators can be obtained by solving two non-linear equations. We also propose simple modified moment estimators for the unknown parameters which are explicit and can therefore be used effectively as an initial guess for the computation of the maximum likelihood estimators. We then present the asymptotic distributions of the maximum likelihood estimators and use them to construct confidence intervals for the parameters. We also discuss likelihood ratio tests for some hypotheses of interest. Monte Carlo simulations are then carried out to examine the performance of the proposed estimators. Finally, a numerical data analysis is performed in order to illustrate all the methods of inference discussed here.     

Consistent and asymptotically normal estimators for cyclically time-dependent linear models

We consider a general class of time series linear models where parameters switch according to a known fixed calendar. These parameters are estimated by means of quasi-generalized least squares estimators. conditions for strong consistency and asymptotic normality are given. Applications to cyclical ARMA models with non constant periods are considered.     

半参数回归模型的稳健补偿最小二乘估计

本文研究了一类半参数回归模型，利用稳健补偿最小二乘估计法，得到了稳健补偿最小二乘估计量，以及它们的影响函数及渐近方差一协方差，对结果的分析表明了该法优于补偿最小二乘法，而且具有稳定性．     

On conditional least squares estimation for affine diffusions based on continuous time observations

Statistical Inference for Stochastic Processes - We study asymptotic properties of conditional least squares estimators for the drift parameters of two-factor affine diffusions based on continuous...     

Robust Frequency Estimation Using Elemental Sets

Abstract

The extraction of sinusoidal signals from time-series data is a classic problem of ongoing interest in the statistics and signal processing literatures. Obtaining least squares estimates is difficult because the sum of squares has local minima O(1/n) apart in the frequencies. In practice the frequencies are often estimated using ad hoc and inefficient methods. Problems of data quality have received little attention. An elemental set is a subset of the data containing the minimum number of points such that the unknown parameters in the model can be identified. This article shows that, using a variant of the classical method of Prony, parameter estimates for a sum of sinusoids can be obtained algebraically from an elemental set. Elemental set methods are used to construct finite algorithm estimators that approximately minimize the least squares, least trimmed sum of squares, or least median of squares criteria. The elemental set estimators prove able in simulations to resolve the frequencies to the correct local minima of the objective functions. When used as the first stage of an MM estimator, the constructed estimators based on the trimmed sum of squares and least median of squares criteria produce final estimators which have high breakdown properties and which are simultaneously efficient when no outliers are present. The approach can also be applied to sums of exponentials, and sums of damped sinusoids. The article includes simulations with one and two sinusoids and two data examples.     

Estimation of time-varying ARMA models with Markovian changes in regime

In this paper, we consider the estimation of time-varying ARMA models subject to Markovian changes in regime. We give explicit conditions ensuring consistency and asymptotic normality, as well as the limiting covariance matrix, of least squares and quasi-generalized least-squares estimators.     

Parameter estimation for the Langevin equation with stationary-increment Gaussian noise

We study the Langevin equation with stationary-increment Gaussian noise. We show the strong consistency and the asymptotic normality with Berry–Esseen bound of the so-called second moment estimator of the mean reversion parameter. The conditions and results are stated in terms of the variance function of the noise. We consider both the case of continuous and discrete observations. As examples we consider fractional and bifractional Ornstein–Uhlenbeck processes. Finally, we discuss the maximum likelihood and the least squares estimators.     

Testing for random effects in linear mixed models for longitudinal data under moment conditions

In this paper, we consider whether the random effect exists in linear mixed models （LMMs） when only moment conditions are assumed. Based on the estimators of parameters and their asymptotic properties, a Wald-type test is constructed. It is consistent against global alternatives and is sensitive to the local alternatives converging to the null hypothesis at parametric rates, a fastest possibly rate for goodness-of-fit testing. Moreover, a simulation study shows the performance of the test is good. The procedure also applies to a real data.     

Asymptotics of Estimating Equations under Natural Conditions

In a variety of statistical problems one needs to solve an equation in order to get an estimator. We consider the large sample properties of such estimators generated from samples that are not necessarily identically distributed. Very general assumptions that lead to the existence, strong consistency, and asymptotic normality of the estimators are given. A number of results that are useful in verifying the general assumptions are given and an example illustrates their use. General applications to maximum likelihood, iteratively reweighted least squares, and robust estimation are discussed briefly.     

变系数模型中的逐元B-Spline估计法

给出了一种用于估计变系数模型中未知函数的逐元B-Spline方法,建立了估计量的局部渐近偏差,方差和渐近正态分布,开发了一种快速选择估计量窗宽的方法,通过Monte Carlo模拟研究了估计量的有限样本性质.     

Parameter estimation for the stochastic SIS epidemic model

In this paper we estimate the parameters in the stochastic SIS epidemic model by using pseudo-maximum likelihood estimation (pseudo-MLE) and least squares estimation. We obtain the point estimators and $100 (1-\alpha )\%$ confidence intervals as well as $100 (1-\alpha )\%$ joint confidence regions by applying least squares techniques. The pseudo-MLEs have almost the same form as the least squares case. We also obtain the exact as well as the asymptotic $100 (1-\alpha )\%$ joint confidence regions for the pseudo-MLEs. Computer simulations are performed to illustrate our theory.     

Bridge estimators and the adaptive lasso under heteroscedasticity

In this paper we investigate penalized least squares methods in linear regression models with heteroscedastic error structure. It is demonstrated that the basic properties with respect to model selection and parameter estimation of bridge estimators, Lasso and adaptive Lasso do not change if the assumption of homoscedasticity is violated. However, these estimators do not have oracle properties in the sense of Fan and Li (2001) if the oracle is based on weighted least squares. In order to address this problem we introduce weighted penalized least squares methods and demonstrate their advantages by asymptotic theory and by means of a simulation study.     

Asymptotic results in segmented multiple regression

This paper studies the asymptotic behavior of the least squares estimators in segmented multiple regression. For a model with more than one partitioning variable, each of which has one or more change-points, we study the asymptotic properties of the estimated change-points and regression coefficients. Using techniques in empirical process theory, we prove the consistency of the least squares estimators and also establish the asymptotic normality of the estimated regression coefficients. For the estimated change-points, we obtain their consistency at the rates of or 1/n, with or without continuity constraints, respectively. The change-points estimated under the continuity constraints are also shown to asymptotically have a multivariate normal distribution. For the case where the regression mean functions are not assumed to be continuous at the change-points, the asymptotic distribution of the estimated change-points involves a step function process, whose distribution does not follow a well-known distribution.     

Least median of squares estimation by optimization heuristics with an application to the CAPM and a multi-factor model

For estimating the parameters of models for financial market data, the use of robust techniques is of particular interest. Conditional forecasts, based on the capital asset pricing model, and a factor model are considered. It is proposed to consider least median of squares estimators as one possible alternative to ordinary least squares. Given the complexity of the objective function for the least median of squares estimator, the estimates are obtained by means of optimization heuristics. The performance of two heuristics is compared, namely differential evolution and threshold accepting. It is shown that these methods are well suited to obtain least median of squares estimators for real world problems. Furthermore, it is analyzed to what extent parameter estimates and conditional forecasts differ between the two estimators. The empirical analysis considers daily and monthly data on some stocks from the Dow Jones Industrial Average Index.     

Double k-Class Estimators in Regression Models with Non-spherical Disturbances

In this paper, we consider a family of feasible generalised double k-class estimators in a linear regression model with non-spherical disturbances. We derive the large sample asymptotic distribution of the proposed family of estimators and compare its performance with the feasible generalized least squares and Stein-rule estimators using the mean squared error matrix and risk under quadratic loss criteria. A Monte-Carlo experiment investigates the finite sample behaviour of the proposed family of estimators.     

Parameter Estimation of Multivariate Regression Model with Fuzzy Random Errors

In many real-world problems, observations are usually described by approximate values due to fuzzy uncertainty, unlikeprobabilistic uncertainty that has nothing to do with experimentation. The combination of statistical model and fuzzy set theory is helpful to improve the identification and analysis of complex systems. As an extension ofstatistical techniques, this study is an investigation of the relationship between fuzzy multiple explanatory variables and fuzzy response with numeric coefficients and the fuzzy random error term. In this work we describe a parameter estimation procedure carrying out the least-squares method in a complete metric space of fuzzy numbers to determine the coefficients based on the extension principle. We demonstrate how the fuzzy least squares estimators present large sample statistical properties, including asymptotic normality, strong consistency and confidence region. The estimators are also examined via asymptotic relative efficiency concerning traditional least squares estimators. Different from the construction of error term in Kim et al.\cite{21}, it is more reasonable in the proposed model since the problems of inconsistency in referring to fuzzy variable and producing the negative spreads may be avoided. The experimental study verifies that the proposed fuzzy least squares estimators achieve the meaning consistent with the theory identification for large sample data set and better generalization regarding one single variable model.     

Comparisons between simultaneous and componentwise splines for varying coefficient models

In this paper, we study the properties of the simultaneous and componentwise splines for the varying coefficient model with repeatedly measured (longitudinal) dependent variable and time invariant covariates. The proposed simultaneous smoothing spline estimators are mainly obtained from the penalized least squares with adjustment for the variations of covariates in the penalized terms. We do this mainly to avoid the penalized terms being influenced by the scales of the covariates and the random smoothing parameters appearing in the estimators, which complicates the derivation of the asymptotic properties of the estimators. It is shown in this study that our estimators have smaller variances than the componentwise ones. Through a Monte Carlo simulation and two empirical examples, the simultaneous smoothing splines are all found to be more accurate in the variances.     

On the asymptotic behavior of the parameter estimators for some diffusion processes: application to neuronal models

We consider a sample of i.i.d. times and we interpret each item as the first-passage time (FPT) of a diffusion process through a constant boundary. The problem is to estimate the parameters characterizing the underlying diffusion process through the experimentally observable FPT’s. Recently in Ditlevsen and Lánsky (Phys Rev E 71, 2005) and Ditlevsen and Lánsky (Phys Rev E 73, 2006) closed form estimators have been proposed for neurobiological applications. Here we study the asymptotic properties (consistency and asymptotic normality) of the class of moment type estimators for parameters of diffusion processes like those in Ditlevsen and Lánsky (Phys Rev E 71, 2005) and Ditlevsen and Lánsky (Phys Rev E 73, 2006). Furthermore, to make our results useful for application instances we establish upper bounds for the rate of convergence of the empirical distribution of each estimator to the normal density. Applications are also considered by means of simulated experiments in a neurobiological context.      

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In this paper an efficient estimation methodology for the partially linear models with random effects is proposed. For this, we use the generalized least square estimate (GLSE) and the B-splines methods to estimate the unknowns, and employ the penalized least square method to obtain the estimators of the random effects item. Further, we also consider the estimation for the variance components. Compared with the existing methods, our proposed methodology performs well. The asymptotic properties of the estimators are obtained. A simulation study is carried out to assess the performance of our proposed methodology.     

Bias-corrected statistical inference for partially linear varying coefficient errors-in-variables models with restricted condition

In this paper, we consider the statistical inference for the partially liner varying coefficient model with measurement error in the nonparametric part when some prior information about the parametric part is available. The prior information is expressed in the form of exact linear restrictions. Two types of local bias-corrected restricted profile least squares estimators of the parametric component and nonparametric component are conducted, and their asymptotic properties are also studied under some regularity conditions. Moreover, we compare the efficiency of the two kinds of parameter estimators under the criterion of Lo?ner ordering. Finally, we develop a linear hypothesis test for the parametric component. Some simulation studies are conducted to examine the finite sample performance for the proposed method. A real dataset is analyzed for illustration.