Complete monotonicity of some entropies

It is well-known that the Shannon entropies of some parameterized probability distributions are concave functions with respect to the parameter. In this paper we consider a family of such distributions (including the binomial, Poisson, and negative binomial distributions) and investigate their Shannon, Rényi, and Tsallis entropies with respect to complete monotonicity.     

Random locations of periodic stationary processes

We consider a family of random locations, called intrinsic location functionals, of periodic stationary processes. This family includes but is not limited to the location of the path supremum and first/last hitting times. We first show that the set of all possible distributions of intrinsic location functionals for periodic stationary processes is the convex hull generated by a specific group of distributions. We then focus on two special subclasses of these random locations. For the first subclass, the density has a uniform lower bound; for the second subclass, the possible distributions are closely related to the concept of joint mixability.     

Conditional tail risk measures for the skewed generalised hyperbolic family

This paper deals with the estimation of loss severity distributions arising from historical data on univariate and multivariate losses. We present an innovative theoretical framework where a closed-form expression for the tail conditional expectation (TCE) is derived for the skewed generalised hyperbolic (GH) family of distributions. The skewed GH family is especially suitable for equity losses because it allows to capture the asymmetry in the distribution of losses that tends to have a heavy right tail. As opposed to the widely used Value-at-Risk, TCE is a coherent risk measure, which takes into account the expected loss in the tail of the distribution. Our theoretical TCE results are verified for different distributions from the skewed GH family including its special cases: Student-t, variance gamma, normal inverse gaussian and hyperbolic distributions. The GH family and its special cases turn out to provide excellent fit to univariate and multivariate data on equity losses. The TCE risk measure computed for the skewed family of GH distributions provides a conservative estimator of risk, addressing the main challenge faced by financial companies on how to reliably quantify the risk arising from the loss distribution. We extend our analysis to the multivariate framework when modelling portfolios of losses, allowing the multivariate GH distribution to capture the combination of correlated risks and demonstrate how the TCE of the portfolio can be decomposed into individual components, representing individual risks in the aggregate (portfolio) loss.     

The generalized hypergeometric family of distributions

Summary This is an expository summary of the authors' report on classification of the generalized hypergeometric (GHg for short) family of distributions (Sibuya and Shimizu (1981),Keio Science and Technology Report, to appear). Emphasis is laid on the definition of the distributions based on some conventional rules, and on the complete classification of the multivariate GHg distributions, whose types are found to be rather limited in spite of their quite general definition. Previous classifications and namings are summarized and compared with the new one.     

A Functional Equation Whose Unknown is \mathcal{P}([0,1]) Valued

We study a functional equation whose unknown maps a Euclidean space into the space of probability distributions on [0,1]. We prove existence and uniqueness of its solution under suitable regularity and boundary conditions, we show that it depends continuously on the boundary datum, and we characterize solutions that are diffuse on [0,1]. A canonical solution is obtained by means of a Randomly Reinforced Urn with different reinforcement distributions having equal means. The general solution to the functional equation defines a new parametric collection of distributions on [0,1] generalizing the Beta family.     

Multivariate Tweedie distributions and some related capital-at-risk analyses

We study a multivariate extension of the univariate exponential dispersion Tweedie family of distributions. The class, referred to as the multivariate Tweedie family (MTwF), on the one hand includes multivariate Poisson, gamma, inverse Gaussian, stable and compound Poisson distributions and on the other hand introduces a high variety of new dependent probabilistic models unstudied so far. We investigate various properties of MTwF and discuss its possible applications to financial risk management.     

Family of multivariate generalized t distributions

In this paper, we introduce a new family of multivariate distributions as the scale mixture of the multivariate power exponential distribution introduced by Gómez et al. (Comm. Statist. Theory Methods 27(3) (1998) 589) and the inverse generalized gamma distribution. Since the resulting family includes the multivariate t distribution and the multivariate generalization of the univariate GT distribution introduced by McDonald and Newey (Econometric Theory 18 (11) (1988) 4039) we call this family as the “multivariate generalized t-distributions family”, or MGT for short. We show that this family of distributions belongs to the elliptically contoured distributions family, and investigate the properties. We give the stochastic representation of a random variable distributed as a multivariate generalized t distribution. We give the marginal distribution, the conditional distribution and the distribution of the quadratic forms. We also investigate the other properties, such as, asymmetry, kurtosis and the characteristic function.     

Loss Estimation for Spherically Symmetrical Distributions

In this paper we consider the problem of estimating the quadratic loss of point estimators of a location parameter for a family of spherically symmetric distributions. We compare the unbiased loss estimator of the minimax estimator with a new shrinkage type loss estimator. Conditions on the distributions for the domination of competing estimators are given. It is shown that, in addition to the class of scale mixtures of normal distributions, there exists a more general family for which the domination results hold.     

Best upper quantile evaluations for NWU distributions

We present optimal upper mean-variance evaluations of quantiles in classes of distributions succeeding a fixed one in the superadditive order. In particular, we provide explicit bounds for the NWU family and show that these are attained by geometric distributions.     

Completeness and unbiased estimation of mean vector in the multivariate group sequential case

We consider estimation after a group sequential test about a multivariate normal mean, such as a χ² test or a sequential version of the Bonferroni procedure. We derive the density function of the sufficient statistics and show that the sample mean remains to be the maximum likelihood estimator but is no longer unbiased. We propose an alternative Rao-Blackwell type unbiased estimator. We show that the family of distributions of the sufficient statistic is not complete, and there exist infinitely many unbiased estimators of the mean vector and none has uniformly minimum variance. However, when restricted to truncation-adaptable statistics, completeness holds and the Rao-Blackwell estimator has uniformly minimum variance.     

On equivalence of weak convergence and moment convergence of life distributions

We investigate the equivalence of weak convergence and moment convergence of life distributions in a family which is larger than the HNBUE family. Also, we point out that within the HNBUE family the exponential distribution can be characterized by one value of its Laplace-Stieltjes transform.     

On the Lagrangian Katz family of distributions as a claim frequency model

The Panjer (Katz) family of distributions is defined by a particular first-order recursion which is built on the basis of two parameters. It is known to characterize the Poisson, negative binomial and binomial distributions. In insurance, its main usefulness is to yield a simple recursive algorithm for the aggregate claims distribution. The present paper is concerned with the more general Lagrangian Katz family of distributions. That family satisfies an extended recursion which now depends on three parameters. To begin with, this recursion is derived through a certain first-crossing problem and two applications in risk theory are described. The distributions covered by the recursion are then identified as the generalized Poisson, generalized negative binomial and binomial distributions. A few other properties of the family are pointed out, including the index of dispersion, an extended Panjer algorithm for compound sums and the asymptotic tail behaviour. Finally, the relevance of the family is illustrated with several data sets on the frequency of car accidents.     

Uniform Mixtures Via Posterior Means

The problem of identification of uniform mixtures via posterior means is studied. For linear posterior means a complete solution is given. It determines a family of prior distributions involving beta of both kinds and gamma. Identifiability via any consistent posterior mean is also investigated.     

Fisher-Rao geometry of Dirichlet distributions

In this paper, we study the geometry induced by the Fisher-Rao metric on the parameter space of Dirichlet distributions. We show that this space is a Hadamard manifold, i.e. that it is geodesically complete and has everywhere negative sectional curvature. An important consequence for applications is that the Fréchet mean of a set of Dirichlet distributions is uniquely defined in this geometry.     

Fiducial inference in the pivotal family of distributions

In this paper a family, called the pivotal family, of distributions is considered. A pivotal family is determined by a generalized pivotal model. Analytical results show that a great many parametric families of distributions are pivotal. In a pivotal family of distributions a general method of deriving fiducial distributions of parameters is proposed. In the method a fiducial model plays an important role. A fiducial model is a function of a random variable with a known distribution, called the pivotal random element, when the observation of a statistic is given. The method of this paper includes some other methods of deriving fiducial distributions. Specially the first fiducial distribution given by Fisher can be derived by the method. For the monotone likelihood ratio family of distributions, which is a pivotal family, the fiducial distributions have a frequentist property in the Neyman-Pearson view. Fiducial distributions of regular parametric functions also have the above frequentist property. Some advantages of the fiducial inference are exhibited in four applications of the fiducial distribution. Many examples are given, in which the fiducial distributions cannot be derived by the existing methods.     

垂直密度表示及其应用

该文论述了Vertical Density Representation (VDR)的历史发展, 现状及其在随机数生成, 多元密度构造等领域的应用及在非正态多元统计分析的潜在应用.     

A family of premium principles based on mixtures of TVaRs

Risk-adjusted distributions are commonly used in actuarial science to define premium principles. In this paper, we claim that an appropriate risk-adjusted distribution, besides satisfying other desirable properties, should be well-behaved under conditioning with respect to the original risk distribution. Based on a sequence of such risk-adjusted distributions, we introduce a family of premium principles that gradually incorporate the degree of risk-aversion of the insurer in the risk loading. Members of this family are particular distortion premium principles that can be represented as mixtures of TVaRs, where the weights in the mixture reflect the attitude toward risk of the insurer. We make a systematic study of this family of premium principles.     

Smooth distributions are finitely generated

A subbundle of variable dimension inside the tangent bundle of a smooth manifold is called a smooth distribution if it is the pointwise span of a family of smooth vector fields. We prove that all such distributions are finitely generated, meaning that the family may be taken to be a finite collection. Further, we show that the space of smooth sections of such distributions need not be finitely generated as a module over the smooth functions. Our results are valid in greater generality, where the tangent bundle may be replaced by an arbitrary vector bundle.     

Probabilistic analysis of the Davis Putnam procedure for solving the satisfiability problem

We look at the instance distributions used by Goldberg [3] for showing that the Davis Putnam Procedure has polynomial average complexity and show that, in a sense, all these distributions are unreasonable. We then present a ‘reasonable’ family of instance distributions F and show that for each distribution in F a variant of the Davis Putnam Procedure without the pure literal rule requires exponential time with probability 1. In addition, we show that adding subsumption still results in exponential complexity with probability 1.     

Convergence Rate of the Distributions of Normalized Maximum Likelihood Estimators for Irregular Parametric Families

We study the convergence rate of the distributions of normalized maximum likelihood estimators defined by a parametric family of discontinuous multidimensional densities in the case of a vector parameter.