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HigherOrder Asymptotics Unleashed Software Design for Nonlinear Heteroscedastic Regression
, 2001
"... One of the main criticisms against the use of higherorder asymptotics is that the algebraic expressions involved are far too complex to be derived by hand in a reasonable amount of time. A further drawback is that the results are so closely tuned to the specic problem at hand that they almost alway ..."
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One of the main criticisms against the use of higherorder asymptotics is that the algebraic expressions involved are far too complex to be derived by hand in a reasonable amount of time. A further drawback is that the results are so closely tuned to the specic problem at hand that they almost always exclude the possibility of transferring available computer code to a dierent though similar problem. The aim of this paper is to show that higherorder asymptotics can be implemented in a general and exible way so as to provide easytouse and selfcontained software useful in routine data analysis. The programming strategy we develop easily applies to many parametric models. We illustrate it by describing the design of the core routines of the nlreg section of the SPLUS library HOA (HOA, 2000) which implements higherorder solutions for nonlinear heteroscedastic regression models. Keywords: Asymptotic Theory, HigherOrder Solution, Symbolic Computation, SPLUS. 1 Introduction The mai...
Accurate Parametric Inference for Small Samples
, 2008
"... We outline how modern likelihood theory, which provides essentially exact inferences in a variety of parametric statistical problems, may routinely be applied in practice. Although the likelihood procedures are based on analytical asymptotic approximations, the focus of this paper is not on theory ..."
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We outline how modern likelihood theory, which provides essentially exact inferences in a variety of parametric statistical problems, may routinely be applied in practice. Although the likelihood procedures are based on analytical asymptotic approximations, the focus of this paper is not on theory but on implementation and applications. Numerical illustrations are given for logistic regression, nonlinear models, and linear nonnormal models, and we describe a sampling approach for the third of these classes. In the case of logistic regression, we argue that approximations are often more appropriate than ‘exact’ procedures, even when these exist.
HigherOrder Asymptotics Unleashed
, 2000
"... One of the main criticism against the use of higherorder asymptotics is that the algebraic expression involved are far too complex to be derived by hand in a reasonable amount of time. The results are furthermore so closely tuned to the specific problem at stake that they exclude the possibility of ..."
Abstract
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One of the main criticism against the use of higherorder asymptotics is that the algebraic expression involved are far too complex to be derived by hand in a reasonable amount of time. The results are furthermore so closely tuned to the specific problem at stake that they exclude the possibility of reusing available computer code in a different though similar problem. The aim of this paper is to show that higherorder asymptotics can be implemented in a general and flexible way such as to provide easytouse and selfcontained software to be used in routine data analysis. We will demonstrate this by illustrating the main programming strategy that underlies the SPLUS library HOA. This library implements higherorder solutions for three main parametric model classes: logistic and loglinear models, linear nonnormal models and nonlinear heteroscedastic regression models.
Printed in Great Britain Effects of the reference set on frequentist inferences
"... We employ secondorder likelihood asymptotics to investigate how ideal frequentist inferences depend on the probability model for the data through more than the likelihood function, referring to this as the effect of the reference set. There are two aspects of higherorder corrections to firstorder ..."
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We employ secondorder likelihood asymptotics to investigate how ideal frequentist inferences depend on the probability model for the data through more than the likelihood function, referring to this as the effect of the reference set. There are two aspects of higherorder corrections to firstorder likelihood methods, namely (i) that involving effects of fitting nuisance parameters and leading to the modified profile likelihood, and (ii) another part pertaining to limitation in adjusted information. Generally, each of these involves a firstorder adjustment depending on the reference set. However, we show that, for some important settings, likelihoodirrelevant model specifications have a secondorder effect on both of these adjustments; this result includes specification of the censoring model for survival data. On the other hand, for sequential experiments the likelihoodirrelevant specification of the stopping rule has a secondorder effect on adjustment (i) but a firstorder effect on adjustment (ii). These matters raise the issue of what are ‘ideal ’ frequentist inferences, since consideration of ‘exact ’ frequentist inferences will not suffice. We indicate that to second order ideal frequentist inferences may be based on the distribution of the ordinary likelihood ratio statistic, without commonly considered adjustments thereto. Some key words: Censoring model; Higherorder asymptotics; Likelihood asymptotics; Likelihood principle; Modified directed deviance; Modified profile likelihood; Sequential experiment.
Institution that will administer the grant
, 1508
"... For NSERC office use only Form 101 Application for a Grant Send to NSERC with your attachments, if applicable ..."
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For NSERC office use only Form 101 Application for a Grant Send to NSERC with your attachments, if applicable
Asymptotically Normal Vectors by
, 2007
"... Abstract We obtain the Edgeworth expansion for P(n 1/2 ( ˆ θ − θ) < x) and its derivatives, and the tilted Edgeworth (or saddlepoint or small sample) expansion for P ( ˆ θ < x) and its derivatives where ˆθ is any vector estimate having the standard cumulant expansions in powers of n −1. ..."
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Abstract We obtain the Edgeworth expansion for P(n 1/2 ( ˆ θ − θ) < x) and its derivatives, and the tilted Edgeworth (or saddlepoint or small sample) expansion for P ( ˆ θ < x) and its derivatives where ˆθ is any vector estimate having the standard cumulant expansions in powers of n −1.
THE 2000 WALD MEMORIAL LECTURES ASYMPTOTICS AND THE THEORY OF INFERENCE
"... Asymptotic analysis has always been very useful for deriving distributions in statistics in cases where the exact distribution is unavailable. More importantly, asymptotic analysis can also provide insight into the inference process itself, suggesting what information is available and how this infor ..."
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Asymptotic analysis has always been very useful for deriving distributions in statistics in cases where the exact distribution is unavailable. More importantly, asymptotic analysis can also provide insight into the inference process itself, suggesting what information is available and how this information may be extracted. The development of likelihood inference over the past twentysome years provides an illustration of the interplay between techniques of approximation and statistical theory. 1. Introduction. The
Tests Concerning Equicorrelation Matrices with Grouped Normal Data
"... Key Words: intraclass correlation; likelihood ratio test; maximum likelihood estimation; score test; smallsample inference. This paper considers three practical hypotheses involving the equicorrelation matrix for grouped normal data. We obtain statistics and computing formulae for common test proce ..."
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Key Words: intraclass correlation; likelihood ratio test; maximum likelihood estimation; score test; smallsample inference. This paper considers three practical hypotheses involving the equicorrelation matrix for grouped normal data. We obtain statistics and computing formulae for common test procedures such as the score test and the likelihood ratio test. In addition, statistics and computing formulae are obtained for various small sample procedures as proposed in Skovgaard (2001). The properties of the tests for each of the three hypotheses are compared using Monte Carlo simulations. 1 1
Multivariate Analysis Tests Concerning Equicorrelation Matrices with Grouped Normal Data
"... This article considers three practical hypotheses involving the equicorrelation matrix for grouped normal data. We obtain statistics and computing formulae for common test procedures such as the score test and the likelihood ratio test. In addition, statistics and computing formulae are obtained for ..."
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This article considers three practical hypotheses involving the equicorrelation matrix for grouped normal data. We obtain statistics and computing formulae for common test procedures such as the score test and the likelihood ratio test. In addition, statistics and computing formulae are obtained for various small sample procedures as proposed in Skovgaard (2001). The properties of the tests for each of the three hypotheses are compared using Monte Carlo simulations.