On the application of improved symplectic integrators in Hamiltonian Monte Carlo

We explore the construction of new symplectic numerical integration schemes to be used in Hamiltonian Monte Carlo and study their efficiency. Integration schemes from Blanes et al., and a new scheme are considered as candidates to the commonly used leapfrog method. All integration schemes are tested within the framework of the No-U-Turn sampler (NUTS), both for a logistic regression model and a student t-model. The results show that the leapfrog method is inferior to all the new methods both in terms of asymptotic expected acceptance probability for a model problem and the efficient sample size per computing time for the realistic models.     

Exponentiated-exponential geometric regression model

A regression model, based on the exponentiated-exponential geometric distribution, is defined and studied. The regression model can be applied to count data with under-dispersion or over-dispersion. Some forms of its modifications to truncated or inflated data are mentioned. Some tests to discriminate between the regression model and its competitors are discussed. Real numerical data sets are used to illustrate the applications of the regression model.     

A Starship Estimation Method for the Generalized λ Distributions

A development of the 'starship' method (Owen, 1988), a computer intensive estimation method, is presented for two forms of generalized λ distributions (gλd). The method can be used for the full parameter space and is flexible, allowing choice of both the form of the generalized λ distribution and of the nature of fit required. Some examples of its use in fitting data and approximating distributions are given. Some simulation studies explore the sampling distribution of the parameter estimates produced by this method for selected values of the parameters and consider comparisons with two other methods, for one of the gλd distributional forms, not previously so investigated. In the forms and parameter regions available to the other methods, it is demonstrated that the starship compares favourably. Although the differences between the methods, where available, tend to disappear with largersamples, the parameter coverage, flexibility and adaptability of the starship method make it attractive. However, the paper also demonstrates that care is needed when fitting and using such quantile-defined distributional families that are rich in shape, but have complex properties.     

Distribution of multivariate quadratic forms under certain covariance structures

Necessary and sufficient conditions are given for the covariance structure of all the observations in a multivariate factorial experiment under which certain multivariate quadratic forms are independent and distributed as a constant times a Wishart. It is also shown that exact multivariate test statistics can be formed for certain covariance structures of the observations when the assumption of equal covariance matrices for each normal population is relaxed. A characterization is given for the dependency structure between random vectors in which the sample mean and sample covariance matrix have certain properties.     

On the functional relationship between entropy and variance with related applications

The functional relationship between entropy and variance is investigated for some well-known distributions. The distributions considered here are the reparameterized versions of the original forms. Such a reparameterization is necessary as in each case we have a common variance. The related graphs of entropy as a function of variance are used for certain comparisons. Further, within the class of distributions having a common variance, a measure of affinity between these distributions is proposed using entropy. A few aspects of the sampling distributions of an estimator of entropy, when the samples are either from the normal or from the exponential distributions, are discussed with a view to possible applications in the testing of hypotheses for related parameters     

The measurement of capital utilisation rates

A strategy using spline function interpolation is developed f o r estimating capital utilisation rates . Cobb-Douglas, CES and translog functional forms are used in estimation. Tests for functional forms are conducted leading t o t h e s e l e c t i o n of the Cobb-Douglas form. Quarterly series of estimated utilisation rates and excess capacity measures are presented.     

The echelon Markov and Chebyshev inequalities

Abstract

A multivariate version of the sharp Markov inequality is derived, when associated probabilities are extended to segments of the supports of non-negative random variables, where the probabilities take echelon forms. It is shown that when some positive lower bounds of these probabilities are available, the multivariate Markov inequality without the echelon forms is improved. The corresponding results for Chebyshev’s inequality are also obtained.     

Asymptotic distributions of quadratic forms occurring in changepoint problems

A stochastic-process approach is used to derive the asymptotic distributions of quadratic forms occurring in the analysis of changepoint data.     

Extremes of Determinants and Optimality of Canonical Variables

Extremes of quadratic forms have been presented by several authors (Okamoto, 1969; Rao, 1973; Seber, 1984). The obvious multivariate extension of the extreme of quadratic forms is the extreme of the determinants as well as the ratios of the determinants. In this paper we develop some supremums of the determinants and the ratios of the determinants. A new optimality and equations of canonical variables are obtained.     

Variance estimation in the central limit theorem for Markov chains

This article concerns the variance estimation in the central limit theorem for finite recurrent Markov chains. The associated variance is calculated in terms of the transition matrix of the Markov chain. We prove the equivalence of different matrix forms representing this variance. The maximum likelihood estimator for this variance is constructed and it is proved that it is strongly consistent and asymptotically normal. The main part of our analysis consists in presenting closed matrix forms for this new variance. Additionally, we prove the asymptotic equivalence between the empirical and the maximum likelihood estimation (MLE) for the stationary distribution.     

The residual effect of a growth-decay mechanism and the distributions of covariance structures

The product of two independent or dependent scalar normal variables, sums of products, sample covariances, and general bilinear forms are considered. Their distributions are shown to belong to a class called generalized Laplacian. A growth-decay mechanism is also shown to produce such a generalized Laplacian. Sets of necessary and sufficient conditions are derived for bilinear forms to belong to this class. As a generalization, the distributions of rectangular matrices associated with multivariate normal random vectors are also discussed.     

On the independence between two generalized second degree polynomial statistics and their covariance

The condition of independence between two generalized second degree polynomial statistics is worked out in this paper under the assumption that the covariance matrix of the underlying normal distribution may be singular. The covariance between two such forms is also worked out without the assumption of normality. The results are indicated in compact matrix form.     

Stress−strength reliability with general form distributions

In this article, we present characterizations, associated with the stress?strength reliability, of distributions with some general exponential and general inverse exponential forms. We obtain a general form for stress?strength reliability, when the distributions of the stress and the strength are non identical and independently distributed with either form. We obtain different point and interval estimators of stress?strength reliability. The theoretical results obtained can be directly applied whenever the distributions of the stress and the strength possess the forms discussed. A simulation study is carried out showing satisfactory performance of the estimators obtained. Moreover, a real data example is presented.     

Kronecker Products in ANOVA—A First Step

In the complete balanced model for the analysis of variance, the equivalence of sums of squares and quadratic forms is seen to imply well-fitting patterns involving Kronecker products of identity matrices and scalar multiples of matrices with all elements equal to 1. The questions of symmetry, idempotency, and orthogonality so central to this topic are answered by simple multiplications; ranks are determined from simple traces. The associations between the forms of the two-factor model are presented here in a way that is accessible to first-year students and makes generalizations to higher order models transparent. The lack of patterns in incomplete or unbalanced models is noted. Additional steps in design and analysis are suggested in the references.     

DETECTION AND TESTING OF DIFFEFENT TYPES OF OUTLIER IN LINEAR STRAUCTURAL RELATIONASHIPS

The linear structural model provides one way of modelling a linear relationship between two random variables. It is well known that problems of unidentifiability arise for unreplicated observations and normal error structure. As in all data sets, outliers can arise and methods are needed for detecting and testing them. An outlier-generating model of mean–slippage type can be used to characterise four different forms of outlier manifestation. It is interesting to find that the unidentifiability problem provides no obstacle for detecting or testing the outliers for three of the four forms. Detection principles, and specific discordancy tests, are derived and illustrated by application to some data on physical measurements of Pacific squid.     

Hazard functions and life distributions in discrete time

The problem of studying lifelength distributions in discrete time is considered for certain forms of hazard functions. A class of life distributions that consists of the geometric, the Waring and the negative hypergeometric distributions is shown to result when the hazard function is inversely proportional to some linear function of time.     

Sufficient reduction methods for multivariate time-dependent health surveillance data

ABSTRACT

Early detection with a low false alarm rate (FAR) is the main aim of outbreak detection as used in public health surveillance or in regard to bioterrorism. Multivariate surveillance is preferable to univariate surveillance since correlation between series (CBS) is recognized and incorporated. Sufficient reduction has proved a promising method for handling CBS, but has not previously been used when correlation within series (CWS) is present. Here we develop sufficient reduction methods for reducing a p-dimensional multivariate series to a univariate series of statistics shown to be sufficient to monitor a sudden, but persistent, shift in the multivariate series mean. Correlation both within and between series is taken into account, as public health data typically exhibit both forms of association. Simultaneous and lagged changes and different shift sizes are investigated. A one-sided exponentially weighted moving average chart is used as a tool for detection of a change. The performance of the proposed method is compared with existing sufficient reduction methods, the parallel univariate method and both VarR and Z charts. A simulation study using bivariate normal autoregressive data shows that the new method gives shorter delays and a lower FAR than other methods, which have high FARs when CWS is clearly present.     

Modelling market shares by segments using volatility

This paper presents the results of market share modelling for individual segments of the UK tea market using scanner panel data. The study is novel in its introduction of the use of volatility as one of the bases for segmentation, others being usage, loyalty or switching between product types and product forms. The segmentation is undertaken on an a priori, quasi-experimental basis, allowing nested tests of constancy of elasticities across segments. The estimated equations (using seemingly unrelated regressions) benefit from extensive specification, including four diff erent forms for the price variable, four variables for promotion, and six for product characteristic, distribution and macroeconomic variables. Tests for the constancy of the parameters across segments show the segmentation to be successful.     

Testing for Serial Independence: Beyond the Portmanteau Approach

Portmanteau tests are typically used to test serial independence even if, by construction, they are generally powerful only in presence of pairwise dependence between lagged variables. In this article, we present a simple statistic defining a new serial independence test, which is able to detect more general forms of dependence. In particular, differently from the Portmanteau tests, the resulting test is powerful also under a dependent process characterized by pairwise independence. A diagram, based on p-values from the proposed test, is introduced to investigate serial dependence. Finally, the effectiveness of the proposal is evaluated in a simulation study and with an application on financial data. Both show that the new test, used in synergy with the existing ones, helps in the identification of the true data-generating process. Supplementary materials for this article are available online.     

Calibrated Bayes factors for model comparison

A Bayes factor between two models can be greatly affected by the prior distributions on the model parameters. When prior information is weak, very dispersed proper prior distributions are known to create a problem for the Bayes factor when competing models differ in dimension, and it is of even greater concern when one of the models is of infinite dimension. Therefore, we propose an innovative method which uses training samples to calibrate the prior distributions so that they achieve a reasonable level of ‘information’. Then the calibrated Bayes factor can be computed over the remaining data. This method makes no assumption on model forms (parametric or nonparametric) and can be used with both proper and improper priors. We illustrate, through simulation studies and a real data example, that the calibrated Bayes factor yields robust and reliable model preferences under various situations.