Estimation of parameters in a generalized GMANOVA model based on an outer product analogy and least squares

This paper investigates estimation of parameters in a combination of the multivariate linear model and growth curve model, called a generalized GMANOVA model. Making analogy between the outer product of data vectors and covariance yields an approach to directly do least squares to covariance. An outer product least squares estimator of covariance (COPLS estimator) is obtained and its distribution is presented if a normal assumption is imposed on the error matrix. Based on the COPLS estimator, two-stage generalized least squares estimators of the regression coefficients are derived. In addition, asymptotic normalities of these estimators are investigated. Simulation studies have shown that the COPLS estimator and two-stage GLS estimators are alternative competitors with more efficiency in the sense of sample mean, standard deviations and mean of the variance estimates to the existing ML estimator in finite samples. An example of application is also illustrated.     

Regression Shrinkage Methods and Autoregressive Time Series Prediction

The pros and cons of applying regression shrinkage prediction arguments and methods to autoregressive time series forecasting are discussed. Simulation evidence of the performance of a Stein regression prediction formula suggests that the overall dominance of the shrunken predictor over least squares in regression no longer holds in time series samples of a reasonable length. Rather, shrinkage appears the better of the two, with respect to prediction mean squared error, only for weaker relationships and seems to be inferior to the least squares predictor when the autoregressive relationship is strong.     

Generalized least squares with misspecified serial correlation structures

Summary. The regression literature contains hundreds of studies on serially correlated disturbances. Most of these studies assume that the structure of the error covariance matrix Ω is known or can be estimated consistently from data. Surprisingly, few studies investigate the properties of estimated generalized least squares (GLS) procedures when the structure of Ω is incorrectly identified and the parameters are inefficiently estimated. We compare the finite sample efficiencies of ordinary least squares (OLS), GLS and incorrect GLS (IGLS) estimators. We also prove new theorems establishing theoretical efficiency bounds for IGLS relative to GLS and OLS. Results from an exhaustive simulation study are used to evaluate the finite sample performance and to demonstrate the robustness of IGLS estimates vis-à-vis OLS and GLS estimates constructed for models with known and estimated (but correctly identified) Ω. Some of our conclusions for finite samples differ from established asymptotic results.     

Estimation of patterned covariance in the multivariate linear models: an outer product least-squares approach

In this article, we present a framework of estimating patterned covariance of interest in the multivariate linear models. The main idea in it is to estimate a patterned covariance by minimizing a trace distance function between outer product of residuals and its expected value. The proposed framework can provide us explicit estimators, called outer product least-squares estimators, for parameters in the patterned covariance of the multivariate linear model without or with restrictions on regression coefficients. The outer product least-squares estimators enjoy the desired properties in finite and large samples, including unbiasedness, invariance, consistency and asymptotic normality. We still apply the framework to three special situations where their patterned covariances are the uniform correlation, a generalized uniform correlation and a general q-dependence structure, respectively. Simulation studies for three special cases illustrate that the proposed method is a competent alternative of the maximum likelihood method in finite size samples.     

Geometric characterization of planar regression

The geometric characterization of linear regression in terms of the ‘concentration ellipse’ by Galton [Galton, F., 1886, Family likeness in stature (with Appendix by Dickson, J.D.H.). Proceedings of the Royal Society of London, 40, 42–73.] and Pearson [Pearson, K., 1901, On lines and planes of closest fit to systems of points in space. Philosophical Magazine, 2, 559–572.] was extended to the case of unequal variances of the presumably uncorrelated errors in the experimental data [McCartin, B.J., 2003, A geometric characterization of linear regression. Statistics, 37(2), 101–117.]. In this paper, this geometric characterization is further extended to planar (and also linear) regression in three dimensions where a beautiful interpretation in terms of the concentration ellipsoid is developed.     

Missing data in linear models with correlated errors

One common method for analyzing data in experimental designs when observations are missing was devised by Yates (1933), who developed his procedure based upon a suggestion by R. A. Fisher. Considering a linear model with independent, equi-variate errors, Yates substituted algebraic values for the missing data and then minimized the error sum of squares with respect to both the unknown parameters and the algebraic values. Yates showed that this procedure yielded the correct error sum of squares and a positively biased hypothesis sum of squares.

Others have elaborated on this technique. Chakrabarti (1962) gave a formal proof of Fisher's rule that produced a way to simplify the calculations of the auxiliary values to be used in place of the missing observations. Kshirsagar (1971) proved that the hypothesis sum of squares based on these values was biased, and developed an easy way to compute that bias. Sclove     

On the equality of usual and amemiya's partially generalized least squares estimator

Equivalent conditions are derived for the equality of GLSE (generalized least squares estimator) and partially GLSE (PGLSE), the latter introduced by Amemiya (1983). By adopting a more general approach the ordinary least squares estimator (OLSE) can shown to be a special PGLSE. Furthcrmore, linearly restricted estimators proposed by Balestra (1983) are investigated in this context. To facilitate the comparison of estimators extensive use of oblique and orthogonal projectors is made.     

Bootstrapping an Econometric Model: Some Empirical Results

The bootstrap, like the jackknife, is a technique for estimating standard errors. The idea is to use Monte Carlo simulation, based on a nonparametric estimate of the underlying error distribution. The bootstrap will be applied to an econometric model describing the demand for capital, labor, energy, and materials. The model is fitted by three-stage least squares. In sharp contrast with previous results, the coefficient estimates and the estimated standard errors perform very well. However, the model's forecasts show serious bias and large random errors, significantly understated by the conventional standard error of forecast.     

Testing inference in heteroskedastic linear regressions: a comparison of two alternative approaches

We consider the issue of performing testing inferences on the parameters that index the linear regression model under heteroskedasticity of unknown form. Quasi-t test statistics use asymptotically correct standard errors obtained from heteroskedasticity-consistent covariance matrix estimators. An alternative approach involves making an assumption about the functional form of the response variances and jointly modelling mean and dispersion effects. In this paper we compare the accuracy of testing inferences made using the two approaches. We consider several different quasi-t tests and also z tests performed after estimated generalized least squares estimation which was carried out using three different estimation strategies. The numerical evidence shows that some quasi-t tests are typically considerably less size distorted in small samples than the tests carried out after the jointly modelling of mean and dispersion effects. Finally, we present and discuss two empirical applications.     

An algorithm for weighted bilinear regression

Bilinear models in which the expectation of a two-way array is the sum of products of parameters are widely used in spectroscopy. In this paper we present an algorithm called combined-vector successive overrelaxation (COV-SOR) for bilinear models, and compare it with methods like alternating least squares, singular value decomposition, and the Marquardt procedure. Comparisons are done for missing data also.     

On approximate least squares estimators of parameters of one-dimensional chirp signal

Chirp signals are quite common in many natural and man-made systems such as audio signals, sonar, and radar. Estimation of the unknown parameters of a signal is a fundamental problem in statistical signal processing. Recently, Kundu and Nandi [Parameter estimation of chirp signals in presence of stationary noise. Stat Sin. 2008;75:187–201] studied the asymptotic properties of least squares estimators (LSEs) of the unknown parameters of a simple chirp signal model under the assumption of stationary noise. In this paper, we propose periodogram-type estimators called the approximate least squares estimators (ALSEs) to estimate the unknown parameters and study the asymptotic properties of these estimators under the same error assumptions. It is observed that the ALSEs are strongly consistent and asymptotically equivalent to the LSEs. Similar to the periodogram estimators, these estimators can also be used as initial guesses to find the LSEs of the unknown parameters. We perform some numerical simulations to see the performance of the proposed estimators and compare them with the LSEs and the estimators proposed by Lahiri et al. [Efficient algorithm for estimating the parameters of two dimensional chirp signal. Sankhya B. 2013;75(1):65–89]. We have analysed two real data sets for illustrative purposes.     

Trend estimation of multivariate time series with controlled smoothness

This paper extends the univariate time series smoothing approach provided by penalized least squares to a multivariate setting, thus allowing for joint estimation of several time series trends. The theoretical results are valid for the general multivariate case, but particular emphasis is placed on the bivariate situation from an applied point of view. The proposal is based on a vector signal-plus-noise representation of the observed data that requires the first two sample moments and specifying only one smoothing constant. A measure of the amount of smoothness of an estimated trend is introduced so that an analyst can set in advance a desired percentage of smoothness to be achieved by the trend estimate. The required smoothing constant is determined by the chosen percentage of smoothness. Closed form expressions for the smoothed estimated vector and its variance-covariance matrix are derived from a straightforward application of generalized least squares, thus providing best linear unbiased estimates for the trends. A detailed algorithm applicable for estimating bivariate time series trends is also presented and justified. The theoretical results are supported by a simulation study and two real applications. One corresponds to Mexican and US macroeconomic data within the context of business cycle analysis, and the other one to environmental data pertaining to a monitored site in Scotland.     

Dealing with emrqes in variables in response surface exploration

This paper is concerned with the analysis of data obtained from a designed experiment where the experimental design cannot be implemented exactly as planned, because errors in the levels of the variables cannot be avoided or measured. When the primary interest of the investigator lies In obtaining a satisfactory response surface model for the investigated relationship, the precision of the model estimates is essential for successful model building and accurate prediction of the response. An iterative procedure is proposed which estimates the effect of the variable in errors and obtains efficient weighted least squares estimates of the parameters of Interest.     

The collected works of john w. tukey

We describe novel, analytical, data-analysis, and Monte-Carlo-simulation studies of strongly heteroscedastic data of both small and wide range.Many different types of heteroscedasticity and fixed or variable weighting are incorporated through error-variance models.Attention is given to parameter bias determinations, evaluations of their significances, and to new ways to correct for bias.The error-variance models allow for both additive and independent power-law errors, and the power exponent is shown to be able to be well determined for typical physicalsciences data by the rapidly-converging, general-purpose, extended-least-squares program we use.The fitting and error-variance models are applied to both low-and high-heteroscedasticity situations, including single-response data from radioactive decay.Monte-Carlo simulations of data with similar parameters are used to evaluate the analytical models developed and the various minimization methods em-ployed, such as extended and generalized least squares.Logarithmic and inversion transformations are investigated in detail, and it is shown analytically and by simulations that exponential data with constant percentage errors can be logarithmically transformed to allow a simple parameter-bias-removal procedure.A more-general bias-reduction approach combining direct and inversion fitting is also developed.Distributions of fitting-model and error-variance-model parameters are shown to be typically non-normal, thus invalidating the usual estimates of parameter bias and precision.Errors in conventional confidence-interval estimates are quantified by comparison with accurate simulation results.     

An approximation of the distribution function of the LIML identifiability test statistic using the method of moments

This paper contains an application of the asymptotic expansion of a pFp() function to a problem encountered in econometrics. In particular we consider an approximation of the distribution function of the limited information maximum likelihood (LIML) identifiability test statistic using the method of moments. An expression for the Sth order asymptotic approximation of the moments of the LIML identifiability test statistic is derived and tabulated. The exact distribution function of the test statistic is approximated by a member of the class of F (variance ratio) distribution functions having the same first two integer moments. Some tabulations of the approximating distribution function are included.     

Testing Covariate Effects in Aalen's Linear Hazard Model

The performance of tests in Aalen's linear regression model is studied using asymptotic power calculations and stochastic simulation. Aalen's original least squares test is compared to two modifications: a weighted least squares test with correct weights and a test where the variance is re-estimated under the null hypothesis. The test with re-estimated variance provides the highest power of the tests for the setting of this paper, and the gain is substantial for covariates following a skewed distribution like the exponential. It is further shown that Aalen's choice for weight function with re-estimated variance is optimal in the one-parameter case against proportional alternatives.     

Comparisons of methods of estimation for a pareto distribution of the first kind

This paper compares methods of estimation for the parameters of a Pareto distribution of the first kind to determine which method provides the better estimates when the observations are censored, The unweighted least squares (LS) and the maximum likelihood estimates (MLE) are presented for both censored and uncensored data. The MLE's are obtained using two methods, In the first, called the ML method, it is shown that log-likelihood is maximized when the scale parameter is the minimum sample value. In the second method, called the modified ML (MML) method, the estimates are found by utilizing the maximum likelihood value of the shape parameter in terms of the scale parameter and the equation for the mean of the first order statistic as a function of both parameters. Since censored data often occur in applications, we study two types of censoring for their effects on the methods of estimation: Type II censoring and multiple random censoring. In this study we consider different sample sizes and several values of the true shape and scale parameters.

Comparisons are made in terms of bias and the mean squared error of the estimates. We propose that the LS method be generally preferred over the ML and MML methods for estimating the Pareto parameter γ for all sample sizes, all values of the parameter and for both complete and censored samples. In many cases, however, the ML estimates are comparable in their efficiency, so that either estimator can effectively be used. For estimating the parameter α, the LS method is also generally preferred for smaller values of the parameter (α ≤4). For the larger values of the parameter, and for censored samples, the MML method appears superior to the other methods with a slight advantage over the LS method. For larger values of the parameter α, for censored samples and all methods, underestimation can be a problem.     

Bootstrapping time series models

This paper surveys recent development in bootstrap methods and the modifications needed for their applicability in time series models. The paper discusses some guidelines for empirical researchers in econometric analysis of time series. Different sampling schemes for bootstrap data generation and different forms of bootstrap test statistics are discussed. The paper also discusses the applicability of direct bootstrapping of data in dynamic models and cointegrating regression models. It is argued that bootstrapping residuals is the preferable approach. The bootstrap procedures covered include the recursive bootstrap, the moving block bootstrap and the stationary bootstrap.     

The peculiar shrinkage properties of partial least squares regression

Partial least squares regression has been widely adopted within some areas as a useful alternative to ordinary least squares regression in the manner of other shrinkage methods such as principal components regression and ridge regression. In this paper we examine the nature of this shrinkage and demonstrate that partial least squares regression exhibits some undesirable properties.     

AMEMIYA'S FORM OF THE WEIGHTED LEAST SQUARES ESTIMATOR

Amemiya's estimator is a weighted least squares estimator of the regression coefficients in a linear model with heteroscedastic errors. It is attractive because the heteroscedasticity is not parametrized and the weights (which depend on the error covariance matrix) are estimated nonparametrically. This paper derives an asymptotic expansion for Amemiya's form of the weighted least squares estimator, and uses it to discuss the effects of estimating the weights, of the number of iterations, and of the choice of the initial estimate. The paper also discusses the special case of normally distributed errors and clarifies the particular consequences of assuming normality.