Order preserving functions and generalized convexity

In order to better understand the monotonicity properties which characterize the gradient of pseudoconvex and quasi convex funzions, psedomonotonicity and quasi monotonicity can be introduced. A quite different approach is proposed in this paper, by defining new order relations, whose preservation leads precisely to several kinds of pseudoconvexity and quasi convexity. Some general properties of order preservation are proved; they are useful to state necessary and sufficient conditions of monotonicity for particular orders related with generalized convexity.     

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.     

A note on complete families of distributions

Some methods to establish completeness of families of distributions are discussed. Several well known parametric families are shown to be complete.     

Partial identification of probability distributions with misclassified data

This paper addresses the problem of data errors in discrete variables. When data errors occur, the observed variable is a misclassified version of the variable of interest, whose distribution is not identified. Inferential problems caused by data errors have been conceptualized through convolution and mixture models. This paper introduces the direct misclassification approach. The approach is based on the observation that in the presence of classification errors, the relation between the distribution of the ‘true’ but unobservable variable and its misclassified representation is given by a linear system of simultaneous equations, in which the coefficient matrix is the matrix of misclassification probabilities. Formalizing the problem in these terms allows one to incorporate any prior information into the analysis through sets of restrictions on the matrix of misclassification probabilities. Such information can have strong identifying power. The direct misclassification approach fully exploits it to derive identification regions for any real functional of the distribution of interest. A method for estimating the identification regions and construct their confidence sets is given, and illustrated with an empirical analysis of the distribution of pension plan types using data from the Health and Retirement Study.     

Continua of stochastic dominance relations for unbounded probability distributions

Let P = {F,G,…} be the set of all probability distribution functions with support (0, ∞). An unrestricted stochastic dominance relation>_∞ is defined on P for each real ∞ 1, where F >_∞ G means that ^x_y = 0 (x - y)^{∞ - 1} dG(y) ^x_{n = 0}(x − y)^∞−1 dG(y) for all ∞ 0, with < for some x. These relations are partial orders that increase as ∞ increases with limit relation>_∞. A class U_∞ of utility functions u on (0, ∞) is defined in such a way that F >_∞ G iff udF > udG for all u ε U_∞. The U_∞ decrease as ∞ increases and have a non-empty intersection U_∞. Each u ε U_∞ is an increasing function that has derivatives of all orders that alternate in sign. Criteria which tell when F eventually dominates G in the sense of F >_∞ G are noted. Comparisons with bounded stochastic dominance results are made in several places.     

Continua of stochastic dominance relations for bounded probability distributions

Let P={F,G,…} be the set of probability distribution functions on [0,b]. For each αε[1, ∞), F_αG means that ∫^x_o(x−y^α−1dF(y)∫^x_o(x−y)^α−1dG(y) for all xε[0, b], and F>_αG means that F_αG and F≠G. Each _α is reflexive and transitive and each>_α is asymmetric and transitive. Both _α and>_α increase as α increases but their limits are not complete. A class U_α of utility functions is defined to give F>_αG iff ∫udF>∫udG for all uεU_α. These classes decrease as α increases, and their limit is empty. Similar decreasing classes are defined for each _α, and their limit is essentially the constant functions on (0, b].     

On the entropy and inaccuracy of similarly and oppositely ordered discrete probability distributions (I)

This note deals with the entropy and inaccuracy of similarly and oppositely ordered discrete probability distributions. The concept of inaccuracy range has also been introduced. In particular, the inacuracy of -Power distributions with respect to another distribution has been discussed in detail. It is shown that these inaccuracies are monotonically increasing function of \ for oppositely ordered distributions and decreasing function of for similarly ordered distributions.     

Characterizations of probability distributions via bivariate regression of record values

Bairamov et al. (Aust N Z J Stat 47:543–547, 2005) characterize the exponential distribution in terms of the regression of a function of a record value with its adjacent record values as covariates. We extend these results to the case of non-adjacent covariates. We also consider a more general setting involving monotone transformations. As special cases, we present characterizations involving weighted arithmetic, geometric, and harmonic means.     

Moment inequalities for order statistics from ordered families of distributions

Summary In this paper some inequalities for the variance and covariance of convex monotone functions of order statistics from ordered families of distributions are presented. The considered order relations in the set of distributions are the stochastic ordering relation and the convex ordering relation. Stochastic comparisons of spacings and their sums are also given. As corollaries the results for IFR and DFR distributions are obtained.     

Evaluating probabilistic forecasts of extremes using continuous ranked probability score distributions

Verifying probabilistic forecasts for extreme events is a highly active research area because popular media and public opinions are naturally focused on extreme events, and biased conclusions are readily made. In this context, classical verification methods tailored for extreme events, such as thresholded and weighted scoring rules, have undesirable properties that cannot be mitigated, and the well-known continuous ranked probability score (CRPS) is no exception.In this paper, we define a formal framework for assessing the behavior of forecast evaluation procedures with respect to extreme events, which we use to demonstrate that assessment based on the expectation of a proper score is not suitable for extremes. Alternatively, we propose studying the properties of the CRPS as a random variable by using extreme value theory to address extreme event verification. An index is introduced to compare calibrated forecasts, which summarizes the ability of probabilistic forecasts for predicting extremes. The strengths and limitations of this method are discussed using both theoretical arguments and simulations.     

Minimax estimation of constrained parametric functions for discrete families of distributions

For a vast class of discrete model families where the natural parameter is constrained to an interval, we give conditions for which the Bayes estimator with respect to a boundary supported prior is minimax under squared error loss type functions. Building on a general development of éric Marchand and Ahmad Parsian, applicable to squared error loss, we obtain extensions to various parametric functions and squared error loss type functions. We provide illustrations for various distributions and parametric functions, and these include examples for many common discrete distributions, as well as when the parametric function is a zero-count probability, an odds-ratio, a Binomial variance, and a Negative Binomial variance, among others. The Research of M. Jafari Jozani is supported by a grant of the Institute for Research and Planning in Higher Education, Ministry of Science, Research and Technology, Iran. The Research of é. Marchand is supported by NSERC of Canada.     

On the existence of random measure preserving bijections

This paper establishes the existence of an invariant probability measure on the set Π of measure-preserving bijections of the unit interval. In fact, the measure may be further restricted to be a Baire-measure. This result is compared with Aumann's (1967) and Aumann and Shapley's (1974) impossibility theorems.     

Shortest confidence interval estimation for families of distributions involving two truncation parameters

In this paper we construct shortest confidence intervals for families of distributions involving two unknown truncation parameters. The method used is that of the pivotal quantity.     

A note on the maximum probability property of MLE's in multiparameter exponential families

Summary For multiparameter exponential models a short and direct proof is given that the maximum likelihood estimator is a maximum probability estimator with respect to a certain sequence of convex and bounded sets inR _(k) that are symmetric about the origin; asymptotically these sets are allowed to be unbounded.     

On extremes of mixtures of distributions

In this article we study the max domains of attraction of mixtures of distributions belonging to the max domain of attraction of max stable laws.