The t Copula and Related Copulas

The t copula and its properties are described with a focus on issues related to the dependence of extreme values. The Gaussian mixture representation of a multivariate t distribution is used as a starting point to construct two new copulas, the skewed t copula and the grouped t copula, which allow more heterogeneity in the modelling of dependent observations. Extreme value considerations are used to derive two further new copulas: the t extreme value copula is the limiting copula of componentwise maxima of t distributed random vectors; the t lower tail copula is the limiting copula of bivariate observations from a t distribution that are conditioned to lie below some joint threshold that is progressively lowered. Both these copulas may be approximated for practical purposes by simpler, better-known copulas, these being the Gumbel and Clayton copulas respectively.     

Fisher information in order statistics and their concomitants in bivariate censored samples

We evaluate the Fisher information (FI) contained in a collection of order statistics and their concomitants from a bivariate random sample. Special attention is given to Type II censored samples. We present a general decomposition result and recurrence relations that are useful in finding the FI in all types of censored samples. We also obtain some asymptotic results for the FI. For the bivariate normal parent, we obtain explicit and asymptotic expressions for the elements of the FI matrix for Type II censored samples. We discuss implications of our findings on inference on the bivariate normal parameters, especially on the correlation. The first author’s research was supported in part by National Institutes of Health, USA, Grant # M01 RR00034 and the second author’s research was supported by a training grant from the Egyptian government     

Remarks on the strong law of large numbers for a triangular array of associated random variables

A strong law of large numbers for a triangular array of strictly stationary associated random variables is proved. It is used to derive the pointwise strong consistency of kernel type density estimator of the one-dimensional marginal density function of a strictly stationary sequence of associated random variables, and to obtain an improved version of a result by Van Ryzin (1969) on the strong consistency of density estimator for a sequence of independent and identically distributed random variables.     

On three and five parameter bivariate beta distributions

Summary In this paper two bivariate beta distributions have been studied. The five parameter bivariate beta distribution is derived from the Morgenstern-system of curves while the three parameter distribution is the bivariate Dirichlet distribution. In both cases the distributions of the product and the quotient of random variables are derived and other properties are also studied.     

Statistics of extremes in climatology

Climatology is an area with many applications of extreme–value theory. In applications to climatic data it is necessary to consider the distribution ol extremes for dependent random variables and to make allowance for non–stationarity (seasonal variation). It is shown that also for such data the classical limiting distributions for normalized maxima of independent and identically distributed random variables remain useful candidates to describe the distribution of the largest value in a year.
Parameter estimation of extreme–value distributions is briefly reviewed. Attention is paid to variance–reduction of quantile estimates by making use of more observations than just the annual maxima. Three examples of applications deal with topics like the estimation of large quantiles by combining information from several records in a region, shifts in the distribution of maxima due to changes in measurement practices, and the use of covariates.     

The rate of convergence in distribution of the maxima

Abstract  Assume the normalized maxima, from an independent and identically distributed sequence of random variables, converge in distribution to a non-degenerate random variable with extreme value distribution G ( x ). We derive a pointwise rate for this convergence which holds for all n and x with G(x ) > 0. A closer examination of this result suggests how a uniform rate can be obtained.     

Monotonicity of the (reversed) hazard rate of the (maximum) minimum in bivariate distributions

It is well known that in the case of independent random variables, the (reversed) hazard rate of the (maximum) minimum of two random variables is the sum of the individual (reversed) hazard rates and hence the onotonicity of the (reversed) hazard rate of the marginals is preserved by the monotonicity of the (reversed) hazard rate of the (maximum) minimum. However, for the bivariate distributions this property is not always preserved. In this paper, we study the monotonicity of the (reversed) hazard rate of the (maximum) minimum for two well known families of bivariate distributions viz the Farlie-Gumbel-Morgenstern (FGM) and Sarmanov family. In case of the FGM family, we obtain the (reversed) hazard rate of the (maximum) minimum and provide several examples in some of which the (reversed) hazard rate is monotonic and in others it is non-monotonic. In the case of Sarmanov family the (reversed) hazard rate of the (maximum) minimum may not be expressed in a compact form in general. We consider some examples to illustrate the procedureResearch of the second author is supported by a grant from Natural Sciences and Engineering Research Council and the research of the other two authors is partially supported by a travel grant from the Canadian American Center of the University of Maine     

A Bounded Model of Time Variation in Trend Inflation,Nairu and the Phillips Curve

In this paper, we develop a bivariate unobserved components model for inflation and unemployment. The unobserved components are trend inflation and the non‐accelerating inflation rate of unemployment (NAIRU). Our model also incorporates a time‐varying Phillips curve and time‐varying inflation persistence. What sets this paper apart from the existing literature is that we do not use unbounded random walks for the unobserved components, but rather bounded random walks. For instance, NAIRU is assumed to evolve within bounds. Our empirical work shows the importance of bounding. We find that our bounded bivariate model forecasts better than many alternatives, including a version of our model with unbounded unobserved components. Our model also yields sensible estimates of trend inflation, NAIRU, inflation persistence and the slope of the Phillips curve. Copyright © 2015 John Wiley & Sons, Ltd.     

Some characterizations of the bivariate normal distribution

Summary In this note assuming the conditional normality of one component of a bivariate random vector given the value of the other component some characterizations of the joint normality of the random vector are given.     

THE ROLE OF CONDITIONAL HETEROSKEDASTICITY IN IDENTIFYING AND ESTIMATING LINEAR TRIANGULAR SYSTEMS,WITH APPLICATIONS TO ASSET PRICING MODELS THAT INCLUDE A MISMEASURED FACTOR

A new estimator is proposed for linear triangular systems, where identification results from the model errors following a bivariate and diagonal GARCH(1,1) process with potentially time‐varying error covariances. This estimator applies when traditional instruments are unavailable. I demonstrate its usefulness on asset pricing models like the capital asset pricing model and Fama–French three‐factor model. In the context of a standard two‐pass cross‐sectional regression approach, this estimator improves the pricing performance of both models. Set identification bounds and an associated estimator are also provided for cases where the conditions supporting point identification fail. Copyright © 2013 John Wiley & Sons, Ltd.     

基于时间价值的国际集装箱运输路径的优化研究

在国际物流中，对集装箱货主而言，选择最佳的出口港不仅仅是考虑陆上运输方式的成本优化，而是要考虑到最终用户手中总成本的优化问题，即需要考虑货物的时间价值。文章以广西防城港经济腹地国际集装箱运输为研究对象，利用货主决策模型，讨论了基于时问价值的货流路径优化问题，为防城港的发展提出了一些建议。     

Maximum likelihood estimation of random effects models

Iterated GLS has a remarkable property when applied to the random effects model in its usual parameterization. The values for the parameter that measures relative variance, obtained through successive iterations, form a monotonic sequence. This property provides convenient checks for multiple maxima of the likelihood function and for existence of a local maximum that satisfies the non-negativity condition.     

Measures of multivariate asymptotic dependence and their relation to spectral expansions

Asymptotic dependence can be interpreted as the property that realizations of the single components of a random vector occur simultaneously with a high probability. Information about the asymptotic dependence structure can be captured by dependence measures like the tail dependence parameter or the residual dependence index. We introduce these measures in the bivariate framework and extend them to the multivariate case afterwards. Within the extreme value theory one can model asymptotic dependence structures by Pickands dependence functions and spectral expansions. Both in the bivariate and in the multivariate case we also compute the tail dependence parameter and the residual dependence index on the basis of this statistical model. They take a specific shape then and are related to the Pickands dependence function and the exponent of variation of the underlying density expansion.     

The M/M/1 queue with gated random order of service

We analyse an M / M /1 queueing model with gated random order of service discipline. In this service discipline there is a waiting room, in which arriving customers are collected, and a service queue. Each time the service queue becomes empty, all customers in the waiting room are instantaneously put in random order in the service queue. We find the joint stationary distribution of the number of customers in the waiting room and the service queue. Furthermore, we obtain the bivariate Laplace–Stieltjes transform of the joint distribution of the sojourn times of a customer in the waiting room and the service queue.     

Another look at risk apportionment

This paper presents a general result on the random selection of an element from an ordered sequence of risks and uses this result to derive additive and cross risk apportionment. Preferences favoring an improvement of the sampling distribution in univariate or bivariate first-order stochastic dominance are those exhibiting additive or cross risk apportionment. The univariate additive and multiplicative risk apportionment concepts are then related to the notion of bivariate cross risk apportionment by viewing the single-attribute utility function of an aggregate position (sum or product of attributes) as a 2-attribute utility function. The results derived in the present paper allow one to further explore the connections between the different concepts of risk apportionment proposed so far in the literature.     

Partial observability in bivariate probit models

This study investigates random utility models in which the observed binary outcome does not reflect the binary choice of a single decision-maker, but rather the joint unobserved binary choices of two decision-makers. Under the usual normality assumptions, the model that arises for the observed binary outcome is not a univariate probit model, but rather a bivariate probit model in which only one of the four possible outcomes is observed. Estimation and identification issues are discussed, and the implications for sample selectivity problems are noted.     

Usual stochastic ordering of the sample maxima from dependent distribution-free random variables

In this paper, we discuss stochastic comparison of the largest order statistics arising from two sets of dependent distribution-free random variables with respect to multivariate chain majorization, where the dependency structure can be defined by Archimedean copulas. When a distribution-free model with possibly two parameter vectors has its matrix of parameters changing to another matrix of parameters in a certain mathematical sense, we obtain the first sample maxima is larger than the second sample maxima with respect to the usual stochastic order, based on certain conditions. Applications of our results for scale proportional reverse hazards model, exponentiated gamma distribution, Gompertz–Makeham distribution, and location-scale model, are also given. Meanwhile, we provide two numerical examples to illustrate the results established here.     

Characterizations and Modelling of Multivariate Lack of Memory Property

In this article we establish characterizations of multivariate lack of memory property in terms of the hazard gradient (whenever exists), the survival function and the cumulative hazard function. Based on one of these characterizations we establish a method of generating bivariate lifetime distributions possessing bivariate lack of memory property (BLMP) with specified marginals. It is observed that the marginal distributions have to satisfy certain conditions to be stated. The method generates absolutely continuous bivariate distributions as well as those containing a singular component. Bivariate exponential distributions due to Proschan and Sullo (Reliability and biometry, pp 423–440, 1974), Freund (in J Am Stat Assoc 56:971–977, 1961), Block and Basu (J Am Stat Assoc 89:1091–1097, 1974) and Marshall and Olkin (J Am Math Assoc 62:30–44, 1967) are generated as particular cases among others using the proposed method. Some other distributions generated using the method may be of practical importance. Shock models leading to bivariate distributions possessing BLMP are given. Some closure properties of a class of univariate failure rate functions that can generate distributions possessing BLMP and of the class of bivariate survival functions having BLMP are studied.     

Parameter estimation for a bivariate lifetime distribution in reliability with multivariate extensions

Bivariate lifetime distributions are considered which describe physically motivated dependencies like those proposed by Freund (1961) and Marshall and Olkin (1967a). Such distributions arise in reliability problems with two-component systems. Generalizations of some previous models are investigated and the maximum likelihood estimates for a combined bivariate exponential distribution are given. The case of dependent random censorship is considered in connection with two-component series systems. Some simulations show how censorship affects the parameter estimates.     

A theory of hyperfinite processes: the complete removal of individual uncertainty via exact LLN

The aim of this paper is to provide a viable measure–theoretic framework for the study of random phenomena involving a large number of economic entities. The work is based on the fact that processes which are measurable with respect to hyperfinite Loeb product spaces capture the limiting behaviors of triangular arrays of random variables and thus constitute the `right' class for general stochastic modeling. The primary concern of the paper is to characterize those hyperfinite processes satisfying the exact law of large numbers by using the basic notions of conditional expectation, orthogonality, uncorrelatedness and independence together with some unifying multiplicative properties of random variables. The general structure of the processes is also analyzed via a biorthogonal expansion of the Karhunen–Loéve type and via the representation in terms of the simpler hyperfinite Loeb counting spaces. A universality property for atomless Loeb product spaces is formulated to show the abundance of processes satisfying the law. Generalizations to a hyperfinite number of continuous (or discrete) parameter stochastic processes are considered. The various necessary and sufficient conditions for the validity of the law provide a rather complete understanding about the cancelation of individual risks or uncertainty in general settings. Some explicit asymptotic interpretations are also given.