Maximum Likelihood Estimators and Likelihood Ratio Criteria for Multivariate Elliptically Contoured Distributions

1982 ◽  
Author(s):  
T. W. Anderson ◽  
Kai-Tai Fang
Keyword(s):  
Maximum Likelihood ◽  
Likelihood Ratio ◽  
Maximum Likelihood Estimators ◽  
Elliptically Contoured Distributions ◽  
Elliptically Contoured

Residual Diagrams Based on a Remarkably Simple Result Concerning the Variances of Maximum Likelihood Estimators

2002 ◽  
Vol 27 (1) ◽  
pp. 19-30 ◽  
Author(s):  
Erling B. Andersen
Keyword(s):  
Maximum Likelihood ◽  
Item Response Theory ◽  
Item Response ◽  
Likelihood Ratio ◽  
Rasch Model ◽  
Maximum Likelihood Estimators ◽  
Likelihood Ratio Tests ◽  
Random Errors ◽  
Simple Result ◽  
Polytomous Rasch Model

A remarkably simple result concerning variances of maximum likelihood (ML) estimators is presented. This result allows for construction of residual diagrams to evaluate whether ML estimators derived from independent samples can be assumed to be equal apart from random errors. Such diagrams can be used to supplement, for example, likelihood ratio tests for the equality of the parameters in independent samples. As an application, a model from item response theory is studied, the polytomous Rasch model. Residual diagrams for the equality of the variances in a one-way analysis of variance are also briefly mentioned.

STATISTICAL INFERENCE IN COINTEGRATED VECTOR AUTOREGRESSIVE MODELS WITH NONLINEAR TIME TRENDS IN COINTEGRATING RELATIONS

2001 ◽  
Vol 17 (2) ◽  
pp. 327-356 ◽  
Author(s):  
Pentti Saikkonen
Keyword(s):  
Maximum Likelihood ◽  
Statistical Inference ◽  
Likelihood Ratio ◽  
Lagrange Multiplier ◽  
Time Trends ◽  
Maximum Likelihood Estimators ◽  
Limiting Distributions ◽  
Vector Autoregressive Models ◽  
Vector Autoregressive ◽  
Short Run

This paper continues the work of Saikkonen (2001, Econometric Theory 17, 296–326) and develops an asymptotic theory of statistical inference in cointegrated vector autoregressive models with nonlinear time trends in cointegrating relations and general nonlinear parameter restrictions. Inference on parameters in cointegrating relations and short-run dynamics is studied separately. It is shown that Gaussian maximum likelihood estimators of parameters in cointegrating relations have mixed normal limiting distributions and that related Wald, Lagrange multiplier, and likelihood ratio tests for general nonlinear hypotheses have usual asymptotic chi-square distributions. These results are shown to hold even if parameters in the short-run dynamics are not identified. In that case suitable estimators of the information matrix have to be used to justify the application of Wald and Lagrange multiplier tests, whereas the likelihood ratio test is free of this difficulty. Similar results are also obtained when inference on parameters in the short-run dynamics is studied, although then Gaussian maximum likelihood estimators have usual normal limiting distributions. All results of the paper are proved without assuming existence of second partial derivatives of the likelihood function, and in some cases even differentiability with respect to nuisance parameters is not required.

Asymptotic Inference on the Moving Average Impact Matrix in Cointegrated 1(1) VAR Systems

1997 ◽  
Vol 13 (1) ◽  
pp. 79-118 ◽  
Author(s):  
Paolo Paruolo
Keyword(s):  
Maximum Likelihood ◽  
Likelihood Ratio ◽  
Moving Average ◽  
Maximum Likelihood Estimators ◽  
Asymptotic Distributions ◽  
Likelihood Ratio Tests ◽  
Common Trend ◽  
Average Impact ◽  
The Common ◽  
Definition Of

This paper addresses the problem of inference on the moving average impact matrix and on its row and column spaces in cointegrated 1(1) VAR processes. The choice of bases (i.e., the identification) of these spaces, which is of interest in the definition of the common trend structure of the system, is discussed. Maximum likelihood estimators and their asymptotic distributions are derived, making use of a relation between properly normalized bases of orthogonal spaces, a result that may be of separate interest. Finally, Wald-type tests are given, and their use in connection with existing likelihood ratio tests is discussed.

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