Search

There are few computational and methodological tools available for the analysis of general multi-state models with interval censoring. Here, we propose a general framework for parametric inference with interval censored multi-state data. Our framework can accommodate any parametric model for the transition times, and covariates may be included in various ways. We present a general method for constructing the likelihood, which we have implemented in a ready-to-use R package, smms, available on GitHub. The R package also computes the required high-dimensional integrals in an efficient manner. Further, we explore connections between our modelling framework and existing approaches: our models fall under the class of semi-Markovian multi-state models, but with a different, and sparser parameterisation than what is often seen. We illustrate our framework through a dataset monitoring heart transplant patients. Finally, we investigate the effect of some forms of misspecification of the model assumptions through simulations.

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.

Have great wars become less violent over time, and is there something we might identify as the long peace? We investigate statistical versions of such questions, by examining the number of battle-deaths in the Correlates of War dataset, with 95 interstate wars from 1816 to 2007. Previous research has found this series of wars to be stationary, with no apparent change over time. We develop a framework to find and assess a change-point in this battle-deaths series. Our change-point methodology takes into consideration the power law distribution of the data, models the full battle-deaths distribution, as opposed to focusing merely on the extreme tail, and evaluates the uncertainty in the estimation. Using this framework, we find evidence that the series has not been as stationary as past research has indicated. Our statistical sightings of better angels indicate that 1950 represents the most likely change-point in the battle-deaths series – the point in time where the battle-deaths distribution might have changed for the better.

In parametric copula setups, where both the margins and copula have parametric forms, two-stage maximum likelihood estimation, often referred to as inference functions for margins, is used as an attractive alternative to the full maximum likelihood estimation strategy. Exploiting the existing model robust inference of two-stage maximum likelihood estimation, a copula information criterion (CIC) for model selection is developed. In a nutshell, CIC aims for the model that minimizes the Kullback–Leibler divergence from the real data generating mechanism. CIC does not assume that the chosen parametric model captures this true model, unlike what is assumed for AIC. In this sense, CIC is analogous to the Takeuchi Information Criterion (TIC), which is defined for the full maximum likelihood. If the additional assumption that a candidate model is correctly specified is made, then CIC for that model simplifies to AIC. Additionally, CIC can easily be extended to the conditional copula setup where covariates are parametrically linked to the copula model. As a numerical illustration, simulation studies were performed to show that the better model according to CIC also has better prediction performance in general. The result also shows that the bias correction term of CIC penalizes the misspecified model more heavily. This bias correction term has a strong positive relationship with the prediction performance of the model. So, a model with bad prediction performance is being penalized more by CIC. Although this behavior of the bias correction part is an important conceptual advance of CIC, this is not sufficient to make CIC outperform AIC in practice. This is because each misspecified model has the bias correction term and they grow at different speeds, depending on the model. The difference between CIC and AIC becomes minimal as sample size grows because the log-likelihood part outgrows the bias correction part.

Aalen’s linear hazard rate regression model is a useful and increasingly popular alternative to Cox’ multiplicative hazard rate model. It postulates that an individual has hazard rate function $$h(s)=z_1\alpha _1(s)+\cdots +z_r\alpha _r(s)$$ h ( s ) = z 1 α 1 ( s ) + ⋯ + z r α r ( s ) in terms of his covariate values $$z_1,\ldots ,z_r$$ z 1 , … , z r . These are typically levels of various hazard factors, and may also be time-dependent. The hazard factor functions $$\alpha _j(s)$$ α j ( s ) are the parameters of the model and are estimated from data. This is traditionally accomplished in a fully nonparametric way. This paper develops methodology for estimating the hazard factor functions when some of them are modelled parametrically while the others are left unspecified. Large-sample results are reached inside this partly parametric, partly nonparametric framework, which also enables us to assess the goodness of fit of the model’s parametric components. In addition, these results are used to pinpoint how much precision is gained, using the parametric-nonparametric model, over the standard nonparametric method. A real-data application is included, along with a brief simulation study.

Changes in the body condition of Antarctic minke whales (Balaenoptera bonaerensis) have been investigated in a number of studies, but remain contested. Here we provide a new analysis of body condition measurements, with particularly careful attention to the statistical model building and to model selection issues. We analyse body condition data for a large number (4704) of minke whales caught between 1987 and 2005. The data consist of five different variables related to body condition (fat weight, blubber thickness and girth) and a number of temporal, spatial and biological covariates. The body condition variables are analysed using linear mixed-effects models, for which we provide sound biological motivation. Further, we conduct model selection with the focused information criterion (FIC), reflecting the fact that we have a clearly specified research question, which leads us to a clear focus parameter of particular interest. We find that there has been a substantial decline in body condition over the study period (the net declines are estimated to 10% for fat weight, 7% for blubber thickness and 3% for the girth). Interestingly, there seems to be some differences in body condition trends between males and females and in different regions of the Antarctic. The decline in body condition could indicate major changes in the Antarctic ecosystem, in particular, increased competition from some larger krill-eating whale species.

We introduce and develop a general paradigm for combining information across diverse data sources. In broad terms, suppose 𝜑 is a parameter of interest, built up via components 𝜓1,…,𝜓𝑘 from data sources 1, … , k. The proposed scheme has three steps. First, the independent inspection (II) step amounts to investigating each separate data source, translating statistical information to a confidence distribution (CD) 𝐶𝑗(𝜓𝑗) for the relevant focus parameter 𝜓𝑗 associated with data source j. Second, confidence conversion (CC) techniques are used to translate the CDs to confidence log-likelihood functions. Finally, the focused fusion (FF) step uses relevant and context-driven techniques to construct a confidence distribution for the primary focus parameter 𝜑=𝜑(𝜓1,…,𝜓𝑘), acting on the combined confidence log-likelihood. In traditional setups, the II-CC-FF strategy amounts to versions of meta-analysis, and turns out to be competitive against state-of-the-art methods. Its potential lies in applications to harder problems, however. Illustrations are presented, related to actual applications.

A central point of disagreement, in certain long-standing discussions about a particular whaling dataset in the Scientific Committee of the International Whaling Commission, has directly involved model selection issues for linear mixed effect models. The biological question under discussion is associated with a clearly defined parameter of primary interest, a focus parameter, which makes model selection with the Focused Information Criterion (FIC) more appropriate than other selection methods. Since the existing FIC methodology has not covered the case of linear mixed effects models, this article sets up the required framework and develops the necessary formulae for the relevant FIC. Our new criterion requires the asymptotic distribution of estimators derived for a given candidate linear mixed model but with behaviour examined under a wider linear mixed model. These results, needed here to build our FIC, also have independent interest.

When using the Focused Information Criterion (FIC) for assessing and ranking candidate models with respect to how well they do for a given estimation task, it is customary to produce a so-called FIC plot. This plot has the different point estimates along the y-axis and the root-FIC scores on the x-axis, these being the estimated root-mean-square scores. In this paper we address the estimation uncertainty involved in each of the points of such a FIC plot. This needs careful assessment of each of the estimators from the candidate models, taking also modelling bias into account, along with the relative precision of the associated estimated mean squared error quantities. We use confidence distributions for these tasks. This leads to fruitful CD&ndash, FIC plots, helping the statistician to judge to what extent the seemingly best models really are better than other models, etc. These efforts also lead to two further developments. The first is a new tool for model selection, which we call the quantile-FIC, which helps overcome certain difficulties associated with the usual FIC procedures, related to somewhat arbitrary schemes for handling estimated squared biases. A particular case is the median-FIC. The second development is to form model averaged estimators with weights determined by the relative sizes of the median- and quantile-FIC scores.

Datasets encountered when examining deeper issues in ecology and evolution are often complex. This calls for careful strategies for both model building, model selection, and model averaging. Our paper aims at motivating, exhibiting, and further developing focused model selection criteria. In contexts involving precisely formulated interest parameters, these versions of FIC, the focused information criterion, typically lead to better final precision for the most salient estimates, confidence intervals, etc. as compared to estimators obtained from other selection methods. Our methods are illustrated with real case studies in ecology; one related to bird species abundance and another to the decline in body condition for the Antarctic minke whale. ispartof: Frontiers In Ecology And Evolution vol:7 pages:1-13 status: published

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.

Suppose observations y 1 , … , y n stem from a parametric model f ( y , θ ) , with the parameter taking one value θ L for y 1 , … , y τ and another value θ R for y τ + 1 , … , y n . This article provides and examines two different general strategies for not merely estimating the break point τ but also to complement such an estimate with full confidence distributions, both for the change-point τ and for associated measures of differences between the two levels of θ . The first idea worked with involves testing homogeneity for the two segments to the left and the right of a candidate change-point value at a fine-tuned level of significance. Carrying out such a scheme requires having a goodness-of-fit test for constancy of the θ parameter over a segment of indices, and we also develop classes of such tests. These also have some independent interest. The second general method uses the log-likelihood function, profiled over the other parameters, and we show how this may lead to confidence inference for τ . Our methods are illustrated for four real data stories, with these meeting different types of challenges.

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 also 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.

[No abstract]

In this paper, we extend the focused information criterion (FIC) to copula models. Copulas are often used for applications where the joint tail behavior of the variables is of particular interest, and selecting a copula that captures this well is then essential. Traditional model selection methods such as the Akaike information criterion (AIC) and the Bayesian information criterion (BIC) aim at finding the overall best‐fitting model, which is not necessarily the one best suited for the application at hand. The FIC, on the other hand, evaluates and ranks candidate models based on the precision of their point estimates of a context‐given focus parameter. This could be any quantity of particular interest, for example, the mean, a correlation, conditional probabilities, or measures of tail dependence. We derive FIC formulae for the maximum likelihood estimator, the two‐stage maximum likelihood estimator, and the so‐called pseudo‐maximum‐likelihood (PML) estimator combined with parametric margins. Furthermore, we confirm the validity of the AIC formula for the PML estimator combined with parametric margins. To study the numerical behavior of FIC, we have carried out a simulation study, and we have also analyzed a multivariate data set pertaining to abalones. The results from the study show that the FIC successfully ranks candidate models in terms of their performance, defined as how well they estimate the focus parameter. In terms of estimation precision, FIC clearly outperforms AIC, especially when the focus parameter relates to only a specific part of the model, such as the conditional upper‐tail probability.

This article is concerned with inference in parametric copula setups, where both the marginals and the copula have parametric forms. For such models, two-stage maximum likelihood estimation, often referred to as inference function for margins, is used as an attractive alternative to the full maximum likelihood estimation strategy. Previous studies of the two-stage maximum likelihood estimator have largely been based on the assumption that the chosen parametric model captures the true model that generated data. We study the impact of dropping this true model assumption, both theoretically and numerically. We first show that the two-stage maximum likelihood estimator is consistent for a well-defined least false parameter value, different from the analogous least false parameter associated with the full maximum likelihood procedure. Then we demonstrate limiting normality of the full vector of estimators, with concise matrix notation for the variance matrices involved. Along with consistent estimators for these, we have built a model-robust machinery for inference in parametric copula models. The special case where the parametric model is assumed to hold corresponds to situations studied earlier in the literature, with simpler formulas for variance matrices. As a numerical illustration, we perform a set of simulations. We also analyze five-dimensional Norwegian precipitation data. We find that the variance of the copula parameter estimate can both increase and decrease, by dropping the true model assumption. In addition, we observe that the two-stage maximum likelihood estimator is still highly efficient when the true model assumption is dropped and thus the model robust asymptotic variance formulas are used. Additionally, we discover that using highly misspecified models can lead to situations where the asymptotic variance of the two-stage maximum likelihood estimator is lower than that of full maximum likelihood estimator. Our results are also used to analyze the mean squared error properties for both the full and the two-stage maximum likelihood estimators of any focus parameter.

Certain recent advances in statistical methodology have promising potential for fruitful use in general biology and the fisheries sciences. This paper reviews and discusses some of the relevant themes, including accurate modelling via focused model selection techniques, dynamic goodness-of-fit testing of processes evolving over time, finding break points for phenomena experiencing regime shifts, prediction uncertainty, and optimal combination of information across diverse sources via confidence distributions. The methods are illustrated for the Hjort liver quality index time series. Its roots lie in the classic study by Hjort in 1914, where liver quality of the Northeast Arctic cod (Gadus morhua) for 1880–1912 was reported on and studied, along with related factors, making it one of the first teleost time series ever published. The series has been extended both backwards and forwards in time, to 1859–2012, due to comprehensive archival and calibration efforts of Kjesbu et al. in 2014, yielding one of the longest time series of marine science. Our study offers a detailed examination of this series and how it relates to and interacts with associated factors, including winter temperatures, length distribution parameters, cod mortality, and a certain index related to availability of food. We identify certain mild nonstationary aspects of the time series, show that there is a regime shift around 1990 in the ways the liver series interacts with winter temperatures, and demonstrate that mortality and food availability play important roles.

Various statistical models involve a certain function, say f , like the mean regression as a function of a covariate, the hazard rate as a function of time, the spectral density of a time series as a function of frequency, or an intensity as a function of geographical position, etc. Such functions are often modelled parametrically, whether for frequentist or Bayesian uses, and under weak conditions there are so-called Bernshteĭn–von Mises theorems implying that these two approaches are large-sample equivalent. Results of this nature do not necessarily hold up in nonparametric and high-dimensional setups, however. The aim of the present paper is to exhibit a unified framework and methodology for both frequentist and Bayesian nonparametric analysis, involving priors that set f constant over windows, and where the number m of such windows grows with sample size n . Applications include nonparametric regression, maximum likelihood with nonparametrically varying parameter functions, hazard rates being functions of covariates, and nonparametric analysis of stationary time series. We work out conditions on the number and sizes of the windows under which Bernshteĭn–von Mises type theorems can be established, with the prior changing with sample size via the growing number of windows. These conditions entail e.g. that if m ∝ n α , then α ∈ ( 1 4 , 1 2 ) is required. Illustrations of the general methodology are given, including setups with nonparametric regression, hazard rate estimation, and inference about frequency spectra for stationary time series.

Should one rely on a parametric or nonparametric model when analysing a given data set? This classic question cannot be answered by traditional model selection criteria like AIC and BIC, since a nonparametric model has no likelihood. The purpose of the present paper is to develop a focused information criterion (FIC) for comparing general non-nested parametric models with a nonparametric alternative. It relies in part on the notion of a focus parameter, a population quantity of particular interest in the statistical analysis. The FIC compares and ranks candidate models based on estimated precision of the different model-based estimators for the focus parameter. It has earlier been developed for several classes of problems, but mainly involving parametric models. The new FIC, including also nonparametrics, is novel also in the mathematical context, being derived without the local neighbourhood asymptotics underlying previous versions of FIC. Certain average-weighted versions, called AFIC, allowing several focus parameters to be considered simultaneously, are also developed. We concentrate on the standard i.i.d. setting and certain direct extensions thereof, but also sketch further generalisations to other types of data. Theoretical and simulation-based results demonstrate desirable properties and satisfactory performance.

This paper develops a hybrid likelihood (HL) method based on a compromise between parametric and nonparametric likelihoods. Consider the setting of a parametric model for the distribution of an observation Y with parameter θ. Suppose there is also an estimating function m(·, μ) identifying another parameter μ via Em(Y, μ) = 0, at the outset defined independently of the parametric model. To borrow strength from the parametric model while obtaining a degree of robustness from the empirical likelihood method, we formulate inference about θ in terms of the hybrid likelihood function Hn (θ) = Ln (θ)1-a Rn (μ(θ)) a . Here a ∈ [0,1) represents the extent of the compromise, Ln is the ordinary parametric likelihood for θ, Rn is the empirical likelihood function, and μ is considered through the lens of the parametric model. We establish asymptotic normality of the corresponding HL estimator and a version of the Wilks theorem. We also examine extensions of these results under misspecification of the parametric model, and propose methods for selecting the balance parameter a. ispartof: STATISTICA SINICA vol:28 issue:4 pages:2389-2407 ispartof: location:China (Republic : 1949- ) status: published

Statistical prediction methods typically require some form of fine‐tuning of tuning parameter(s), with K‐fold cross‐validation as the canonical procedure. For ridge regression, there exist numerous procedures, but common for all, including cross‐validation, is that one single parameter is chosen for all future predictions. We propose instead to calculate a unique tuning parameter for each individual for which we wish to predict an outcome. This generates an individualized prediction by focusing on the vector of covariates of a specific individual. The focused ridge—fridge—procedure is introduced with a two‐part contribution: First we define an oracle tuning parameter minimizing the mean squared prediction error of a specific covariate vector, and then we propose to estimate this tuning parameter by using plug‐in estimates of the regression coefficients and error variance parameter. The procedure is extended to logistic ridge regression by using parametric bootstrap. For high‐dimensional data, we propose to use ridge regression with cross‐validation as the plug‐in estimate, and simulations show that fridge gives smaller average prediction error than ridge with cross‐validation for both simulated and real data. We illustrate the new concept for both linear and logistic regression models in two applications of personalized medicine: predicting individual risk and treatment response based on gene expression data. The method is implemented in the R package fridge.

Cox’s proportional hazards regression model is the standard method for modelling censored life-time data with covariates. In its standard form, this method relies on a semiparametric proportional hazards structure, leaving the baseline unspecified. Naturally, specifying a parametric model also for the baseline hazard, leading to fully parametric Cox models, will be more efficient when the parametric model is correct, or close to correct. The aim of this paper is two-fold. (a) We compare parametric and semiparametric models in terms of their asymptotic relative efficiencies when estimating different quantities. We find that for some quantities the gain of restricting the model space is substantial, while it is negligible for others. (b) To deal with such selection in practice we develop certain focused and averaged focused information criteria (FIC and AFIC). These aim at selecting the most appropriate proportional hazards models for given purposes. Our methodology applies also to the simpler case without covariates, when comparing Kaplan–Meier and Nelson–Aalen estimators to parametric counterparts. Applications to real data are also provided, along with analyses of theoretical behavioural aspects of our methods.

The arguments leading to the classical AIC does not hold for the case of parametric copulae models when using the pseudo maximum likelihood procedure. We derive a proper correction, and name it the Copula Information Criterion (CIC). The CIC is, as the copula, invariant to the underlying marginal distributions, but is fundamentally dierent from the AIC formula erroneously used by practitioners. We further observe that bias-correction-based information criteria do not exist for a large class of commonly used copulae, such as the Gumbel or Joe copulae. This motivates the investigation of median unbiased information criteria in the fully parametric setting, which then could be ported over to the current, more complex, semi-parametric setting.

Watson and Holmes propose ways of investigating robustness of statistical decisions by examining certain neighbourhoods around a posterior distribution. This may partly amount to ad hoc modelling of extra uncertainty. Instead of creating neighbourhoods around the posterior a posteriori, we argue that it might be more fruitful to model a layer of extra uncertainty first, in the model building process, and then allow the data to determine how big the resulting neighbourhoods ought to be. We develop and briefly illustrate a general strategy along such lines.

Suppose estimates are available for correlations between pairs of variables but that the matrix of correlation estimates is not positive definite. In various applications, having a valid correlation matrix is important in connection with follow-up analyses that might, for example, involve sampling from a valid distribution. We present new methods for adjusting the initial estimates to form a proper, that is, nonnegative definite, correlation matrix. These are based on constructing certain pseudo-likelihood functions, formed by multiplying together exact or approximate likelihood contributions associated with the individual correlations. Such pseudo-likelihoods may then be maximized over the range of proper correlation matrices. They may also be utilized to form pseudo-posterior distributions for the unknown correlation matrix, by factoring in relevant prior information for the separate correlations. We illustrate our methods on two examples from a financial time series and genomic pathway analysis.

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.

To obtain a distribution representing the inferred uncertain knowledge about the parameter directly from the data without access to any prior distribution was the glorious goal that Fisher succeeded in reaching with his fiducial distribution for a scalar parameter, but for vector parameters he got into trouble. He regarded the fiducial distribution as an ordinary probability distribution subject to the usual Kolmogorovian laws for sigma-additive probability measures. Fisher did not develop any mathematical theory of fiducial probability, but chose to illustrate his thoughts by examples. It was soon found that even in Fisher's examples with more than one parameter, there were no unique fiducial distributions, and inconsistencies and paradoxes were identified. After re-visiting Fisher's ideas over 1930–1935, which underlie our confidence distributions, we summarise and discuss the big debate over the fiducial argument, which died out only after Fisher's death in 1962, leaving the statistical community to regard it as badly flawed and Fisher's biggest blunder. The chapter ends with three attempts at saving the fiducial argument. Despite the potential problems with multivariate fiducial distributions, their marginals are often exact or approximate confidence distributions. The initial argument Fisher introduced the fiducial argument in 1930. In the introduction to Fisher (1930) he indicates that he had solved a very important problem that had escaped “the most eminent men of their time” since Bayes introduced his theorem and the method of inverse probability was established by Laplace. Later he stated that he really had a solution to “more than 150 years of disputation between the pros and cons of inverse probability [that] had left the subject only more befogged by doubt and frustration” (Fisher discussing Neyman in Neyman [1934, p. 617]). This was a view he held throughout his life. Fisher saw the fiducial probability as a probability measure over the parameter inherited from the probability model of the sampling experiment and the observed data. His early interpretation of fiducial probability was very similar to Neyman's coverage probability for intervals. Later he thought of fiducial distributions as a representation of the information in the data as seen on the background of the statistical model. It would then serve as an epistemic probability distribution for rational people. This is close to the Bayesian understanding of a posterior distribution, at least for people agreeing on the prior distribution.

This chapter is an invitation to the central themes of the book: confidence, likelihood, probability and confidence distributions. We sketch the historical backgrounds and trace various sources of influence leading to the present and somewhat bewildering state of ‘modern statistics’, which perhaps to the confusion of many researchers working in the applied sciences is still filled with controversies and partly conflicting paradigms regarding even basic concepts. Introduction The aim of this book is to prepare for a synthesis of the two main traditions of statistical inference: those of the Bayesians and of the frequentists. Sir Ronald Aylmer Fisher worked out the theory of frequentist statistical inference from around 1920. From 1930 onward he developed his fiducial argument, which was intended to yield Bayesian-type results without the often ill-founded prior distributions needed in Bayesian analyses. Unfortunately, Fisher went wrong on the fiducial argument. We think, nevertheless, that it is a key to obtaining a synthesis of the two, partly competing statistical traditions. Confidence, likelihood and probability are words used to characterise uncertainty in most everyday talk, and also in more formal contexts. The Intergovernmental Panel on Climate Change (IPCC), for example, concluded in 2007, “Most of the observed increase in global average temperature since the mid-20th century is very likely due to the observed increase in anthropogenic greenhouse gas concentrations” (Summary for Policymakers, IPCC, 2007). They codify ‘very likely’ as having probability between 0.90 and 0.95 according to expert judgment. In its 2013 report IPCC is firmer and more precise in its conclusion. The Summary for Policymakers states, “It is extremely likely that more than half of the observed increase in global surface temperature from 1951 to 2010 was caused by the anthropogenic increase in greenhouse gas concentrations and other anthropogenic forcings together” (IPCC, 2013, p. 17). By extremely likely they mean more than 95% certainty. We would have used ‘confidence’ rather than ‘likelihood’ to quantify degree of belief based on available data. We will use the term ‘likelihood’ in the technical sense usual in statistics. Confidence, likelihood and probability are pivotal words in the science of statistics. Mathematical probability models are used to build likelihood functions that lead to confidence intervals. Why do we need three words, and actually additional words such as credibility and propensity, to measure uncertainty and frequency of chance events?

This lively book lays out a methodology of confidence distributions and puts them through their paces. Among other merits, they lead to optimal combinations of confidence from different sources of information, and they can make complex models amenable to objective and indeed prior-free analysis for less subjectively inclined statisticians. The generous mixture of theory, illustrations, applications and exercises is suitable for statisticians at all levels of experience, as well as for data-oriented scientists. Some confidence distributions are less dispersed than their competitors. This concept leads to a theory of risk functions and comparisons for distributions of confidence. Neyman–Pearson type theorems leading to optimal confidence are developed and richly illustrated. Exact and optimal confidence distribution is the gold standard for inferred epistemic distributions. Confidence distributions and likelihood functions are intertwined, allowing prior distributions to be made part of the likelihood. Meta-analysis in likelihood terms is developed and taken beyond traditional methods, suiting it in particular to combining information across diverse data sources.

There is a large number of applications where family relationships need to be determined from DNA data. In forensic science, competing ideas are in general verbally formulated as the two hypotheses of a test. For the most common paternity case, the null hypothesis states that the alleged father is the true father against the alternative hypothesis that the father is an unrelated man. A likelihood ratio is calculated to summarize the evidence. We propose an alternative framework whereby a model and the hypotheses are formulated in terms of parameters representing identity-by-descent probabilities. There are several advantages to this approach. Firstly, the alternative hypothesis can be completely general. Specifically, the alternative does not need to specify an unrelated man. Secondly, the parametric formulation corresponds to the approach used in most other applications of statistical hypothesis testing and so there is a large theory of classical statistics that can be applied. Theoretical properties of the test statistic under the null hypothesis are studied. An extension to trios of individuals has been carried out. The methods are exemplified using simulations and a real dataset of 27 Spanish Romani individuals.

Model selection methods provide a way to select one model among a set of models in a statistically valid way. Such methods include tools for variable selection in regression models. Asymptotic properties such as consistency and efficiency, the specific use of the model, or properties regarding minimization of a certain risk function such as the expected prediction error, may help to decide which method to choose. Model selection is a special case of model averaging where the estimators obtained from different models are combined in a weighted average. Model averaging avoids the selection of one model. The choice of the weights may be determined by a model selection method or may come from a priori knowledge in a Bayesian framework.

This article addresses issues of model criticism and model comparison in Bayesian contexts, and focuses on the use of the so-called posterior predictive p values (ppp). These involve a general discrepancy or conflict measure and depend on the prior, the model, and the data. They are used in statistical practice to quantify the degree of surprise or conflict in data and to compare different combinations of prior and model. The distribution of such ppp values is far from uniform however, as we demonstrate for different models, making their interpretation and comparison a difficult matter. We propose a natural calibration of the ppp values, where the resulting cppp values are uniform on the unit interval under model conditions. The cppp values, which in general rely on a double-simulation scheme for their computation, may then be used to assess and compare different priors and models. Our methods also make it possible to compare parametric and nonparametric model specifications, in that genuine “measures of ...