Pub Date : 1993-09-01DOI: 10.1111/J.2517-6161.1993.TB01951.X
Xizhi Wu, Zhen Luo
To examine global and local influence, and their relations for regression models, we study the perturbation-formed surface of a variable, such as the maximum likelihood estimate of a parameter, by evaluating the second derivative or curvature of the surface. (The corresponding slope of this surface is related to the curvature of the likelihood displacement surface of Cook.) Examples show that this approach, with the aid of plots, is helpful not only to discover influential cases including those hidden in an individual global sense but also to understand the nature of influence
{"title":"Second‐Order Approach to Local Influence","authors":"Xizhi Wu, Zhen Luo","doi":"10.1111/J.2517-6161.1993.TB01951.X","DOIUrl":"https://doi.org/10.1111/J.2517-6161.1993.TB01951.X","url":null,"abstract":"To examine global and local influence, and their relations for regression models, we study the perturbation-formed surface of a variable, such as the maximum likelihood estimate of a parameter, by evaluating the second derivative or curvature of the surface. (The corresponding slope of this surface is related to the curvature of the likelihood displacement surface of Cook.) Examples show that this approach, with the aid of plots, is helpful not only to discover influential cases including those hidden in an individual global sense but also to understand the nature of influence","PeriodicalId":17425,"journal":{"name":"Journal of the royal statistical society series b-methodological","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1993-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88250534","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1993-09-01DOI: 10.1111/J.2517-6161.1993.TB01944.X
D. Lindley, N. Singpurwalla
SUMMARY The life testing of items that exhibit a distribution of times to failure is undertaken for making decisions such as design qualification and reliability demonstration. In such contexts, procedures based on the Bayesian paradigm have assumed a common prior distribution of item reliability by both the consumer and the manufacturer. In this paper we consider the adversarial situation wherein both parties agree on a statistical model for lifetimes but use different prior distributions. We require that the consumer's criteria for acceptance and rejection be known to the manufacturer. We illustrate our approach via the case of exponentially distributed life lengths and relate it to the approach specified in standard MIL STD 781C.
{"title":"Adversarial life testing","authors":"D. Lindley, N. Singpurwalla","doi":"10.1111/J.2517-6161.1993.TB01944.X","DOIUrl":"https://doi.org/10.1111/J.2517-6161.1993.TB01944.X","url":null,"abstract":"SUMMARY The life testing of items that exhibit a distribution of times to failure is undertaken for making decisions such as design qualification and reliability demonstration. In such contexts, procedures based on the Bayesian paradigm have assumed a common prior distribution of item reliability by both the consumer and the manufacturer. In this paper we consider the adversarial situation wherein both parties agree on a statistical model for lifetimes but use different prior distributions. We require that the consumer's criteria for acceptance and rejection be known to the manufacturer. We illustrate our approach via the case of exponentially distributed life lengths and relate it to the approach specified in standard MIL STD 781C.","PeriodicalId":17425,"journal":{"name":"Journal of the royal statistical society series b-methodological","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1993-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90388315","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1993-09-01DOI: 10.1111/J.2517-6161.1993.TB01474.X
R. Eubank, W. Thomas
SUMMARY Diagnostic tests and plots are proposed for detecting heteroscedasticity in nonparametric regression. The large and small sample power properties are studied for a class of test statistics for the hypothesis of homogeneous variances. New diagnostic plots are also developed and illustrated.
{"title":"Detecting Heteroscedasticity in Nonparametric Regression","authors":"R. Eubank, W. Thomas","doi":"10.1111/J.2517-6161.1993.TB01474.X","DOIUrl":"https://doi.org/10.1111/J.2517-6161.1993.TB01474.X","url":null,"abstract":"SUMMARY Diagnostic tests and plots are proposed for detecting heteroscedasticity in nonparametric regression. The large and small sample power properties are studied for a class of test statistics for the hypothesis of homogeneous variances. New diagnostic plots are also developed and illustrated.","PeriodicalId":17425,"journal":{"name":"Journal of the royal statistical society series b-methodological","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1993-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78438000","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1993-09-01DOI: 10.1111/J.2517-6161.1993.TB01467.X
J. Besag, P. Green
on Wednesday, May 6th, 1992, Professor B. W. Silverman in the Chair] SUMMARY Markov chain Monte Carlo (MCMC) algorithms, such as the Gibbs sampler, have provided a Bayesian inference machine in image analysis and in other areas of spatial statistics for several years, founded on the pioneering ideas of Ulf Grenander. More recently, the observation that hyperparameters can be included as part of the updating schedule and the fact that almost any multivariate distribution is equivalently a Markov random field has opened the way to the use of MCMC in general Bayesian computation. In this paper, we trace the early development of MCMC in Bayesian inference, review some recent computational progress in statistical physics, based on the introduction of auxiliary variables, and discuss its current and future relevance in Bayesian applications. We briefly describe a simple MCMC implementation for the Bayesian analysis of agricultural field experiments, with which we have some practical experience.
{"title":"Spatial Statistics and Bayesian Computation","authors":"J. Besag, P. Green","doi":"10.1111/J.2517-6161.1993.TB01467.X","DOIUrl":"https://doi.org/10.1111/J.2517-6161.1993.TB01467.X","url":null,"abstract":"on Wednesday, May 6th, 1992, Professor B. W. Silverman in the Chair] SUMMARY Markov chain Monte Carlo (MCMC) algorithms, such as the Gibbs sampler, have provided a Bayesian inference machine in image analysis and in other areas of spatial statistics for several years, founded on the pioneering ideas of Ulf Grenander. More recently, the observation that hyperparameters can be included as part of the updating schedule and the fact that almost any multivariate distribution is equivalently a Markov random field has opened the way to the use of MCMC in general Bayesian computation. In this paper, we trace the early development of MCMC in Bayesian inference, review some recent computational progress in statistical physics, based on the introduction of auxiliary variables, and discuss its current and future relevance in Bayesian applications. We briefly describe a simple MCMC implementation for the Bayesian analysis of agricultural field experiments, with which we have some practical experience.","PeriodicalId":17425,"journal":{"name":"Journal of the royal statistical society series b-methodological","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1993-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80159115","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1993-09-01DOI: 10.1111/J.2517-6161.1993.TB01468.X
W. Gilks, D. Clayton, D. Spiegelhalter, N. Best, A. McNeil, L. Sharples, A. Kirby
SUMMARY We review applications of Gibbs sampling in medicine, involving longitudinal, spatial, covariate measurement and survival models. Applications in immunology, pharmacology, transplantation, cancer screening, industrial epidemiology and genetic epidemiology are discussed.
{"title":"Modelling Complexity: Applications of Gibbs Sampling in Medicine","authors":"W. Gilks, D. Clayton, D. Spiegelhalter, N. Best, A. McNeil, L. Sharples, A. Kirby","doi":"10.1111/J.2517-6161.1993.TB01468.X","DOIUrl":"https://doi.org/10.1111/J.2517-6161.1993.TB01468.X","url":null,"abstract":"SUMMARY We review applications of Gibbs sampling in medicine, involving longitudinal, spatial, covariate measurement and survival models. Applications in immunology, pharmacology, transplantation, cancer screening, industrial epidemiology and genetic epidemiology are discussed.","PeriodicalId":17425,"journal":{"name":"Journal of the royal statistical society series b-methodological","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1993-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86029449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1993-09-01DOI: 10.1111/J.2517-6161.1993.TB01949.X
N. Gordon, Adrian F. M. Smith
SUMMARY A new recursive estimation procedure is proposed for the location of a dynamic linear model with non-normal errors. The procedure is a modification of a modal approximation algorithm, which is shown to be prone to instabilities. The modification is motivated by a notion of posterior modal consistency. Many researchers have considered the problem of sequential updating of the first two posterior moments of the location vector of a dynamic linear time series model with non-normal errors. Such models are motivated by considerations of realism or robustness (in particular accommodation of outliers). In this paper, we shall re- examine, for the scalar case, a posterior modal approximation scheme discussed by West (1981) and Fahrmeir and Kaufmann (1991), which in practice has been found to be prone to producing wild instabilities in the recursive estimates. We identify the cause of this and present a modified form of recursive approximation which avoids this problem. We assume fully specified measurement and system models, a standard assumption for the autonomous tracking filters that we have in mind as applications. An extension to unknown measurement variance, along the lines of West (1981), is straightforward.
{"title":"Approximate Non-Gaussian Bayesian Estimation and Modal Consistency","authors":"N. Gordon, Adrian F. M. Smith","doi":"10.1111/J.2517-6161.1993.TB01949.X","DOIUrl":"https://doi.org/10.1111/J.2517-6161.1993.TB01949.X","url":null,"abstract":"SUMMARY A new recursive estimation procedure is proposed for the location of a dynamic linear model with non-normal errors. The procedure is a modification of a modal approximation algorithm, which is shown to be prone to instabilities. The modification is motivated by a notion of posterior modal consistency. Many researchers have considered the problem of sequential updating of the first two posterior moments of the location vector of a dynamic linear time series model with non-normal errors. Such models are motivated by considerations of realism or robustness (in particular accommodation of outliers). In this paper, we shall re- examine, for the scalar case, a posterior modal approximation scheme discussed by West (1981) and Fahrmeir and Kaufmann (1991), which in practice has been found to be prone to producing wild instabilities in the recursive estimates. We identify the cause of this and present a modified form of recursive approximation which avoids this problem. We assume fully specified measurement and system models, a standard assumption for the autonomous tracking filters that we have in mind as applications. An extension to unknown measurement variance, along the lines of West (1981), is straightforward.","PeriodicalId":17425,"journal":{"name":"Journal of the royal statistical society series b-methodological","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1993-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90735863","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1993-09-01DOI: 10.1111/J.2517-6161.1993.TB01943.X
R. Taplin
We propose a computationally efficient method for calculating the likelihoods of a time series under many submodels, each of which assumes a patch of outliers or level shifts. We assume a state space representation of the time series model with a Bayesian-type treatment of anomalies. The calculations form the basis for an efficient and robust estimation procedure. The method is also applicable to linear regression with correlated errors and is illustrated with two examples
{"title":"Robust Likelihood Calculation for Time Series","authors":"R. Taplin","doi":"10.1111/J.2517-6161.1993.TB01943.X","DOIUrl":"https://doi.org/10.1111/J.2517-6161.1993.TB01943.X","url":null,"abstract":"We propose a computationally efficient method for calculating the likelihoods of a time series under many submodels, each of which assumes a patch of outliers or level shifts. We assume a state space representation of the time series model with a Bayesian-type treatment of anomalies. The calculations form the basis for an efficient and robust estimation procedure. The method is also applicable to linear regression with correlated errors and is illustrated with two examples","PeriodicalId":17425,"journal":{"name":"Journal of the royal statistical society series b-methodological","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1993-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81655850","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1993-09-01DOI: 10.1111/J.2517-6161.1993.TB01471.X
N. Fisher, A. D. Lunn, S. Davies
In an earlier paper, Fisher defined the notion of a median direction for data from a unimodal distribution of three-dimensional unit vectors and studied its statistical properties. Corresponding notions of a median axis for bipolar axial or great circle data were also suggested but not analysed. This paper gives a statistical treatment of spherical median axes, including their asymptotic relative efficiencies, and a comparison with the customary principal or polar axis estimators using asymptotic relative efficiencies
{"title":"Spherical median axes","authors":"N. Fisher, A. D. Lunn, S. Davies","doi":"10.1111/J.2517-6161.1993.TB01471.X","DOIUrl":"https://doi.org/10.1111/J.2517-6161.1993.TB01471.X","url":null,"abstract":"In an earlier paper, Fisher defined the notion of a median direction for data from a unimodal distribution of three-dimensional unit vectors and studied its statistical properties. Corresponding notions of a median axis for bipolar axial or great circle data were also suggested but not analysed. This paper gives a statistical treatment of spherical median axes, including their asymptotic relative efficiencies, and a comparison with the customary principal or polar axis estimators using asymptotic relative efficiencies","PeriodicalId":17425,"journal":{"name":"Journal of the royal statistical society series b-methodological","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1993-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82283572","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1993-09-01DOI: 10.1111/J.2517-6161.1993.TB01950.X
J. Kunert, B. P. Utzig
SUMMARY There is concern that the usual analysis of crossover designs with more than two treatments is subject to bias due to correlations between the measurements on the same experimental units. It has been shown by Kunert in the special case of balanced Latin squares that this bias can be present but that it is limited. Extending the work of Kunert we show in the present paper that there is a constant X* which guarantees that the estimate for the variance of any treatment contrast from the usual model multiplied by X* has an expectation which is at least as big as the true variance. This result holds for any within-unit covariance structure and it is valid for a class of commonly applied designs, allowing for fewer periods than treatments. The constant X* depends on the number of units, periods and treatments but not on the data or the unknown covariance matrix. We also deal with the effect that our result can have on tests for hypotheses about treatment contrasts.
{"title":"Estimation of Variance in Crossover Designs","authors":"J. Kunert, B. P. Utzig","doi":"10.1111/J.2517-6161.1993.TB01950.X","DOIUrl":"https://doi.org/10.1111/J.2517-6161.1993.TB01950.X","url":null,"abstract":"SUMMARY There is concern that the usual analysis of crossover designs with more than two treatments is subject to bias due to correlations between the measurements on the same experimental units. It has been shown by Kunert in the special case of balanced Latin squares that this bias can be present but that it is limited. Extending the work of Kunert we show in the present paper that there is a constant X* which guarantees that the estimate for the variance of any treatment contrast from the usual model multiplied by X* has an expectation which is at least as big as the true variance. This result holds for any within-unit covariance structure and it is valid for a class of commonly applied designs, allowing for fewer periods than treatments. The constant X* depends on the number of units, periods and treatments but not on the data or the unknown covariance matrix. We also deal with the effect that our result can have on tests for hypotheses about treatment contrasts.","PeriodicalId":17425,"journal":{"name":"Journal of the royal statistical society series b-methodological","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1993-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83373409","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1993-09-01DOI: 10.1111/J.2517-6161.1993.TB01472.X
G. Macaskill
SUMMARY The problem of partial non-linear regression is used to provide an example of McCullagh and Tibshirani's profile likelihood adjustment for more than one parameter of interest, where for one linear nuisance parameter an expression for the adjusted profile likelihood can be derived.
{"title":"A Note on Adjusted Profile Likelihoods in Non-linear Regression","authors":"G. Macaskill","doi":"10.1111/J.2517-6161.1993.TB01472.X","DOIUrl":"https://doi.org/10.1111/J.2517-6161.1993.TB01472.X","url":null,"abstract":"SUMMARY The problem of partial non-linear regression is used to provide an example of McCullagh and Tibshirani's profile likelihood adjustment for more than one parameter of interest, where for one linear nuisance parameter an expression for the adjusted profile likelihood can be derived.","PeriodicalId":17425,"journal":{"name":"Journal of the royal statistical society series b-methodological","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1993-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87229597","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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