In the context of inference about a scalar parameter in the presence of nuisance parameters, some simple modifications for the signed root of the log-likelihood ratio statistic R are developed that reduce the order of...
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In the context of inference about a scalar parameter in the presence of nuisance parameters, some simple modifications for the signed root of the log-likelihood ratio statistic R are developed that reduce the order of error in the standard normal approximation to the distribution of R from O ( n -1/2 ) to O ( n -1 ). Barndorff-Nielsen has introduced a variable U such that the error in the standard normal approximation to the distribution of R + R -1 log( U/R ) is of order O ( n -3/2 ), but calculation of U requires the specification of an exact or approximate ancillary statistic A. This paper proposes an alternative variable to U , denoted by T , that is available without knowledge of A and satisfies T = U + O p ( n -1 ) in general. Thus the standard normal approximation to the distribution of R + R -1 log( T/R ) has error of order O ( n -1 ), and it can be used to construct approximate confidence limits having coverage error of order O ( n -1 ). In certain cases, however, T and U are identical. The derivation of T involves the Bayesian approach to constructing confidence limits considered by Welch and Peers, and Peers. Similar modifications for the signed root of the conditional likelihood ratio statistic are also developed, and these modifications are seen to be useful when a large number of nuisance parameters are present. Several examples are presented, including inference for natural parameters in exponential models and inference about location-scale models with type II censoring. In each case, the relationship between T and U is discussed. Numerical examples are also given, including inference for regressionmodels, inference about the means of log-normal distributions and inference for exponential lifetime models with type I censoring, where Barndorff-Nielsen's variable U is not available.
M-estimation of the multiple linear regression model under restrictions on the parameters is considered. Using a penalty function approach an iterative procedure for obtaining the estimates is proposed. The iterations...
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M-estimation of the multiple linear regression model under restrictions on the parameters is considered. Using a penalty function approach an iterative procedure for obtaining the estimates is proposed. The iterations are based on an iterative reweighted least squares procedure to which a modification for the restrictions is added. The influence function for the restricted estimator is derived. It is finally noted that the penalty function approach can be used to define a ridge type M-estimator.
Inference for a scalar parameter in the pressence of nuisance parameters requires high dimensional integrations of the joint density of the pivotal quantities. Recent development in asymptotic methods provides accurat...
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Inference for a scalar parameter in the pressence of nuisance parameters requires high dimensional integrations of the joint density of the pivotal quantities. Recent development in asymptotic methods provides accurate approximations for significance levels and thus confidence intervals for a scalar component parameter. In this paper, a simple, efficient and accurate numerical procedure is first developed for the location model and is then extended to the location-scale model and the linear regression model. This numerical procedure only requires a fine tabulation of the parameter and the observed log likelihood function, which can be either the full, marginal or conditional observed log likelihood function, as input and output is the corresponding significance function. Numerical results showed that this approximation is not only simple but also very accurate. It outperformed the usual approximations such as the signed likelihood ratio statistic, the maximum likelihood estimate and the score statistic.
Monthly runoff data of the southeast of North America were analysed by a linear regression model. The model is to be validated for the prognosis of runoff and the description of runoff and its influencing meteorologic...
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Monthly runoff data of the southeast of North America were analysed by a linear regression model. The model is to be validated for the prognosis of runoff and the description of runoff and its influencing meteorological and geoecological variables. The area studied was a region in the southeast of North America chosen on the basis of the climatological criteria of the Koppen-Geiger climate classification. A further differentiation was also made between the four seasons and the two predominant types of land cover. Every single examination of the individual influencing variables revealed reasonable results according to the physical context between runoff and its influencing variables. The comparison between measured runoff values and estimated runoff values also showed an excellent coincidence in 35 out of 39 cases examined.
In this paper, a new pre-test forecast for the linear regression model is developed which is based on stochastic, possibly incorrect, restrictions on the parameter vector. This procedure turns out to be an improvement...
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In this paper, a new pre-test forecast for the linear regression model is developed which is based on stochastic, possibly incorrect, restrictions on the parameter vector. This procedure turns out to be an improvement over forecasts depending on pre-test estimators. Finally, the mean square error matrix of our forecast is given.
Consider the linear regression model, y(i) = x(i)(T)beta0 + e(i), i = 1, ..., n, and an M-estimate beta of beta0 obtained by minimizing SIGMArho(y(i) - x(i)(T)beta), where rho is a convex function. Let S(n) = SIGMAx(i...
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Consider the linear regression model, y(i) = x(i)(T)beta0 + e(i), i = 1, ..., n, and an M-estimate beta of beta0 obtained by minimizing SIGMArho(y(i) - x(i)(T)beta), where rho is a convex function. Let S(n) = SIGMAx(i)x(i)T and r(n) = S(n)1/2(beta - beta0) - S(n)-1/2 SIGMAx(i)h(e(i)), where, with a suitable choice of h(.), the expression SIGMAx(i)h(e(i)) provides a linear representation of beta. Bahadur (1966) obtained the order of r(n) as n --> infinity when beta0 is a one-dimensional location parameter representing the median, and Babu (1989) proved a similar result for the general regression parameter estimated by the LAD (least absolute deviations) method. We obtain the stochastic order of r(n) as n --> infinity for a general M-estimate as defined above, which agrees with the results of Bahadur and Babu in the special cases considered by them.
Dodge and Jureckova (1987) showed that the estimation of linearregression parameter vector by a convex combination of least squares and least absolute deviation estimators could be adapted so that the resulting estim...
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Dodge and Jureckova (1987) showed that the estimation of linearregression parameter vector by a convex combination of least squares and least absolute deviation estimators could be adapted so that the resulting estimator achieves the minimum asymptotic variance in the model under consideration. The present paper considers the computational aspects of this adaptive estimator;an algorithm based on the iteratively reweighted least squares method is recommended and formally described. Technical details and an effect of the choice of a normalizing constant, appearing in the definition of the estimator, are also discussed. The behavior of the procedure is demonstrated on example.
This paper considers estimation of β in the regressionmodel y =Xβ+μ, where the error components in μ have the jointly multivariate Student‐t distribution. A family of James‐Stein type estimators (characterised ...
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The problem of combining linear and ellipsoidal restrictions in linearregression is investigated. Necessary and sufficient conditions for compactness of the restriction set are proved assuring the existence of a mini...
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