Exponential rates for kernel density estimation under association

The estimation of density based on positive dependent samples has been studied recently with consistency and asymptotic normality results being obtained. In with regard to the characterization on decrease rates the results have been scarce. We prove two versions of an exponential inequality: one assuming stationarity and association alone and the other under a further assumption on the joint distributions of the sample. These inequalities are then used to prove exponential decrease rates for the kernel estimator of the density with a uniform version over compact sets. The conditions assumed impose convenient decrease rates on the covariance structure of the sample. Some examples supposing geometrical or polynomial decrease rates on the covariances that fulfill our assumptions are presented in the last section. Explicit almost sure rates are derived for geometrically decreasing covariances. Under the extra assumption on the joint distributions the rates are close to the best known ones for independent variables.     

A note on reliability properties of k-record statistics

LetY _k,n denote the nth (upper) k-record value of an infinite sequence of independent, identically distributed random variables with common continuous distribution function F. We show that if the nth k-record valueY _k,n has an increasing failure rate (IFR), thenY _l,n (l<k) andY _k+1,n+1(n≤k+1) also have IFR distributions. On the other hand, ifY _k,n has a decreasing failure rate (DFR), thenY _l,n (1>k) has also a DFR distribution. We also present some results concerning log convexity and log concavity ofY _k,n .     

Patterns and dynamics of population heterogeneity in mixtures models

Mixtures of distributions are a common modelling tool for durations of social phenomena, especially when the population is believed to be heterogeneous. We discuss heterogeneity patterns which can be captured by various mixing distributions in continuous and discrete time. Particular attention is given to recidivism data which Kaplan modeled by beta-mixtures of geometric distributions. We also investigate the dynamics of heterogeneity, measured via the variance of the mixing distribution, over the duration. It is shown that not all mixture models display decreasing heterogeneity over time.     

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.     

Noncentral bimatrix variate generalised beta distributions

In this paper, we determine the density functions of nonsymmetrised doubly noncentral matrix variate beta type I and II distributions. The nonsymmetrised density functions of doubly noncentral and noncentral bimatrix variate generalised beta type I and II distributions are also obtained.     

Interarrival times in a counting process and bird watching

We determine simple approximate relations between the distributions of the interarrival times and the total time of a stopped counting process, under the assumption of small intensities. These relations suggest applying recent results for nonparametric estimation of monotone and convex densities. The results are applied to estimating the distribution of the period of stay of migrating birds.     

Generalized trapezoidal distributions

We present a construction and basic properties of a class of continuous distributions of an arbitrary form defined on a compact (bounded) set by concatenating in a continuous manner three probability density functions with bounded support using a modified mixture technique. These three distributions may represent growth, stability and decline stages of a physical or mental phenomenon. Revised: April 2002     

Assessing skewness in financial markets

It is a matter of common observation that investors value substantial gains but are averse to heavy losses. Obvious as it may sound, this translates into an interesting preference for right-skewed return distributions, whose right tails are heavier than their left tails. Skewness is thus not only a way to describe the shape of a distribution, but also a tool for risk measurement. We review the statistical literature on skewness and provide a comprehensive framework for its assessment. Then, we present a new measure of skewness, based on the decomposition of variance in its upward and downward components. We argue that this measure fills a gap in the literature and show in a simulation study that it strikes a good balance between robustness and sensitivity.     

A Tutorial on Palm Distributions for Spatial Point Processes

This tutorial provides an introduction to Palm distributions for spatial point processes. Initially, in the context of finite point processes, we give an explicit definition of Palm distributions in terms of their density functions. Then we review Palm distributions in the general case. Finally, we discuss some examples of Palm distributions for specific models and some applications.     

Smoothing histograms by means of lattice-and continuous distributions

Summary Using lattice distributions or an auxiliary density function each satisfying certain moment conditions a general type of estimator for a one dimensional density functionf is developed. This estimator can be looked at as a smoothed histogram. As a measure of quality the exact order of magnitude for the mean squared error is established (pointwise and uniformly) in terms of the size of an iid sample drawn fromf and depending on a design parameter. The methods in deriving the asymptotic behaviour of the mean squared error are based on Edgeworth expansions for the auxiliary distributions.     

Mixtures of t-distributions for finance and forecasting

We explore convenient analytic properties of distributions constructed as mixtures of scaled and shifted t-distributions. Particularly desirable for econometric applications are closed-form expressions for antiderivatives (e.g., the cumulative density function). We illustrate the usefulness of these distributions in two applications. In the first application, we produce density forecasts of U.S. inflation and show that these forecasts are more accurate, out-of-sample, than density forecasts obtained using normal or standard t-distributions. In the second application, we replicate the option-pricing exercise of Abadir and Rockinger [Density functionals, with an option-pricing application. Econometric Theory 19, 778–811.] and obtain comparably good results, while gaining analytical tractability.     

Forecasting in GARCH models with polynomially modified innovations

Orthogonal polynomials can be used to modify the moments of the distribution of a random variable. In this paper, polynomially adjusted distributions are employed to model the skewness and kurtosis of the conditional distributions of GARCH models. To flexibly capture the skewness and kurtosis of data, the distributions of the innovations that are polynomially reshaped include, besides the Gaussian, also leptokurtic laws such as the logistic and the hyperbolic secant. Modeling GARCH innovations with polynomially adjusted distributions can effectively improve the precision of the forecasts. This strategy is analyzed in GARCH models with different specifications for the conditional variance, such as the APARCH, the EGARCH, the Realized GARCH, and APARCH with time-varying skewness and kurtosis. An empirical application on different types of asset returns shows the good performance of these models in providing accurate forecasts according to several criteria based on density forecasting, downside risk, and volatility prediction.     

Factor substitution,employment density and suburbanization

A model of an urban area producing one good for export is presented and solved to yield the employment density function. The production technology is a constant elasticity of substitution (σ) production function, unlike other models which use a Cobb-Douglas function. The shape of the employment density function proves to be sensitive to the elasticity of substitution and can diverge markedly from the usually assumed negative exponential form. The employment density function is a constant when σ = 0; always declines for positive σ but at an increasing rate if $\text{σ ≤}\text{1}\text{2}$; at either an increasing rate near the city center and decreasing thereafter or always at a decreasing rate if $\text{1}\text{2}\text{< σ < 1}$; and at a decreasing rate if σ ≥ 1. When the employment density function is differentiated with respect to σ, the sign of the derivative is positive near the city center and generally negative in the outer regions; the city seems to centralize. A decline in transport costs does not always suburbanize employment; it is contingent on the nature of the production technology.     

Forecasting Heavy‐Tailed Densities with Positive Edgeworth and Gram‐Charlier Expansions*

This article presents a new semi‐nonparametric (SNP) density function, named Positive Edgeworth‐Sargan (PES). We show that this distribution belongs to the family of (positive) Gram‐Charlier (GC) densities and thus it preserves all the good properties of this type of SNP distributions but with a much simpler structure. The in‐ and out‐of‐sample performance of the PES is compared with symmetric and skewed GC distributions and other widely used densities in economics and finance. The results confirm the PES as a good alternative to approximate financial returns distribution, specially when skewness is not severe.     

Matricvariate and matrix multivariate T distributions and associated distributions

Several distributions are studied, simultaneously in the real, complex, quaternion and octonion cases. Specifically, these are the central, nonsingular matricvariate and matrix multivariate T and beta type II distributions and the joint density of the singular values are obtained for real normed division algebras.     

Minimax estimation under convex loss when the parameter interval is bounded

In the present paper families of truncated distributions with a Lebesgue density forx=(x ₁,...,x _n ) ε ℝ ⁿ are considered, wheref ₀:ℝ → (0, ∞) is a known continuous function andC _n (ϑ) denotes a normalization constant. The unknown truncation parameterϑ which is assumed to belong to a bounded parameter intervalΘ=[0,d] is to be estimated under a convex loss function. It is studied whether a two point prior and a corresponding Bayes estimator form a saddle point when the parameter interval is sufficiently small.     

A class of flat prior distributions for the Poisson‐gamma hierarchical model

A wide class of prior distributions for the Poisson‐gamma hierarchical model is proposed. Prior distributions in this class carry vague information in the sense that their tails exhibit slow decay. Conditions for the propriety of the resulting posterior density are determined, as well as for the existence of posterior moments of the Poisson rate of either an observed or an unobserved unit.     

Distributional properties of portfolio weights

In this paper, we prove several distributional properties for optimal portfolio weights. The weights are estimated by replacing the parameters with the sample counterparts. All results for finite samples are made assuming normally distributed returns. We calculate the exact covariances for the weights obtained by the expected quadratic utility. Additionally we derive the multivariate density function of the global minimum variance portfolio and the univariate density of the tangency portfolio. We obtain the conditional density for the Sharpe ratio optimal weights and show that the expectations of the Sharpe ratio optimal weights do not exist. Moreover, we determine the asymptotic distributions of the estimated weights assuming that the returns follow a multivariate stationary Gaussian process.     

制备工艺对氧化锌陶瓷超声性能的影响

本文研究了烧结温度和成型压力对氧化锌陶瓷密度及超声波传播速度的影响。试验结果表明,随着烧结温度提高,超声波在氧化锌陶瓷中的传播速度下降;随着成型压力增大,超声波在陶瓷中的传播速度呈上升趋势。     

Generalized Bayes minimax estimators of location vectors for spherically symmetric distributions with residual vector

We consider estimation of the mean vector, $\theta $ , of a spherically symmetric distribution with known scale parameter under quadratic loss and when a residual vector is available. We show minimaxity of generalized Bayes estimators corresponding to superharmonic priors with a non decreasing Laplacian of the form $\pi (\Vert \theta \Vert ^{2})$ , under certain conditions on the generating function $f(\cdot )$ of the sampling distributions. The class of sampling distributions includes certain variance mixtures of normals.