Split Hamiltonian Monte Carlo

We show how the Hamiltonian Monte Carlo algorithm can sometimes be speeded up by “splitting” the Hamiltonian in a way that allows much of the movement around the state space to be done at low computational cost. One context where this is possible is when the log density of the distribution of interest (the potential energy function) can be written as the log of a Gaussian density, which is a quadratic function, plus a slowly-varying function. Hamiltonian dynamics for quadratic energy functions can be analytically solved. With the splitting technique, only the slowly-varying part of the energy needs to be handled numerically, and this can be done with a larger stepsize (and hence fewer steps) than would be necessary with a direct simulation of the dynamics. Another context where splitting helps is when the most important terms of the potential energy function and its gradient can be evaluated quickly, with only a slowly-varying part requiring costly computations. With splitting, the quick portion can be handled with a small stepsize, while the costly portion uses a larger stepsize. We show that both of these splitting approaches can reduce the computational cost of sampling from the posterior distribution for a logistic regression model, using either a Gaussian approximation centered on the posterior mode, or a Hamiltonian split into a term that depends on only a small number of critical cases, and another term that involves the larger number of cases whose influence on the posterior distribution is small.     

Robust models in probability sampling

In the estimation of a population mean or total from a random sample, certain methods based on linear models are known to be automatically design consistent, regardless of how well the underlying model describes the population. A sufficient condition is identified for this type of robustness to model failure; the condition, which we call 'internal bias calibration', relates to the combination of a model and the method used to fit it. Included among the internally bias-calibrated models, in addition to the aforementioned linear models, are certain canonical link generalized linear models and nonparametric regressions constructed from them by a particular style of local likelihood fitting. Other models can often be made robust by using a suboptimal fitting method. Thus the class of model-based, but design consistent, analyses is enlarged to include more realistic models for certain types of survey variable such as binary indicators and counts. Particular applications discussed are the estimation of the size of a population subdomain, as arises in tax auditing for example, and the estimation of a bootstrap tail probability.     

Estimating the frequency of familial dilated cardiomyopathy in the presence of misclassification errors

Dilated cardiomyopathy is a disease of unknown cause characterized by dilation and impaired function of one or both ventricles. Most cases are believed to be sporadic, although familial forms have been detected. The familial form has been estimated to have a relative frequency of about 25%. Since, except for familial history, familial form has no other characteristics that could help in classifying the two diseases, the estimate of the frequency of the familial form should take into account a possible misclassification error. In our study, 100 cases were randomly selected in a prospective series of 350 patients. Out of them, 28 index cases were included in the analysis: 12 were known to be familial, and 88 were believed to be sporadic. After extensive clinical examination of the relatives, 3 patients supposed to have a sporadic form were found to have a familial form. 13 cases had a confirmed sporadic disease. Models in the Log-Linear Product class (LLP) have been used to separate classification errors from underlying patterns of disease incidence. The most conservative crude estimate of the misclassification error is 16.1% (CI 0.22- 23.27%), which leads to a crude estimate of the frequency of the familiar form of about 60%. An estimate of the disease frequency, adjusted for taking into consideration the sampling plan, is 40.93% (CI 32.29-44.17%). The results are consistent with the hypothesis that genetic factors are still underestimated, although they represent a major cause of the disease.     

A modified likelihood ratio test for homogeneity in finite mixture models

Testing for homogeneity in finite mixture models has been investigated by many researchers. The asymptotic null distribution of the likelihood ratio test (LRT) is very complex and difficult to use in practice. We propose a modified LRT for homogeneity in finite mixture models with a general parametric kernel distribution family. The modified LRT has a χ-type of null limiting distribution and is asymptotically most powerful under local alternatives. Simulations show that it performs better than competing tests. They also reveal that the limiting distribution with some adjustment can satisfactorily approximate the quantiles of the test statistic, even for moderate sample sizes.     

Designing fractional two-level experiments with nested error structures

A common feature of experiments with a random blocking factor and splitplot experiments is their nested error structure. This paper proposes a general strategy to handle fractional two-level experiments with such error structures. The strategy aims to create error strata with sufficient numbers of contrasts to separate active effects from inactive effects. The strategy also details the construction of treatment generators, given the constraints of a predetermined error structure. The key elements of the strategy are illustrated with a chemical experiment that has 16 factors and 32 runs blocked according to working days, and a cheese-making experiment that has 11 factors and 128 runs, divided over milk supplies as whole plots, curds productions as subplots and sets of identically treated cheeses as sub-subplots.     

Evaluating the power of Minitab's data subsetting lack of fit test in multiple linear regression

Minitab's data subsetting lack of fit test (denoted XLOF) is a combination of Burn and Ryan's test and Utts' test for testing lack of fit in linear regression models. As an alternative to the classical or pure error lack of fit test, it does not require replicates of predictor variables. However, due to the uncertainty about its performance, XLOF still remains unfamiliar to regression users while the well-known classical lack of fit test is not applicable to regression data without replicates. So far this procedure has not been mentioned in any textbooks and has not been included in any other software packages. This study assesses the performance of XLOF in detecting lack of fit in linear regressions without replicates by comparing the power with the classic test. The power of XLOF is simulated using Minitab macros for variables with several forms of curvature. These comparisons lead to pragmatic suggestions on the use of XLOF. The performance of XLOF was shown to be superior to the classical test based on the results. It should be noted that the replicates required for the classical test made itself unavailable for most of the regression data while XLOF can still be as powerful as the classic test even without replicates.     

Influence diagnostics in the capital asset pricing model under elliptical distributions

In this paper we consider the Capital Asset Pricing Model under Elliptical (symmetric) Distributions. This class of distributions, which contains the normal distribution, t, contaminated normal and power exponential, among others, offers a more flexible framework for modelling asset prices or returns. In order to analyze the sensibility to possible outliers and/or atypical returns of the maximum likelihood estimators, the local influence method was implemented. The results are illustrated by using a set of shares from companies who trade in the Chilean Stock Market. Our main conclusion is that symmetric distributions having heavier tails than those of the normal distribution, especially the t distribution with small degrees of freedom, show a better fit and allow the reduction of the influence of atypical returns in the maximum likelihood estimators.     

Forecasting mortality rates via density ratio modeling

The authors propose a semiparametric approach to modeling and forecasting age‐specific mortality in the United States. Their method is based on an extension of a class of semiparametric models to time series. It combines information from several time series and estimates the predictive distribution conditional on past data. The conditional expectation, which is the most commonly used predictor in practice, is the first moment of this distribution. The authors compare their method to that of Lee and Carter.