This paper is devoted to the multivariate estimation of a vector of Poisson means. A novel loss f... more This paper is devoted to the multivariate estimation of a vector of Poisson means. A novel loss function that penalises bad estimates of each of the parameters and the sum (or equivalently the mean) of the parameters is introduced. Under this loss function, a class of minimax estimators that uniformly dominate the maximum likelihood estimator is derived. Crucially, these methods have the property that for estimating a given component parameter, the full data vector is utilised. Estimators in this class can be fine-tuned to limit shrinkage away from the maximum likelihood estimator, thereby avoiding implausible estimates of the sum of the parameters. Further light is shed on this new class of estimators by showing that it can be derived by Bayesian and empirical Bayesian methods. In particular, we exhibit a generalisation of the Clevenson-Zidek estimator, and prove its admissibility. Moreover, a class of prior distributions for which the Bayes estimators uniformly dominate the maximum likelihood estimator under the new loss function is derived. A section is included involving weighted loss functions, notably also leading to a procedure improving uniformly on the maximum likelihood method in an infinite-dimensional setup. Importantly, some of our methods lead to constructions of new multivariate models for both rate parameters and count observations. Finally, estimators that shrink the usual estimators towards a data based point in the parameter space are derived and compared.
Statistical Inference for Stochastic Processes, Dec 1, 2005
Two characterisations of a random mean from a Dirichlet process, as a limit of finite sums of a s... more Two characterisations of a random mean from a Dirichlet process, as a limit of finite sums of a simple symmetric form and as a solution of a certain stochastic equation, are developed and investigated. These are used to reach results on and new insights into the distributions of such random means. In particular, identities involving functional transforms and recursive moment formulae are established. Furthermore, characterisations for several choices of the Dirichlet process parameter (leading to symmetric, unimodal, stable, and finite mixture distributions) are provided. Our methods lead to exact simulation recipes for prior and posterior random means, an approximation algorithm for the exact densities of these means, and limiting normality theorems for posterior distributions. The theory also extends to mixtures of Dirichlet processes and to the case of several random means simultaneously.
This paper is devoted to the multivariate estimation of a vector of Poisson means. A novel loss f... more This paper is devoted to the multivariate estimation of a vector of Poisson means. A novel loss function that penalises bad estimates of each of the parameters and the sum (or equivalently the mean) of the parameters is introduced. Under this loss function, a class of minimax estimators that uniformly dominate the maximum likelihood estimator is derived. Crucially, these methods have the property that for estimating a given component parameter, the full data vector is utilised. Estimators in this class can be fine-tuned to limit shrinkage away from the maximum likelihood estimator, thereby avoiding implausible estimates of the sum of the parameters. Further light is shed on this new class of estimators by showing that it can be derived by Bayesian and empirical Bayesian methods. In particular, we exhibit a generalisation of the Clevenson-Zidek estimator, and prove its admissibility. Moreover, a class of prior distributions for which the Bayes estimators uniformly dominate the maximum likelihood estimator under the new loss function is derived. A section is included involving weighted loss functions, notably also leading to a procedure improving uniformly on the maximum likelihood method in an infinite-dimensional setup. Importantly, some of our methods lead to constructions of new multivariate models for both rate parameters and count observations. Finally, estimators that shrink the usual estimators towards a data based point in the parameter space are derived and compared.
The best known methods for estimating hazard rate functions in survival analysis models are eithe... more The best known methods for estimating hazard rate functions in survival analysis models are either purely parametric or purely nonparametric. The parametric ones are sometimes too biased while the nonparametric ones are sometimes too variable. In the present paper a certain semiparametric approach to hazard rate estimation, proposed in Hjort (1991), is developed further, aiming to combine parametric and nonparametric features. It uses a dynamic local likelihood approach to fit the locally most suitable member in a given parametric class of hazard rates, and amounts to a version of non parametric parameter smoothing within the parametric class. Thus the parametric hazard rate estimate at time 8 inserts a parameter estimate that also depends on 8. We study bias and variance properties of the resulting estimator and methods for choosing the local smoothing parameter. It is shown that dynamic likelihood estimation often leads to better performance than the purely nonparametric methods, while also having capacity for not losing much to the parametric methods in cases where the model being smoothed is adequate.
This paper develops a class of Bayesian non-and semiparametric methods for estimating regression ... more This paper develops a class of Bayesian non-and semiparametric methods for estimating regression curves and surfaces. The main idea is to model the regression as locally linear, and then place suitable local priors on the local parameters. The method requires the posterior distribution of the local parameters given local data, and this is found via a suitably defined local likelihood function. When the width of the local data window is large the methods reduce to familiar fully parametric Bayesian methods, and when the width is small the estimators are essentially nonparametric. When noninformative reference priors are used the resulting estimators coincide with recently developed well-performing local weighted least squares methods for nonparametric regression. Each local prior distribution needs in general a centre parameter and a variance parameter. Of particular interest are versions of the scheme that are more or less automatic and objective in the sense that they do not require subjective specifications of prior parameters. We therefore develop empirical Bayes methods to obtain the variance parameter and a hierarchical Bayes method to account for uncertainty in the choice of centre parameter. There are several possible versions of the general programme, and a number of its specialisations are discussed. Some of these are shown to be capable of outperforming standard nonparametric regression methods, particularly in situations with several covariates.
We introduce a model for recurrent event data subject to left-, right-, and intermittent-censorin... more We introduce a model for recurrent event data subject to left-, right-, and intermittent-censoring. The observations consist of binary sequences (along with covariates) for each individual under study. These sequences are modeled as generated by latent Ornstein-Uhlenbeck processes being above or below certain thresholds. Features of the latent process and the thresholds are taken as functions of covariates, allowing the researcher to distinguish factors that have an effect on the frailty, from those that have an effect on the variability, of the observational unit. Inference is achieved by a quasi-likelihood approach, for which consistency and asymptotic normality is established. An advantage of our model is that particularities regarding the censoring need not be taken actively into account, and that it is well suited for situations where the individuals under study are irregularly and asynchronously observed. The motivation for our model came from a dataset pertaining to the incidence of diarrhoea among Brazilian children growing up under rather harsh conditions. We analyze these data with our model and contrast the results with an intensity-based counting process analysis of the same data.
Recently, Efron and Tibshirani (1996) proposed a semiparametric density estimator, which works by... more Recently, Efron and Tibshirani (1996) proposed a semiparametric density estimator, which works by multiplying an initial kernel type estimate with a parametric exponential type correction factor, chosen so as to match certain empirical moments. While Efron and Tibshirani investigate and illustrate many aspects of their method, the basic questions of performance, and comparison with other density estimators, were not directly addressed in their article. The purpose of this paper is to provide formulae for bias and variance and hence mean squared error for the estimator. This additional insight into the method makes it easy to compare its performance with that of other recently proposed semiparametric constructions. A brief comparison study is carried out here. It indicates that the new method, used with lower order polynomials in the exponential correction term, is often better than the kernel estimator, in a reasonable neighbourhood around the normal distribution, but that its performance as a density estimator is more than equalled by other methods.
The Olympic 500 meter sprint competition is the 'Formula One event' of speed skating, and is watc... more The Olympic 500 meter sprint competition is the 'Formula One event' of speed skating, and is watched by millions of television viewers. A draw decides who should start in inner lane and who in outer lane. Many skaters dread the last inner lane, where they need to tackle heavier centrifugal forces than their companions in the last outer lane, at maximum speed around 55 km/hour, at a time when fatigue may set in. The aim of this article is to investigate this potential difference between last inner and last outer lane. For this purpose data from eleven Sprint World Championships 1984-1994 are exploited. A bivariate mixed effects model is used that in addition to the inner-outer lane information takes account of different ice and weather conditions on different days, unequal levels for different skaters, and the passing times for the first 100 meter. The underlying 'unfairness parameter', estimated with optimal precision, is about 0.05 seconds, and is indeed significantly different from zero; it is about three times as large as its estimated standard deviation. Results from the work reported on here played a decisive role in leading the International Skating Union and the International Olympic Committee to change the rules for the 500 meter sprint event; as of the Nagano 1998 Olympic Games, the sprinters are to skate twice, with one start in inner lane and one in outer lane. The best average result determines the final list, and the best skaters from the first run are paired to skate last in the second run. It has also been decided that the same rules shall apply for the single distance 500 meter World Championships; these are arranged yearly from 1996 onwards.
A family of random probabilities is defined and studied. This family contains the Dirichlet proce... more A family of random probabilities is defined and studied. This family contains the Dirichlet process as a special case, corresponding to an inner point in the appropriate parameter space. The extension makes it possible to have.random means with larger or smaller skewnesses as compared to skewnesses under the Dirichlet prior, and also in other ways amounts to additional modelling flexibility. The usefulness of such random probabilities for use in nonparametric Bayesian statistics is discussed. The posterior distribution is complicated, but inference can nevertheless be carried out via simulation, and some exact formulae are derived for the case of random means. The class of nonparametric priors provides an instructive example where the speed with which the posterior forgets its prior with increasing data sample size depends on special aspects of the prior, which is a different situation from that of parametric inference.
Journal of Statistical Planning and Inference, May 1, 2018
This is the guest editors' general introduction to a Special Issue of the Journal of Statistical ... more This is the guest editors' general introduction to a Special Issue of the Journal of Statistical Planning and Inference, dedicated to confidence distributions and related themes. Confidence distributions (CDs) are distributions for parameters of interest, constructed via a statistical model after analysing the data. As such they serve the same purpose for the frequentist statisticians as the posterior distributions for the Bayesians. There have been several attempts in the literature to put up a clear theory for such confidence distributions, from Fisher's fiducial inference and onwards. There are certain obstacles and difficulties involved in these attempts, both conceptually and operationally, which have contributed to the CDs being slow in entering statistical mainstream. Recently there is a renewed surge of interest in CDs and various related themes, however, reflected in both series of new methodological research, advanced applications to substantive sciences, and dissemination and communication via workshops and conferences. The present special issue of the JSPI is a collection of papers emanating from the Inference With Confidence workshop in Oslo, May 2015. Several of the papers appearing here were first presented at that workshop. The present collection includes however also new research papers from other scholars in the field.
Previous chapters have developed concepts and methodology pertaining to confidence distributions ... more Previous chapters have developed concepts and methodology pertaining to confidence distributions and related inference procedures. Some of these methods take the form of generally applicable recipes, via log-likelihood profiles, deviances and first-order large-sample approximations to the distribution of estimators of the focus estimands in question. Sometimes these recipes are too coarse and are in need of modification and perfection, however, which is the topic of the present chapter. We discuss methods based on mean and bias corrected deviance curves, t-bootstrapping, a certain acceleration and bias correction method, approximations via expansions, prepivoting and modified likelihood profiles. The extent to which these methods lead to improvements is also briefly illustrated and discussed. Introduction Uniformly most powerful exact inference is in the presence of nuisance parameters available only in regular exponential models for continuous data and other models with Neyman structure, as discussed and exemplified in Chapter 5. Exact confidence distributions exist in a wider class of models, but need not be canonical. The estimate of location based on the Wilcoxon statistic, for example, has an exact known distribution in the location model where only symmetry is assumed; see Section 11.4. In more complex models, the statistic on which to base the confidence distribution might be chosen on various grounds: the structure of the likelihood function, perceived robustness, asymptotic properties, computational feasibility, perspective and tradition of the study. In the given model, with finite data, it might be difficult to obtain an exact confidence distribution based on the chosen statistic. As we shall see there are various techniques available for obtaining approximate confidence distributions and confidence likelihoods, however, improving on the first-order ones worked with in Chapters 3–4. Bootstrapping, simulation and asymptotics are useful tools in calculating approximate confidence distributions and in characterising their power properties. When an estimator, often the maximum likelihood estimator of the interest parameter, is used as the statistic on which the confidence distribution is based, bootstrapping provides an estimate of the sampling distribution of the statistic. This empirical sampling distribution can be turned into an approximate confidence distribution in several ways, which we address in the text that follows.
This article extends the scope of empirical likelihood methodology in three directions: to allow ... more This article extends the scope of empirical likelihood methodology in three directions: to allow for plug-in estimates of nuisance parameters in estimating equations, slower than √ n-rates of convergence, and settings in which there are a relatively large number of estimating equations compared to the sample size. Calibrating empirical likelihood confidence regions with plugin is sometimes intractable due to the complexity of the asymptotics, so we introduce a bootstrap approximation that can be used in such situations. We provide a range of examples from survival analysis and nonparametric statistics to illustrate the main results.
This paper is devoted to the multivariate estimation of a vector of Poisson means. A novel loss f... more This paper is devoted to the multivariate estimation of a vector of Poisson means. A novel loss function that penalises bad estimates of each of the parameters and the sum (or equivalently the mean) of the parameters is introduced. Under this loss function, a class of minimax estimators that uniformly dominate the maximum likelihood estimator is derived. Crucially, these methods have the property that for estimating a given component parameter, the full data vector is utilised. Estimators in this class can be fine-tuned to limit shrinkage away from the maximum likelihood estimator, thereby avoiding implausible estimates of the sum of the parameters. Further light is shed on this new class of estimators by showing that it can be derived by Bayesian and empirical Bayesian methods. In particular, we exhibit a generalisation of the Clevenson-Zidek estimator, and prove its admissibility. Moreover, a class of prior distributions for which the Bayes estimators uniformly dominate the maximum likelihood estimator under the new loss function is derived. A section is included involving weighted loss functions, notably also leading to a procedure improving uniformly on the maximum likelihood method in an infinite-dimensional setup. Importantly, some of our methods lead to constructions of new multivariate models for both rate parameters and count observations. Finally, estimators that shrink the usual estimators towards a data based point in the parameter space are derived and compared.
Statistical Inference for Stochastic Processes, Dec 1, 2005
Two characterisations of a random mean from a Dirichlet process, as a limit of finite sums of a s... more Two characterisations of a random mean from a Dirichlet process, as a limit of finite sums of a simple symmetric form and as a solution of a certain stochastic equation, are developed and investigated. These are used to reach results on and new insights into the distributions of such random means. In particular, identities involving functional transforms and recursive moment formulae are established. Furthermore, characterisations for several choices of the Dirichlet process parameter (leading to symmetric, unimodal, stable, and finite mixture distributions) are provided. Our methods lead to exact simulation recipes for prior and posterior random means, an approximation algorithm for the exact densities of these means, and limiting normality theorems for posterior distributions. The theory also extends to mixtures of Dirichlet processes and to the case of several random means simultaneously.
This paper is devoted to the multivariate estimation of a vector of Poisson means. A novel loss f... more This paper is devoted to the multivariate estimation of a vector of Poisson means. A novel loss function that penalises bad estimates of each of the parameters and the sum (or equivalently the mean) of the parameters is introduced. Under this loss function, a class of minimax estimators that uniformly dominate the maximum likelihood estimator is derived. Crucially, these methods have the property that for estimating a given component parameter, the full data vector is utilised. Estimators in this class can be fine-tuned to limit shrinkage away from the maximum likelihood estimator, thereby avoiding implausible estimates of the sum of the parameters. Further light is shed on this new class of estimators by showing that it can be derived by Bayesian and empirical Bayesian methods. In particular, we exhibit a generalisation of the Clevenson-Zidek estimator, and prove its admissibility. Moreover, a class of prior distributions for which the Bayes estimators uniformly dominate the maximum likelihood estimator under the new loss function is derived. A section is included involving weighted loss functions, notably also leading to a procedure improving uniformly on the maximum likelihood method in an infinite-dimensional setup. Importantly, some of our methods lead to constructions of new multivariate models for both rate parameters and count observations. Finally, estimators that shrink the usual estimators towards a data based point in the parameter space are derived and compared.
The best known methods for estimating hazard rate functions in survival analysis models are eithe... more The best known methods for estimating hazard rate functions in survival analysis models are either purely parametric or purely nonparametric. The parametric ones are sometimes too biased while the nonparametric ones are sometimes too variable. In the present paper a certain semiparametric approach to hazard rate estimation, proposed in Hjort (1991), is developed further, aiming to combine parametric and nonparametric features. It uses a dynamic local likelihood approach to fit the locally most suitable member in a given parametric class of hazard rates, and amounts to a version of non parametric parameter smoothing within the parametric class. Thus the parametric hazard rate estimate at time 8 inserts a parameter estimate that also depends on 8. We study bias and variance properties of the resulting estimator and methods for choosing the local smoothing parameter. It is shown that dynamic likelihood estimation often leads to better performance than the purely nonparametric methods, while also having capacity for not losing much to the parametric methods in cases where the model being smoothed is adequate.
This paper develops a class of Bayesian non-and semiparametric methods for estimating regression ... more This paper develops a class of Bayesian non-and semiparametric methods for estimating regression curves and surfaces. The main idea is to model the regression as locally linear, and then place suitable local priors on the local parameters. The method requires the posterior distribution of the local parameters given local data, and this is found via a suitably defined local likelihood function. When the width of the local data window is large the methods reduce to familiar fully parametric Bayesian methods, and when the width is small the estimators are essentially nonparametric. When noninformative reference priors are used the resulting estimators coincide with recently developed well-performing local weighted least squares methods for nonparametric regression. Each local prior distribution needs in general a centre parameter and a variance parameter. Of particular interest are versions of the scheme that are more or less automatic and objective in the sense that they do not require subjective specifications of prior parameters. We therefore develop empirical Bayes methods to obtain the variance parameter and a hierarchical Bayes method to account for uncertainty in the choice of centre parameter. There are several possible versions of the general programme, and a number of its specialisations are discussed. Some of these are shown to be capable of outperforming standard nonparametric regression methods, particularly in situations with several covariates.
We introduce a model for recurrent event data subject to left-, right-, and intermittent-censorin... more We introduce a model for recurrent event data subject to left-, right-, and intermittent-censoring. The observations consist of binary sequences (along with covariates) for each individual under study. These sequences are modeled as generated by latent Ornstein-Uhlenbeck processes being above or below certain thresholds. Features of the latent process and the thresholds are taken as functions of covariates, allowing the researcher to distinguish factors that have an effect on the frailty, from those that have an effect on the variability, of the observational unit. Inference is achieved by a quasi-likelihood approach, for which consistency and asymptotic normality is established. An advantage of our model is that particularities regarding the censoring need not be taken actively into account, and that it is well suited for situations where the individuals under study are irregularly and asynchronously observed. The motivation for our model came from a dataset pertaining to the incidence of diarrhoea among Brazilian children growing up under rather harsh conditions. We analyze these data with our model and contrast the results with an intensity-based counting process analysis of the same data.
Recently, Efron and Tibshirani (1996) proposed a semiparametric density estimator, which works by... more Recently, Efron and Tibshirani (1996) proposed a semiparametric density estimator, which works by multiplying an initial kernel type estimate with a parametric exponential type correction factor, chosen so as to match certain empirical moments. While Efron and Tibshirani investigate and illustrate many aspects of their method, the basic questions of performance, and comparison with other density estimators, were not directly addressed in their article. The purpose of this paper is to provide formulae for bias and variance and hence mean squared error for the estimator. This additional insight into the method makes it easy to compare its performance with that of other recently proposed semiparametric constructions. A brief comparison study is carried out here. It indicates that the new method, used with lower order polynomials in the exponential correction term, is often better than the kernel estimator, in a reasonable neighbourhood around the normal distribution, but that its performance as a density estimator is more than equalled by other methods.
The Olympic 500 meter sprint competition is the 'Formula One event' of speed skating, and is watc... more The Olympic 500 meter sprint competition is the 'Formula One event' of speed skating, and is watched by millions of television viewers. A draw decides who should start in inner lane and who in outer lane. Many skaters dread the last inner lane, where they need to tackle heavier centrifugal forces than their companions in the last outer lane, at maximum speed around 55 km/hour, at a time when fatigue may set in. The aim of this article is to investigate this potential difference between last inner and last outer lane. For this purpose data from eleven Sprint World Championships 1984-1994 are exploited. A bivariate mixed effects model is used that in addition to the inner-outer lane information takes account of different ice and weather conditions on different days, unequal levels for different skaters, and the passing times for the first 100 meter. The underlying 'unfairness parameter', estimated with optimal precision, is about 0.05 seconds, and is indeed significantly different from zero; it is about three times as large as its estimated standard deviation. Results from the work reported on here played a decisive role in leading the International Skating Union and the International Olympic Committee to change the rules for the 500 meter sprint event; as of the Nagano 1998 Olympic Games, the sprinters are to skate twice, with one start in inner lane and one in outer lane. The best average result determines the final list, and the best skaters from the first run are paired to skate last in the second run. It has also been decided that the same rules shall apply for the single distance 500 meter World Championships; these are arranged yearly from 1996 onwards.
A family of random probabilities is defined and studied. This family contains the Dirichlet proce... more A family of random probabilities is defined and studied. This family contains the Dirichlet process as a special case, corresponding to an inner point in the appropriate parameter space. The extension makes it possible to have.random means with larger or smaller skewnesses as compared to skewnesses under the Dirichlet prior, and also in other ways amounts to additional modelling flexibility. The usefulness of such random probabilities for use in nonparametric Bayesian statistics is discussed. The posterior distribution is complicated, but inference can nevertheless be carried out via simulation, and some exact formulae are derived for the case of random means. The class of nonparametric priors provides an instructive example where the speed with which the posterior forgets its prior with increasing data sample size depends on special aspects of the prior, which is a different situation from that of parametric inference.
Journal of Statistical Planning and Inference, May 1, 2018
This is the guest editors' general introduction to a Special Issue of the Journal of Statistical ... more This is the guest editors' general introduction to a Special Issue of the Journal of Statistical Planning and Inference, dedicated to confidence distributions and related themes. Confidence distributions (CDs) are distributions for parameters of interest, constructed via a statistical model after analysing the data. As such they serve the same purpose for the frequentist statisticians as the posterior distributions for the Bayesians. There have been several attempts in the literature to put up a clear theory for such confidence distributions, from Fisher's fiducial inference and onwards. There are certain obstacles and difficulties involved in these attempts, both conceptually and operationally, which have contributed to the CDs being slow in entering statistical mainstream. Recently there is a renewed surge of interest in CDs and various related themes, however, reflected in both series of new methodological research, advanced applications to substantive sciences, and dissemination and communication via workshops and conferences. The present special issue of the JSPI is a collection of papers emanating from the Inference With Confidence workshop in Oslo, May 2015. Several of the papers appearing here were first presented at that workshop. The present collection includes however also new research papers from other scholars in the field.
Previous chapters have developed concepts and methodology pertaining to confidence distributions ... more Previous chapters have developed concepts and methodology pertaining to confidence distributions and related inference procedures. Some of these methods take the form of generally applicable recipes, via log-likelihood profiles, deviances and first-order large-sample approximations to the distribution of estimators of the focus estimands in question. Sometimes these recipes are too coarse and are in need of modification and perfection, however, which is the topic of the present chapter. We discuss methods based on mean and bias corrected deviance curves, t-bootstrapping, a certain acceleration and bias correction method, approximations via expansions, prepivoting and modified likelihood profiles. The extent to which these methods lead to improvements is also briefly illustrated and discussed. Introduction Uniformly most powerful exact inference is in the presence of nuisance parameters available only in regular exponential models for continuous data and other models with Neyman structure, as discussed and exemplified in Chapter 5. Exact confidence distributions exist in a wider class of models, but need not be canonical. The estimate of location based on the Wilcoxon statistic, for example, has an exact known distribution in the location model where only symmetry is assumed; see Section 11.4. In more complex models, the statistic on which to base the confidence distribution might be chosen on various grounds: the structure of the likelihood function, perceived robustness, asymptotic properties, computational feasibility, perspective and tradition of the study. In the given model, with finite data, it might be difficult to obtain an exact confidence distribution based on the chosen statistic. As we shall see there are various techniques available for obtaining approximate confidence distributions and confidence likelihoods, however, improving on the first-order ones worked with in Chapters 3–4. Bootstrapping, simulation and asymptotics are useful tools in calculating approximate confidence distributions and in characterising their power properties. When an estimator, often the maximum likelihood estimator of the interest parameter, is used as the statistic on which the confidence distribution is based, bootstrapping provides an estimate of the sampling distribution of the statistic. This empirical sampling distribution can be turned into an approximate confidence distribution in several ways, which we address in the text that follows.
This article extends the scope of empirical likelihood methodology in three directions: to allow ... more This article extends the scope of empirical likelihood methodology in three directions: to allow for plug-in estimates of nuisance parameters in estimating equations, slower than √ n-rates of convergence, and settings in which there are a relatively large number of estimating equations compared to the sample size. Calibrating empirical likelihood confidence regions with plugin is sometimes intractable due to the complexity of the asymptotics, so we introduce a bootstrap approximation that can be used in such situations. We provide a range of examples from survival analysis and nonparametric statistics to illustrate the main results.
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Papers by Nils Hjort