Autoregresive conditional volatility, skewness and kurtosis

This paper proposes a GARCH-type model allowing for time-varying volatility, skewness and kurtosis. The model is estimated assuming a Gram–Charlier (GC) series expansion of the normal density function for the error term, which is easier to estimate than the non-central t distribution proposed by [Harvey, C. R. & Siddique, A. (1999). Autorregresive Conditional Skewness. Journal of Financial and Quantitative Analysis 34, 465–487). Moreover, this approach accounts for time-varying skewness and kurtosis while the approach by Harvey and Siddique [Harvey, C. R. & Siddique, A. (1999). Autorregresive Conditional Skewness. Journal of Financial and Quantitative Analysis 34, 465–487] only accounts for non-normal skewness. We apply this method to daily returns of a variety of stock indices and exchange rates. Our results indicate a significant presence of conditional skewness and kurtosis. It is also found that specifications allowing for time-varying skewness and kurtosis outperform specifications with constant third and fourth moments.     

Computing the covariance of two Brownian area integrals

We compute the expected product of two correlated Brownian area integrals, a problem that arises in the analysis of a popular sorting algorithm. Along the way we find three different formulas for the expectation of the product of the absolute values of two standard normal random variables with correlation θ . These two formulas are found: (a) via conditioning and the non-central chi-square distribution; (b) via Mehler's formula; (c) by representing the correlated normal random variables in terms of independent normal's and integration using polar coordinates.     

Bivariate non-central negative binomial distribution: Another generalisation

Summary Another bivariate generalisation (Type V) of the non-central negative binomial distribution is considered. This generalisation is constructed (i) as a latent structure model; (ii) as an extension of an accident proneness model investigated by Edwards/Gurland (1961); and (iii) as a reversible stochastic counter model. The third construction gives, as a result, an apparently new formulation of the Edwards/Gurland model. The probabilities, moments, recurrence formulas and some properties are given. An application to the data used by Holgate (1966) is considered.     

A stochastic differential equation approach to the analysis of the 2017 and 2019 UK general election polls

Human dynamics and sociophysics build on statistical models that can shed light on and add to our understanding of social phenomena. We propose a generative model based on a stochastic differential equation that enables us to model the opinion polls leading up to the 2017 and 2019 UK general elections and to make predictions relating to the actual results of the elections. After a brief analysis of the time series of the poll results, we provide empirical evidence that the gamma distribution, which is often used in financial modelling, fits the marginal distribution of this time series. We demonstrate that the proposed poll-based forecasting model may improve upon predictions based solely on polls. The method uses the Euler–Maruyama method to simulate the time series, measuring the prediction error with the mean absolute error and the root mean square error, and as such could be used as part of a toolkit for forecasting elections.     

Compound gamma,beta and F distributions

Summary In this paper a compound gamma distribution has been derived by compounding a gamma distribution with another gamma distribution. The resulting compound gamma distribution has been reduced to the Beta distributions of the first kind and the second kind and to theF distribution by suitable transformations. This includes theLomax distribution as a special case which enjoys a useful property. Moment estimators for two of its parameters are explicitly obtained, which tend to a bivariate normal distribution. The paper contains expressions for a bivariate probability density function, its conditional expectation, conditional variance and the product moment correlation coefficient. Finally, all the parameters of the compound gamma distribution are explicitly expressed in terms of the functions of the moments of the functions of random variables in two different ways. This note is based on a technical report prepared by the author while he was with the Procter and Gamble Company.     

陕西省非中心城市体育院系科研论文的统计分析

主要运用文献资料法和统计分析法,通过Internet在《中国期刊全文数据库》检索,对陕西省非中心城市的六所高校体育院系2005-2010年体育科研论文进行了统计与分析。统计数据表明:近六年来陕西省非中心城市的高校体育院系的科研论文数在不断的增加,科研水平在逐步提高;科研论文的主题主要集中在学校体育、体育理论这两方面,竞技体育、体育产业方面的论文较少;有部分学校也发表了少数非体育类科研论文;这些学校的论文刊登在核心期刊上的论文数较少,分析得出,陕西省非中心城市高校体育院系的科研水平较弱,需要进一步加强与提升。     

Nonlinear regression and curved exponential families. Improvement of the approximation to the asymptotic distribution

Inference in nonlinear models is usually based on the asymptotic normal distribution, based on linearizing the model. The accuracy of this approximation can in many cases be improved by a reparametrization. Systematic methods for doing this will be described. Sometimes a saddlepoint approximation can be used, and this offers several advantages compared to the asymptotic distribution and the Edgeworth expansion. The improved methods are unfortunately not commonly used. It will be discussed why this is so. The methods will be illustrated by a series of examples.     

Robust and Powerful Tests for Nonlinear Deterministic Components

We develop a test for the presence of nonlinear deterministic components in a univariate time series, approximated using a Fourier series expansion, designed to be asymptotically robust to the order of integration of the process and to any weak dependence present. We show that our proposed test has uniformly greater local asymptotic power than the existing tests of Harvey, Leybourne and Xiao (2010) when the shocks are I(1), identical local asymptotic power when the shocks are I(0), and also improved finite sample properties. We also consider the issue of determining the number of Fourier frequencies used to specify any nonlinear deterministic components.     

Full Likelihood Inference in Normal-Gamma Stochastic Frontier Models

The paper takes up inference in the stochastic frontier model with gamma distributed inefficiency terms, without restricting the gamma distribution to known integer values of its shape parameter (the Erlang form). The paper shows that Gibbs sampling with data augmentation can be used in a computationally efficient way to explore the posterior distribution of the model and conduct inference regarding parameters as well as functions of interest related to technical inefficiency.     

On the normality of probability distributions of inflation and GNP forecasts

This study analyzes mean probability distributions reported by ASA-NBER forecasters on two macroeconomic variables, GNP and the GNP implicit price deflator. In the derivation of expectations, a critical assertion has been that the aggregate average expectation can be regarded as coming from a normal distribution. We find that, in fact, this assumption should be rejected in favor of distributions which are more peaked and skewed. For IPD, they are mostly positively skewed, and for nominal GNP the reverse is true. We then show that a non-central scaled t-distribution fit the empirical distributions remarkably well. The practice of using the degree of consensus across a group of predictions as a measure of a typical forecasters' uncertainty about the prediction is called to question.     

The Early History of the Cumulants and the Gram-Charlier Series

The early history of the Gram-Charlier series is discussed from three points of view: (1) a generalization of Laplace's central limit theorem, (2) a least squares approximation to a continuous function by means of Chebyshev-Hermite polynomials, (3) a generalization of Gauss's normal distribution to a system of skew distributions. Thiele defined the cumulants in terms of the moments, first by a recursion formula and later by an expansion of the logarithm of the moment generating function. He devised a differential operator which adjusts any cumulant to a desired value. His little known 1899 paper in Danish on the properties of the cumulants is translated into English in the Appendix.     

Decision rules for the choice of structural equations

In practical econometric analysis we are faced with the problem of how to specify structural equations. The conventional t-test of coefficients is apparently inappropriate. The smallest root, say λ, of a certain determinantal equation provides us with basis for the test of overidentifying restrictions. The preliminary test, based on λ, may give us a possible decision rule for choosing a structural equation from nested alternatives. However, ambiguity remains in specifying the significance level. We propose a decision method called the unbiased decision rule; unbiased in the sense that we attain a correct decision with probability of more than a half. The critical points are found as the medians of non-central F-distributions. The degrees of freedom and the non-centrality parameter of non-central F-distributions are determined by the properties of contending models. We also discuss the implications of the unbiased decision rule in the context of the conventional pre-test.     

Modelling Time Series of Counts in Epidemiology

We review generalized dynamic models for time series of count data. Usually temporal counts are modelled as following a Poisson distribution, and a transformation of the mean depends on parameters which evolve smoothly with time. We generalize the usual dynamic Poisson model by considering continuous mixtures of the Poisson distribution. We consider Poisson‐gamma and Poisson‐log‐normal mixture models. These models have a parameter for each time t which captures possible extra‐variation present in the data. If the time interval between observations is short, many observed zeros might result. We also propose zero inflated versions of the models mentioned above. In epidemiology, when a count is equal to zero, one does not know if the disease is present or not. Our model has a parameter which provides the probability of presence of the disease given no cases were observed. We rely on the Bayesian paradigm to obtain estimates of the parameters of interest, and discuss numerical methods to obtain samples from the resultant posterior distribution. We fit the proposed models to artificial data sets and also to a weekly time series of registered number of cases of dengue fever in a district of the city of Rio de Janeiro, Brazil, during 2001 and 2002.     

Reparameterized inverse gamma regression models with varying precision

In this article, we propose a mean linear regression model where the response variable is inverse gamma distributed using a new parameterization of this distribution that is indexed by mean and precision parameters. The main advantage of our new parametrization is the straightforward interpretation of the regression coefficients in terms of the expectation of the positive response variable, as usual in the context of generalized linear models. The variance function of the proposed model has a quadratic form. The inverse gamma distribution is a member of the exponential family of distributions and has some distributions commonly used for parametric models in survival analysis as special cases. We compare the proposed model to several alternatives and illustrate its advantages and usefulness. With a generalized linear model approach that takes advantage of exponential family properties, we discuss model estimation (by maximum likelihood), black further inferential quantities and diagnostic tools. A Monte Carlo experiment is conducted to evaluate the performances of these estimators in finite samples with a discussion of the obtained results. A real application using minerals data set collected by Department of Mines of the University of Atacama, Chile, is considered to demonstrate the practical potential of the proposed model.     

On Some Statistical Methods for Modelling the Incidence of Poverty

This paper analyzes a common method of estimating the pattern of individual characteristics associated with poverty – the logit/probit technique applied to a dummy variable identifying those individuals who are below a poverty line. We show that this procedure is seriously flawed by a fundamental logical difficulty and further statistical complications when the poverty line is estimated or when generalized poverty measures are used. An alternative approach is proposed, based on a semi-parametric series expansion appproximation to the conditional income distribution. Estimation, testing and summary procedures are established and applied to Hungarian survey data.     

The distribution function of a statistic for testing the equality of scale parameters in two gamma populations

Summary The exact distribution function of the ratio of two sums of gamma variates is derived in this paper. The result applies to ratios of quadratic forms and to a statistic used for testing the equality of scale parameters in two gamma populations.     

Interpoint Distance Test of Homogeneity for Multivariate Mixture Models

Finite mixtures offer a rich class of distributions for modelling of a variety of random phenomena in numerous fields of study. Using the sample interpoint distances (IPDs), we propose the IPD‐test statistic for testing the hypothesis of homogeneity of mixture of multivariate power series distribution or multivariate normal distribution. We derive the distribution of the IPDs that are drawn from a finite mixture of the multivariate power series distribution and multivariate normal distribution. Based on the empirical distribution of the IPDs, we construct a bootstrap test of homogeneity for other multivariate finite mixture models. The IPD test is applied to mixture models for matrix‐valued distributions and a test of homogeneity for Wishart mixture is presented. Numerical comparisons show that IPD test has accurate type I errors and is more powerful in most multivariate cases than the expectation–maximization (EM) test and modified likelihood ratio test.     

The bivariate non-central negative binomial distributions

Summary Four bivariate generalisations (Type I–IV) of the non-central negative binomial distribution (Ong/Lee) are considered. The Type I generalisation is constructed using the latent structure model scheme (Goodman) while the Type II generalisation arises from a variation of this scheme. The Type III generalisation is formed by using the method of random elements in common (Mardia). The Type IV is an extension of the Type I generalisation. Properties of these bivariate distributions including joint central and factorial moments are discussed; several recurrence formulae of the probabilities are given. An application to the childhood accident data of Mellinger et al. is considered with the precision of the Type I maximum likelihood estimates computed.     

On the product of gamma random variables

We give expressions for the distribution and density of a product of gamma or equivalently chi-square random variables. In particular, we give the distribution of the product of two independent gamma variables of mean k in terms of the Bessel functions K ₁, … , K _k .     

Multivariate gamma distributions with one-factorial accompanying correlation matrices and applications to the distribution of the multivariate range

Summary A new representation for the characteristic function of the joint distribution of the Mahalanobis distances betweenk independentN(μ, Σ)-distributed points is given. Especially fork=3 the corresponding distribution function is obtained as a special case of multivariate gamma distributions whose accompanying normal distribution has a positive semidefinite correlation matrix with correlationsϱ _ij=−a _i a _j. These gamma distribution functions are given here by one-dimensional parameter integrals. With some further trivariate gamma distributions third order Bonferroni inequalities are derived for the upper tails of the distribution function of the multivariate range ofk independentN(μ, I)-distributed points. From these inequalities very accurate (conservative) approximations to upperα-level bounds can also be computed for studentized multivariate ranges.