On some tests for a multivariate linear functional relationship with replication

For testing problems of the coefficient vector and the interception of multivariate linear functional relationship with replicated observations, the likelihood ratio test statistics are considered. Their asymptotic distributions are obtained under each null hypothesis respectively.     

Alternative covariance estimates in a replicated measurement error model with correlated,heteroscedastic errors

The article concerns covariance estimates in a replicated measurement error model with correlated, heteroscedastic errors. Freedman has conjectured that using more of the data will improve estimates of covariance matrices and result in a more efficient estimate of the coefficient of the regression model. The paper confirms the conjecture asymptotically for the case that all random variables are normally distributed, but the gain is not substantial.     

Testing for lack of fit in linear multiresponse models based on exact or near replicates

Three procedures for testing the adequacy of a proposed linear multiresponse regression model against unspecified general alternatives are considered. The model has an error structure with a matrix normal distribution which allows the vector of responses for a particular run to have an unknown covariance matrix while the responses for different runs are uncorrelated. Furthermore, each response variable may be modeled by a separate design matrix. Multivariate statistics corresponding to the classical univariate lack of fit and pure error sums of squares are defined and used to determine the multivariate lack of fit tests. A simulation study was performed to compare the power functions of the test procedures in the case of replication. Generalizations of the tests for the case in which there are no independent replicates on all responses are also presented.     

A review of model selection procedures relevant to the weibull distribution

A number of goodness-of-fit and model selection procedures related to the Weibull distribution are reviewed. These procedures include probability plotting, correlation type goodness-of-fit tests, and chi-square goodness-of-fit tests. Also the Kolmogorow-Smirniv, Kuiper, and Cramer-Von Mises test statistics for completely specified hypothesis based on censored data are reviewed, and these test statistics based on complete samples for the unspecified parameters case are considered. Goodness-of-fit tests based on sample spacings, and a goodness-of-fit test for the Weibull process, is also discussed.

Model selection procedures for selecting between a Weibull and gamma model, a Weibull and lognormal model, and for selecting from among all three models are considered. Also tests of exponential versus Weibull and Weibull versus generalized gamma are mentioned.     

Data quality: A statistical perspective

We present the old-but-new problem of data quality from a statistical perspective, in part with the goal of attracting more statisticians, especially academics, to become engaged in research on a rich set of exciting challenges. The data quality landscape is described, and its research foundations in computer science, total quality management and statistics are reviewed. Two case studies based on an EDA approach to data quality are used to motivate a set of research challenges for statistics that span theory, methodology and software tools.     

Equivalence of two tests in varying coefficient partially linear errors in variable model with missing responses

This paper considers two tests on varying coefficient partially linear errors-in-variables models (VCPLM-EV) with missing responses under the linear constraint. The restricted estimator for the parametric component is derived and proven to share asymptotically normal distribution. In order to test the linear constraint, two statistics based on the profile Lagrange multiplier method and the corrected residual sum of squares method respectively, are proposed. It is of interest to obtain that the magnitudes of the two statistics are equal exactly and follow the asymptotical chi-square distribution. This reveals a new type of Wilk’s phenomenon in VCPLM-EV models with missing response. Finally, some numerical examples are carried out to illustrate relevant performances.     

Dickey-fuller tests with trend

The Dickey-Fuller [rcirc]_τ and [pcirc]_τ tests are based on a regression of a variable on its lagged value, an intercept, and a trend term. The distributions of both statistics depend on the coefficient of the trend, and the usual Dickey-Fuller tabulations assume that this coefficient equals zero. This paper provides tabulations for the case that the coefficient of the trend is non-zero.     

Measuring agreement between two statistics with applications to age standardization

Summary.  We detail a general method for measuring agreement between two statistics. An application is two ratios of directly standardized rates which differ only by the choice of the standard. If the statistics have a high value for the coefficient of agreement then the expected squared difference between the statistics is small relative to the variance of the average of the two statistics, and inferences vary little by changing statistics. The estimation of a coefficient of agreement between two statistics is not straightforward because there is only one pair of observed values, each statistic calculated from the data. We introduce estimators of the coefficient of agreement for two statistics and discuss their use, especially as applied to functions of standardized rates.     

Probability of ties and Markov property in discrete order statistics

It is well known that, in the continuous case, the probability that two consecutive order statistics are equal to zero, whereas it is not true when the distribution is discrete. It is, perhaps, for this reason that order statistics from discrete distributions has not been investigated in the literature as much as from a continuous distribution. The main purpose of this paper, therefore, is to obtain the probability of ties when the distribution is discrete. Also it is shown that, in the discrete case, the Markov property does not hold good. However, the order statistics from a geometric distribution forms a Markov chain.     

On the distribution of linear combinations of the components of a Dirichlet random vector

On making use of a result of Imhof, an integral representation of the distribution function of linear combinations of the components of a Dirichlet random vector is obtained. In fact, the distributions of several statistics such as Moran and Geary's indices, the Cliff‐Ord statistic for spatial correlation, the sample coefficient of determination, F‐ratios and the sample autocorrelation coefficient can be similarly determined. Linear combinations of the components of Dirichlet random vectors also turn out to be a key component in a decomposition of quadratic forms in spherically symmetric random vectors. An application involving the sample spectrum associated with series generated by ARMA processes is discussed.     

An evaluation of the reconstructed coefficient of determination and potential adjustments

Previously, a method was proposed for calculating a reconstructed coefficient of determination in the case of right-censored regression using the expectation–maximization (EM) algorithm. This measure is assessed via simulation study for the purpose of evaluating the utility of model fit. Further, several reconstructed adjusted coefficients of determination are proposed and compared via simulation study for the purpose of model selection. The application of these proposed measures is illustrated on a real dataset.     

A RESTRICTED MAXIMUM LIKELIHOOD PROCEDURE FOR ESTIMATING THE VARIANCE FUNCTION OF AN IMMUNOASSAY

Restricted maximum likelihood (REML) is a procedure for estimating a variance function in a heteroscedastic linear model. Although REML has been extended to non-linear models, the case in which the data are dominated by replicated observations with unknown values of the independent variable of interest, such as the concentration of a substance in a blood sample, has not been considered. We derive a REML procedure for an immunoassay and show that the resulting estimator is superior to those currently being used. Some interesting properties of the REML estimator are derived, and its relationship to other estimators is discussed.     

On structure,family and parameter estimation of hierarchical Archimedean copulas

Research on structure determination and parameter estimation of hierarchical Archimedean copulas (HACs) has so far mostly focused on the case in which all appearing Archimedean copulas belong to the same Archimedean family. The present work addresses this issue and proposes a new approach for estimating HACs that involve different Archimedean families. It is based on employing goodness-of-fit test statistics directly into HAC estimation. The approach is summarized in a simple algorithm, its theoretical justification is given and its applicability is illustrated by several experiments, which include estimation of HACs involving up to five different Archimedean families.     

On marginal likelihood inference for the intra-class correlation coefficient

A p-component set of responses have been constructed by a location-scale transformation to a p-component set of error variables, the covariance matrix of the set of error variables being of intra-class covariance structure:all variances being unity, and covariance being equal [IML0001]. A sample of size n has been described as a conditional structural model, conditional on the value of the intra-class correlation coefficient ρ. The conditional technique of structural inference provides the marginal likelihood function of ρ based on the standardized residuals. For the normal case, the marginal likelihood function of ρ is seen to be dependent on the standardized residuals through the sample intra-class correlation coefficient. By the likelihood modulation technique, the nonnull distribution of the sample intra-class correlation coefficient has also been obtained.     

The union-intersection principle applied to the test of dimensionality

In the first section Anderson-Rao-Fujikoshi's test statistics for testing the hypothesis of dimensionality are reviewed and then Olkin-Tomsky's generalized union-intersection principle is applied to show that a new class of test statistics for testing the hypothesis of dimensionality are derived which includes the likelihood ratio test statistics, the trace test statistics and a version of ROY'S maximum root test statistics.     

Inference in a two-parameter generalized order statistics model

As a submodel of generalized order statistics with two unknown model parameters, m-generalized order statistics may serve as a simple model for ordered quantities in a given application. It is shown that the joint distribution of m-generalized order statistics has a representation as a regular exponential family in the model parameters, as it is the case for the comprising model. Utilizing this finding, a minimal sufficient and complete statistic is obtained along with distributional properties. Joint maximum likelihood estimation of the parameters is considered, and strong consistency and asymptotic efficiency of the estimator are established. A test is provided to decide whether a restriction to the submodel is reasonable.     

Wald,LM and LR test statistics of linear hypothese in a strutural equation model

For the linear hypothesis in a strucural equation model, the properties of test statistics based on the two stage least squares estimator (2SLSE) have been examined since these test statistics are easily derived in the instrumental variable estimation framework. Savin (1976) has shown that inequalities exist among the test statistics for the linear hypothesis, but it is well known that there is no systematic inequality among these statistics based on 2SLSE for the linear hypothesis in a structural equation model. Morimune and Oya (1994) derived the constrained limited information maximum likelihood estimator (LIMLE) subject to general linear constraints on the coefficients of the structural equation, as well as Wald, LM and Lr Test statistics for the adequacy of the linear constraints.

In this paper, we derive the inequalities among these three test statistics based on LIMLE and the local power functions based on Limle and 2SLSE to show that there is no test statistic which is uniformly most powerful, and the LR test statistic based on LIMLE is locally unbised and the other test statistics are not. Monte Carlo simulations are used to examine the actual sizes of these test statistics and some numerical examples of the power differences among these test statistics are given. It is found that the actual sizes of these test statistics are greater than the nominal sizes, the differences between the actual and nominal sizes of Wald test statistics are generally the greatest, those of LM test statistics are the smallest, and the power functions depend on the correlations between the endogenous explanatory variables and the error term of the structural equation, the asymptotic variance of estimator of coefficients of the structural equation and the number of restrictions imposed on the coefficients.     

Nonstationary panel data analysis: an overview of some recent developments

This paper overviews some recent developments in panel data asymptotics, concentrating on the nonstationary panel case and gives a new result for models with individual effects. Underlying recent theory are asymptotics for multi-indexed processes in which both indexes may pass to infinity. We review some of the new limit theory that has been developed, show how it can be applied and give a new interpretation of individual effects in nonstationary panel data. Fundamental to the interpretation of much of the asymptotics is the concept of a panel regression coefficient which measures the long run average relation across a section of the panel. This concept is analogous to the statistical interpretation of the coefficient in a classical regression relation. A variety of nonstationary panel data models are discussed and the paper reviews the asymptotic properties of estimators in these various models. Some recent developments in panel unit root tests and stationary dynamic panel regression models are also reviewed.     

Nonstationary panel data analysis: an overview of some recent developments

This paper overviews some recent developments in panel data asymptotics, concentrating on the nonstationary panel case and gives a new result for models with individual effects. Underlying recent theory are asymptotics for multi-indexed processes in which both indexes may pass to infinity. We review some of the new limit theory that has been developed, show how it can be applied and give a new interpretation of individual effects in nonstationary panel data. Fundamental to the interpretation of much of the asymptotics is the concept of a panel regression coefficient which measures the long run average relation across a section of the panel. This concept is analogous to the statistical interpretation of the coefficient in a classical regression relation. A variety of nonstationary panel data models are discussed and the paper reviews the asymptotic properties of estimators in these various models. Some recent developments in panel unit root tests and stationary dynamic panel regression models are also reviewed.     

Immaculating the inconsistent estimator of slope parameter in measurement error model with replicated data

ABSTRACT

The measurement error model with replicated data on study as well as explanatory variables is considered. The measurement error variance associated with the explanatory variable is estimated using the complete data and the grouped data which is used for the construction of the consistent estimators of regression coefficient. These estimators are further used in constructing an almost unbiased estimator of regression coefficient. The large sample properties of these estimators are derived without assuming any distributional form of the measurement errors and the random error component under the setup of an ultrastructural model.