The Copula Information Criteria

We derive two types of Akaike information criterion (AIC)‐like model‐selection formulae for the semiparametric pseudo‐maximum likelihood procedure. We first adapt the arguments leading to the original AIC formula, related to empirical estimation of a certain Kullback–Leibler information distance. This gives a significantly different formula compared with the AIC, which we name the copula information criterion. However, we show that such a model‐selection procedure cannot exist for copula models with densities that grow very fast near the edge of the unit cube. This problem affects most popular copula models. We then derive what we call the cross‐validation copula information criterion, which exists under weak conditions and is a first‐order approximation to exact cross validation. This formula is very similar to the standard AIC formula but has slightly different motivation. A brief illustration with real data is given.     

On Bayesian consistency

We consider a sequence of posterior distributions based on a data-dependent prior (which we shall refer to as a pseudoposterior distribution) and establish simple conditions under which the sequence is Hellinger consistent. It is shown how investigations into these pseudo posteriors assist with the understanding of some true posterior distributions, including Pólya trees, the infinite dimensional exponential family and mixture models.     

Statistical Corrections of Invalid Correlation Matrices

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.     

Focused information criteria for copulas

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.     

Volunteering and Policy Makers: The Political Uses of the UK Conservative Party’s International Development Volunteering Projects

VOLUNTAS: International Journal of Voluntary and Nonprofit Organizations - What motivates a political party to develop overseas development volunteering projects for members? How do such activities...     

Goodness of Fit via Non-parametric Likelihood Ratios

Abstract.  To test if a density f is equal to a specified f ₀, one knows by the Neyman–Pearson lemma the form of the optimal test at a specified alternative f ₁. Any non-parametric density estimation scheme allows an estimate of f . This leads to estimated likelihood ratios. Properties are studied of tests which for the density estimation ingredient use log-linear expansions. Such expansions are either coupled with subset selectors like the Akaike information criterion and the Bayesian information criterion regimes, or use order growing with sample size. Our tests are generalized to testing the adequacy of general parametric models, and to work also in higher dimensions. The tests are related to, but are different from, the 'smooth tests' that go back to Neyman [Skandinavisk Aktuarietidsskrift 20(1937) 149] and that have been studied extensively in recent literature. Our tests are large-sample equivalent to such smooth tests under local alternative conditions, but different from the smooth tests and often better under non-local conditions.     

Correction of Density Estimators that are not Densities

Abstract. Several old and new density estimators may have good theoretical performance, but are hampered by not being bona fide densities; they may be negative in certain regions or may not integrate to 1. One can therefore not simulate from them, for example. This paper develops general modification methods that turn any density estimator into one which is a bona fide density, and which is always better in performance under one set of conditions and arbitrarily close in performance under a complementary set of conditions. This improvement-for-free procedure can, in particular, be applied for higher-order kernel estimators, classes of modern h ⁴ bias kernel type estimators, superkernel estimators, the sinc kernel estimator, the k -NN estimator, orthogonal expansion estimators, and for various recently developed semi-parametric density estimators.     

The Bernstein–von Mises theorem in semiparametric competing risks models

Semiparametric Bayesian models are nowadays a popular tool in event history analysis. An important area of research concerns the investigation of frequentist properties of posterior inference. In this paper, we propose novel semiparametric Bayesian models for the analysis of competing risks data and investigate the Bernstein–von Mises theorem for differentiable functionals of model parameters. The model is specified by expressing the cause-specific hazard as the product of the conditional probability of a failure type and the overall hazard rate. We take the conditional probability as a smooth function of time and leave the cumulative overall hazard unspecified. A prior distribution is defined on the joint parameter space, which includes a beta process prior for the cumulative overall hazard. We first develop the large-sample properties of maximum likelihood estimators by giving simple sufficient conditions for them to hold. Then, we show that, under the chosen priors, the posterior distribution for any differentiable functional of interest is asymptotically equivalent to the sampling distribution derived from maximum likelihood estimation. A simulation study is provided to illustrate the coverage properties of credible intervals on cumulative incidence functions.     

What price semiparametric Cox regression?

Lifetime Data Analysis - 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...