On Discrete oc–Unimodality

A definition of α–unimodality for integer–valued random variables is introduced. Some characterization results in terms of probability mass functions and characteristic functions are established. Properties of generalized discrete uni–modality are investigated. Further, by use of limiting properties, discrete α–unimodality is shown to lead to the concept of α–unimodality for the real–valued random variables due to OLSHEN and SAVAGE (1970).     

Bayesian modeling of joint and conditional distributions

In this paper, we study a Bayesian approach to flexible modeling of conditional distributions. The approach uses a flexible model for the joint distribution of the dependent and independent variables and then extracts the conditional distributions of interest from the estimated joint distribution. We use a finite mixture of multivariate normals (FMMN) to estimate the joint distribution. The conditional distributions can then be assessed analytically or through simulations. The discrete variables are handled through the use of latent variables. The estimation procedure employs an MCMC algorithm. We provide a characterization of the Kullback–Leibler closure of FMMN and show that the joint and conditional predictive densities implied by the FMMN model are consistent estimators for a large class of data generating processes with continuous and discrete observables. The method can be used as a robust regression model with discrete and continuous dependent and independent variables and as a Bayesian alternative to semi- and non-parametric models such as quantile and kernel regression. In experiments, the method compares favorably with classical nonparametric and alternative Bayesian methods.     

Some results on discrete a–monotonicity

Some characterizations and preservation properties of a–monotonicity are investigated. In particular it is shown that a mixture of binomial or Poisson distributions preserves the a–monotonicity of the mixing distribution. Also, a characterization of a class of infinitely divisible distributions, within the discrete α–unimodal class, is given. This characterization is the discrete version of that of ALAMATSAZ (1985) and KOTZ and STEUTEL (1988) in the continuous case.     

A note on characterization by the conditional expectation

Summary Various distributions like exponential, Pareto, power and Burr's distribution are characterized in terms of conditional expectation of a functionh (·) of the absolutely continuous random variable.     

A nonparametric test for equality of distributions with mixed categorical and continuous data

In this paper we consider the problem of testing for equality of two density or two conditional density functions defined over mixed discrete and continuous variables. We smooth both the discrete and continuous variables, with the smoothing parameters chosen via least-squares cross-validation. The test statistics are shown to have (asymptotic) normal null distributions. However, we advocate the use of bootstrap methods in order to better approximate their null distribution in finite-sample settings and we provide asymptotic validity of the proposed bootstrap method. Simulations show that the proposed tests have better power than both conventional frequency-based tests and smoothing tests based on ad hoc smoothing parameter selection, while a demonstrative empirical application to the joint distribution of earnings and educational attainment underscores the utility of the proposed approach in mixed data settings.     

Conditional distributions of generalized order statistics and some characterizations

Generalized order statistics have been introduced in Kamps (1995a). They enable a unified approach to several models of ordered random variables, e.g. (ordinary) order statistics, record values, sequential order statistics, record values from non-identical distributions. The purpose of this paper is to develop conditional distributions of one generalized order statistic given another and to characterize the underlying continuous distribution by different conditional expectations. Well-known results for ordinary order statistics and record values are extended to generalized order statistics. Received: July 1997     

Quasi-maximum likelihood estimation and testing for nonlinear models with endogenous explanatory variables

I propose a quasi-maximum likelihood framework for estimating nonlinear models with continuous or discrete endogenous explanatory variables. Joint and two-step estimation procedures are considered. The joint procedure is a quasi-limited information maximum likelihood procedure, as one or both of the log likelihoods may be misspecified. The two-step control function approach is computationally simple and leads to straightforward tests of endogeneity. In the case of discrete endogenous explanatory variables, I argue that the control function approach can be applied with generalized residuals to obtain average partial effects. I show how the results apply to nonlinear models for fractional and nonnegative responses.     

Ordering univariate distributions by entropy and variance

This paper examines the role of variance and entropy in ordering distributions and random prospects. There is no universal relation between entropy and variance orderings of distributions. But we place their relationship in the context of a stronger ordering relation known as dispersion ordering. Further, some conditions are identified under which variance and entropy order similarly when continuous variables are transformed. We also analyze parametric changes which do not disturb the agreement between these rankings. The results are conveniently tabulated in terms of distribution parameters.     

On stochastic comparisons of minimum order statistics from the location–scale family of distributions

We consider stochastic comparisons of minimum order statistics from the location–scale family of distributions that contain most of the popular lifetime distributions. Under certain assumptions, we show that the minimum order statistic of one set of random variables dominates that of another set of random variables with respect to different stochastic orders. Furthermore, we illustrate our results using some well-known specific distributions.     

Optimal stratification and grouping by dynamic programming

Summary In statistics and their fields of application a number of different problems in respect to stratification and grouping of random variables or their values result in optimization problems of the same structure. By a suitable transformation a global optimal solution of these problems can be determined by dynamic programming. The results are illustrated for discrete and continuous random variables by numerical results.

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Zusammenfassung In der Statistik und ihren Anwendungsgebieten führen verschiedene Aufgabenstellungen bei der Schichtung und Gruppierung von Zufallsgrößen, oder deren Realisationen auf Optimierungsprobleme derselben Struktur. Diese Optimierungsprobleme können durch eine geeignete Transformation auf eine Form gebracht werden, so daß eines der rekursiven Verfahren der dynamischen Optimierung zur Bestimmung eines globalen Optimums angewandt werden kann. An Beispielen diskreter und stetiger Zufallsvariablen wird die Vorgehensweise demonstriert.

     

Consistent estimation of correlations between observed interval variables with skewed distributions

Departures from multinormality due to skewness in observed distributions may result in inconsistent estimates of product-moment correlations between interval variables. Therefore, the robustness of the product-moment correlation estimator against skewness in the distributions of sample data on interval variables has been investigated. This estimator is robust against skewness of maximally about 1 in absolute value. If the observed distributions have larger skewnesses, the sample data on interval variables may be redistributed over normally distributed discrete variables with 10 categories each. The estimated polychoric correlations between these discrete variables represent consistent estimates of the product-moment correlations between the original interval variables in the population.     

Right censoring in a discrete life model

Although there are many sophisticated models for estimation of failure rate based on censored data in continuous distributions, not much work has been done in the discrete case. We introduce a discrete model for life lengths and consider its properties. For this model, we derive the corresponding maximum likelihood estimators of the parameters under Type I and Type II right-censoring. Received May 2000     

Variables sampling plans for the Poisson and the binomial

Variables sampling plans based upon continuous distributions are well known. The usual assumption is that a measurable characteristic associated with a product has a normal distribution, a case which has been treated extensively in the literature. Other continuous distributions, particularly the exponential, have also been used as models. In this paper we discuss variables sampling plans for situations in which the measurable characteristic has either a Poisson or a binomial distribution.     

On decompositions of some random variables

The aim of this paper is to present several stochastic analogs of classical formulas for the gamma function. The obtained results provide representation of some random variables as finite or infinite products of independent random variables. Examples include generalized gamma, normal, beta and other distributions.     

Distributional Properties of Record Values from the Geometric Distribution

Some distributional properties of the record values of non–identically distributed random variables having geometric distributions are discussed. Three theorems dealing with the characterization of the geometric distribution based on these distributional properties are presented. The unique minimum variance unbiased estimators of some functions of the parameters of the distribution and various predictors of the s–th record valued utilizing the first m(m相似文献   

Predicting workload requirements for scheduled health care services, with an application to radiology departments.

A model is developed for predicting workload requirements for scheduled health care services. The model is then applied to an actual planning problem for a radiology department. The probability distribution of future workload is represented by the convolution of two families of random variables such that a compound Poisson process adequately describes workload requirements. The model developed herein can be applied to a wide assortment of capacity-expansion problems that are characterized by discrete demands (e.g. number of jobs) occurring in a given period of time, where the amount of time needed to complete each job is a continuous random variable.     

A residual inaccuracy measure based on the relevation transform

Inaccuracy and information measures based on the cumulative residual entropy are useful in various fields, and are attracting increasing attention in Probability Theory and Statistics. In this paper, we introduce and study an inaccuracy measure concerning the relevation transform of two nonnegative continuous random variables. We investigate various distributional properties and characterization results that are based on the mean residual lifetime and involve the generalized Pareto distribution. A connection with the proportional hazards model is also provided. We obtain comparison results involving the proposed inaccuracy measure and some existing inaccuracy measures. Some illustrative examples are finally given.     

Reconstruction of previous failure times and records

For the sequences of independent identically distributed random variables with continuous distributions, we provide the optimal upper bounds for the increments of order and record statistics under condition that the values of future order statistics and records are known. The bounds are expressed in terms of quantiles and absolute moments centered about the quantiles of the parent distribution. We also describe the distributions which approach the bounds with arbitrary desired accuracy.The second author was supported by the Polish State Committee for Scientific Research (KBN) under Grant 5 P03A 012 20Received November 2003     

Bayesian Enhancement of Speech and Audio Signals which can be Modelled as ARMA Processes

In application areas which involve digitised speech and audio signals, such as coding, digital remastering of old recordings and recognition of speech, it is often desirable to reduce the effects of noise with the aim of enhancing intelligibility and perceived sound quality. We consider the case where noise sources contain non-Gaussian, impulsive elements superimposed upon a continuous Gaussian background. Such a situation arises in areas such as communications channels, telephony and gramophone recordings where impulsive effects might be caused by electromagnetic interference (lightning strikes), electrical switching noise or defects in recording media, while electrical circuit noise or the combined effect of many distant atmospheric events lead to a continuous Gaussian component.
In this paper we discuss the background to this type of noise degradation and describe briefly some existing statistical techniques for noise reduction. We propose new methods for enhancement based upon Markov chain Monte Carlo (MCMC) simulation. Signals are modelled as autoregressive moving-average (ARMA); while noise sources are treated as discrete and continuous mixtures of Gaussian distributions. Results are presented for both real and artificially corrupted data sequences, illustrating the potential of the new methods.     

The relationship between moments of truncated and original distributions plus some other simple structural properties of weighted distributions

A simple relationship between expectations of positive functions of random variables from truncated and original distributions is pointed out. This is a special case of a relationship between moments of weighted and original distributions which is also discussed. Another topic is that of expressing weighted mixture distributions as mixtures of weighted distributions and vice versa. Finally, an error in some structural properties of weighted distributions given in the literature is identified and corrected. Research carried out while the author held a British Science and Engineering Research Council Postdoctoral Research Fellowship in the Department of Statistics, University of Birmingham, U.K.