Bayesian estimation for time-series regressions improved with exact likelihoods

We propose an estimation procedure for time-series regression models under the Bayesian inference framework. With the exact method of Wise [Wise, J. (1955). The autocorrelation function and spectral density function. Biometrika, 42, 151–159], an exact likelihood function can be obtained instead of the likelihood conditional on initial observations. The constraints on the parameter space arising from the stationarity conditions are handled by a reparametrization, which was not taken into consideration by Chib [Chib, S. (1993). Bayes regression with autoregressive errors: A Gibbs sampling approach. J. Econometrics, 58, 275–294] or Chib and Greenberg [Chib, S. and Greenberg, E. (1994). Bayes inference in regression model with ARMA(p, q) errors. J. Econometrics, 64, 183–206]. Simulation studies show that our method leads to better inferential results than their results.     

Exact bootstrap confidence intervals for regression coefficients in small samples

ABSTRACT

Regression analysis is one of the important tools in statistics to investigate the relationships among variables. When the sample size is small, however, the assumptions for regression analysis can be violated. This research focuses on using the exact bootstrap to construct confidence intervals for regression parameters in small samples. The comparison of the exact bootstrap method with the basic bootstrap method was carried out by a simulation study. It was found that on a very small sample (n ≈ 5) under Laplace distribution with the independent variable treated as random, the exact bootstrap was more effective than the standard bootstrap confidence interval.     

Confidence intervals for the regression coefficient in a simple regression model with a balanced two-fold nested error structure

ABSTRACT

In applications using a simple regression model with a balanced two-fold nested error structure, interest focuses on inferences concerning the regression coefficient. This article derives exact and approximate confidence intervals on the regression coefficient in the simple regression model with a balanced two-fold nested error structure. Eleven methods are considered for constructing the confidence intervals on the regression coefficient. Computer simulation is performed to compare the proposed confidence intervals. Recommendations are suggested for selecting an appropriate method.     

A Statistical Model for Shelf Life Estimation Using Sensory Evaluations Scores

Abstract

This article focuses on the problem of estimating the shelf life of food products by modeling the results coming from sensory evaluations. In such studies, trained panelists are asked to judge food attributes by reference to a scale of numbers (scores varying often from 0 to 6). The usual statistical approach for data analysis is to fit a regression line relating the scores and the time of evaluation. The estimate of the shelf life is obtained by solving the regression equation and replacing the score by a cut-off point (which indicates product “failure”) previously chosen by the food company. The procedure used in these sensory evaluations is such that one never knows the exact “time to failure”. Consequently, data arising from these studies are either right or left censored. We propose a model which incorporates these informations and assumes a Weibull for the underlying distribution of the failure time. Simulation studies were implemented. The approach was used in a real data set coming from sensory evaluations of a dehydrated food product.     

Confidence intervals for lognormal regression and a non-parametric alternative

Approximate confidence intervals are given for the lognormal regression problem. The error in the nominal level can be reduced to O(n ^?2), where n is the sample size. An alternative procedure is given which avoids the non-robust assumption of lognormality. This amounts to finding a confidence interval based on M-estimates for a general smooth function of both ? and F, where ? are the parameters of the general (possibly nonlinear) regression problem and F is the unknown distribution function of the residuals. The derived intervals are compared using theory, simulation and real data sets.     

Distribution of the mean reversion estimator in the Ornstein–Uhlenbeck process

ABSTRACT

We derive the exact distribution of the maximum likelihood estimator of the mean reversion parameter (κ) in the Ornstein–Uhlenbeck process using numerical integration through analytical evaluation of a joint characteristic function. Different scenarios are considered: known or unknown drift term, fixed or random start-up value, and zero or positive κ. Monte Carlo results demonstrate the remarkably reliable performance of our exact approach across all the scenarios. In comparison, misleading results may arise under the asymptotic distributions, including the advocated infill asymptotic distribution, which performs poorly in the tails when there is no intercept in the regression and the starting value of the process is nonzero.     

Minimax prediction in the linear model with a relative squared error

Arnold and Stahlecker (Stat Pap 44:107–115, 2003) considered the prediction of future values of the dependent variable in the linear regression model with a relative squared error and deterministic disturbances. They found an explicit form for a minimax linear affine solution d* of that problem. In the paper we generalize this result proving that the decision rule d* is also minimax when the class D{\mathcal{D}} of possible predictors of the dependent variable is unrestricted. Then we show that d* remains minimax in D{\mathcal{D}} when the disturbances are random with the mean vector zero and the known positive definite covariance matrix.     

Interval estimation for a lower-truncated distribution based on the double Type-II censored sample

ABSTRACT

The interval estimation problem is investigated for the parameters of a general lower truncated distribution under double Type-II censoring scheme. The exact, asymptotic and bootstrap interval estimates are derived for the unknown model parameter and the lower truncated threshold bound. One real-life example and a numerical study are presented to illustrate performance of our methods.     

Regression Depth with Censored and Truncated Data

Abstract

In this article, we consider the problem of estimating regression coefficients for a linear model with censored and truncated data based on regression depth. Any line can be given a rank using regression depth and the deepest regression line is the line with the maximum regression depth. We propose a method to define the regression depth of a line in the presence of censoring and truncation. We show how the proposed regression performs through analyzing Stanford heart transplant data and AIDS incubation data.     

TREE-BASED REGRESSION FOR A CIRCULAR RESPONSE

ABSTRACT

The application of conventional statistical methods to directional data generally produces erroneous results. Various regression models for a circular response have been presented in the literature, however these are unsatisfactory either in the limited relationships that can be modeled, or the limitations on the number or type of covariates admissible. One difficulty with circular regression is devising a meaningful regression function. This problem is exacerbated when trying to incorporate both linear and circular variables as covariates. Due to these complexities, circular regression is ripe for exploration via tree-based methods, in which a formal regression function is not needed, but where insight into the general structure and relationship between predictors and the response may be obtained. A basic framework for regression trees, predicting a circular response from a combination of circular and linear predictors, will be presented.     

Shrinkage ridge regression in partial linear models

ABSTRACT

In this paper, shrinkage ridge estimator and its positive part are defined for the regression coefficient vector in a partial linear model. The differencing approach is used to enjoy the ease of parameter estimation after removing the non parametric part of the model. The exact risk expressions in addition to biases are derived for the estimators under study and the region of optimality of each estimator is exactly determined. The performance of the estimators is evaluated by simulated as well as real data sets.     

Quantile regression for panel data models with fixed effects under random censoring

Abstract

The locally weighted censored quantile regression approach is proposed for panel data models with fixed effects, which allows for random censoring. The resulting estimators are obtained by employing the fixed effects quantile regression method. The weights are selected either parametrically, semi-parametrically or non-parametrically. The large panel data asymptotics are used in an attempt to cope with the incidental parameter problem. The consistency and limiting distribution of the proposed estimator are also derived. The finite sample performance of the proposed estimators are examined via Monte Carlo simulations.     

Using Liu-Type Estimator to Combat Collinearity

Abstract

Linear regression model and least squares method are widely used in many fields of natural and social sciences. In the presence of collinearity, the least squares estimator is unstable and often gives misleading information. Ridge regression is the most common method to overcome this problem. We find that when there exists severe collinearity, the shrinkage parameter selected by existing methods for ridge regression may not fully address the ill conditioning problem. To solve this problem, we propose a new two-parameter estimator. We show using both theoretic results and simulation that our new estimator has two advantages over ridge regression. First, our estimator has less mean squared error (MSE). Second, our estimator can fully address the ill conditioning problem. A numerical example from literature is used to illustrate the results.     

Rational Regression Models with Continuous Chebyshev Design

In rational regression models, the G-optimal designs are very difficult to derive in general. Even when an G-optimal design can be found, it has, from the point of view of modern nonparametric regression, certain drawbacks because the optimal design crucially depends on the model. Hence, it can be used only when the model is given in advance. This leads to the problem of finding designs which would be nearly optimal for a broad class of rational regression models. In this article, we will show that the so-called continuous Chebyshev Design is a practical solution to this problem.     

Incremental modelling for compositional data streams

ABSTRACT

Incremental modelling of data streams is of great practical importance, as shown by its applications in advertising and financial data analysis. We propose two incremental covariance matrix decomposition methods for a compositional data type. The first method, exact incremental covariance decomposition of compositional data (C-EICD), gives an exact decomposition result. The second method, covariance-free incremental covariance decomposition of compositional data (C-CICD), is an approximate algorithm that can efficiently compute high-dimensional cases. Based on these two methods, many frequently used compositional statistical models can be incrementally calculated. We take multiple linear regression and principle component analysis as examples to illustrate the utility of the proposed methods via extensive simulation studies.     

Comparison of computer programs for simple linear L 1 regression

A number of efficient computer codes are available for the simple linear L ₁ regression problem. However, a number of these codes can be made more efficient by utilizing the least squares solution. In fact, a couple of available computer programs already do so.

We report the results of a computational study comparing several openly available computer programs for solving the simple linear L ₁ regression problem with and without computing and utilizing a least squares solution.     

Range-preserving unbiased estimators in the poisson case

Abstract

The Kruskal–Wallis test is a popular nonparametric test for comparing k independent samples. In this article we propose a new algorithm to compute the exact null distribution of the Kruskal–Wallis test. Generating the exact null distribution of the Kruskal–Wallis test is needed to compare several approximation methods. The 5% cut-off points of the exact null distribution which StatXact cannot produce are obtained as by-products. We also investigate graphically a reason that the exact and approximate distributions differ, and hope that it will be a useful tutorial tool to teach about the Kruskal–Wallis test in undergraduate course.     

On exact strong laws of large numbers for ratios of random variables and their applications

Abstract

We study the almost sure convergence of weighted sums of ratios of independent random variables satisfying some general, mild conditions. The obtained results are applied to exact laws for order statistics. An exact law for independent random variables which are nonidentically distributed is also proved and applied to ratios of adjacent order statistics for a sample of uniformly distributed random variables.