An alternate development of conditioning

Abstract  The "classical" development of conditioning, due to K olmogorov , does not agree with the "practical" (more intuitive, but unrigorous) way in which probabilists and statisticians actually think about conditioning. This paper describes an alternative to the classical development. It is shown that standard concepts and results can be developed, rigorously, along lines, which correspond to the "practical" approach, and so as to include the classical material as a special case. More specifically, let Xand Y be random variables (r.v.'s) from (Ω, f, P) to ( x, f_x ) and (y. f^y.), respectively. In this paper, the fundamental concept is the conditional probability P(AX = x ), a function of xε x which satisfies a "natural" defining condition. This is used to define a conditional distribution P_y/x, as a mapping x × f^y-R such that, as a function of B, P_ylx=x,(B ) is a probability measure on f^y. Then, for a numerical r.v. Y , conditional expectation E(Y/X) is defined as a mapping x → whose value at x isE(Y/X = x) = ydP_Y/x=i(y ). Basic properties of conditional probabilities, distributions, and expectations, are derived and their existence and uniqueness are discussed. Finally, for a sub-o-algebra and a numerical r.v. Y , the classical conditional expectation E(Y) is obtained as E(Y/X) with X = i , the identity mapping from (Ω, f) to (Ω).     

On characterizing distributions by conditional expectations of functions of order statistics

In the present paper, we give some general theorems on characterizations based on conditional expectations of the functions of order statistics. In addition, we indicate special forms of the theorems for the familiar probability distributions.     

A unified approach to bivariate discrete distributions

Here we introduce a bivariate generalized hypergeometric factorial moment distribution (BGHFMD) through its probability generating function (p.g.f.) whose marginal distributions are the generalized hypergeometric factorial moment distributions introduced by Kemp and Kemp (Bull Int Stat Inst 43:336–338,1969). Well-known bivariate versions of distributions such as binomial, negative binomial and Poisson are special cases of this distribution. A genesis of the distribution and explicit closed form expressions for the probability mass function of the BGHFMD, its factorial moments and the p.g.f.’s of its conditional distributions are derived here. Certain recurrence relations for probabilities, moments and factorial moments of the bivariate distribution are also established.     

Forecasting dynamic return distributions based on ordered binary choice

We present a simple approach to the forecasting of conditional probability distributions of asset returns. We work with a parsimonious specification of ordered binary choice regressions that imposes a connection on sign predictability across different quantiles. The model forecasts the future conditional probability distributions of returns quite precisely when using a past indicator and a past volatility proxy as predictors. The direct benefits of the model are revealed in an empirical application to the 29 most liquid U.S. stocks. The forecast probability distribution is translated to significant economic gains in a simple trading strategy. Our approach can also be useful in many other applications in which conditional distribution forecasts are desired.     

Bridging the gap between a stationary point process and its Palm distribution

In the context of stationary point processes measurements are usually made from a time point chosen at random or from an occurrence chosen at random. That is, either the stationary distribution P or its Palm distribution P° is the ruling probability measure. In this paper an approach is presented to bridge the gap between these distributions. We consider probability measures which give exactly the same events zero probability as P°, having simple relations with P . Relations between P and P° are derived with these intermediate measures as bridges. With the resulting Radon-Nikodym densities several well-known results can be proved easily. New results are derived. As a corollary of cross ergodic theorems a conditional version of the well-known inversion formula is proved. Several approximations of P° are considered, for instance the local characterization of P^o as a limit of conditional probability measures P° N The total variation distance between P° and P₁ can be expressed in terms of the P-distribution function of the forward recurrence time.     

Zero truncated Poisson integer-valued AR(1) model

In this paper, we introduce a new stationary integer-valued autoregressive process of the first order with zero truncated Poisson marginal distribution. We consider some properties of this process, such as autocorrelations, spectral density and multi-step ahead conditional expectation, variance and probability generating function. Stationary solution and its uniqueness are obtained with a discussion to strict stationarity and ergodicity of such process. We estimate the unknown parameters by using conditional least squares estimation, nonparametric estimation and maximum likelihood estimation. The asymptotic properties and asymptotic distributions of the conditional least squares estimators have been investigated. Some numerical results of the estimators are presented and some sample paths of the process are illustrated. Some possible applications of the introduced model are discussed.     

A new statistic and practical guidelines for nonparametric Granger causality testing

In this paper we introduce a new nonparametric test for Granger non-causality which avoids the over-rejection observed in the frequently used test proposed by Hiemstra and Jones [1994. Testing for linear and nonlinear Granger causality in the stock price-volume relation. Journal of Finance 49, 1639–1664]. After illustrating the problem by showing that rejection probabilities under the null hypothesis may tend to one as the sample size increases, we study the reason behind this phenomenon analytically. It turns out that the Hiemstra–Jones test for the null of Granger non-causality, which can be rephrased in terms of conditional independence of two vectors X and Z given a third vector Y, is sensitive to variations in the conditional distributions of X and Z that may be present under the null. To overcome this problem we replace the global test statistic by an average of local conditional dependence measures. By letting the bandwidth tend to zero at appropriate rates, the variations in the conditional distributions are accounted for automatically. Based on asymptotic theory we formulate practical guidelines for choosing the bandwidth depending on the sample size. We conclude with an application to historical returns and trading volumes of the Standard and Poor's index which indicates that the evidence for volume Granger-causing returns is weaker than suggested by the Hiemstra–Jones test.     

The History of the Dirichlet and Liouville Distributions

The Dirichlet distributions have long been the subject of intense scrutiny in statistics and probability. Despite the enormous interest in, and wide-ranging applications of, these distributions, little appears to be known about their history. In this article we review the development of the Dirichlet distributions and their companions, the Liouville distributions. After reviewing some integral formulas of Dirichlet and Liouville, we survey the theory and applications of these distributions in statistics.     

How To Assign Probabilities If You Must

Empirical evidence can sometimes be incorporated in a probabilistic analysis by conditioning with respect to the observations. Usually, the underlying probability distribution and also the conditional distribution are not completely known. The assignment of probabilities will then require a compromise. The making of such a compromise goes beyond mathematical theory: a statistical discussion is needed. It depends on the context whether the result of such discussion is almost compelling, reasonable, or not really agreeable. This is illustrated by means of a simple example from the area of predictive distributional inference.     

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.     

Can the random walk model be beaten in out-of-sample density forecasts? Evidence from intraday foreign exchange rates

It has been documented that random walk outperforms most economic structural and time series models in out-of-sample forecasts of the conditional mean dynamics of exchange rates. In this paper, we study whether random walk has similar dominance in out-of-sample forecasts of the conditional probability density of exchange rates given that the probability density forecasts are often needed in many applications in economics and finance. We first develop a nonparametric portmanteau test for optimal density forecasts of univariate time series models in an out-of-sample setting and provide simulation evidence on its finite sample performance. Then we conduct a comprehensive empirical analysis on the out-of-sample performances of a wide variety of nonlinear time series models in forecasting the intraday probability densities of two major exchange rates—Euro/Dollar and Yen/Dollar. It is found that some sophisticated time series models that capture time-varying higher order conditional moments, such as Markov regime-switching models, have better density forecasts for exchange rates than random walk or modified random walk with GARCH and Student-t innovations. This finding dramatically differs from that on mean forecasts and suggests that sophisticated time series models could be useful in out-of-sample applications involving the probability density.     

Testing conditional symmetry against one-sided alternatives in square contingency tables

For a (k×k) square contingency table with ordered categories, letX(Y) denote the row (column) number. The conditional symmetry model is given byP(X=i, Y=j|X<Y)=P(X=j, Y=i |X>Y), ∀i<j. In this paper, we study the likelihood ratio tests of conditional symmetry in a square contingency table against two particular classes of one-sided alternatives. We obtain the maximum likelihood estimators under each alternative. The asymptotic null distributions of the likelihood ratio statistics are shown to have chi-bar square type distributions. A simulation study is performed by comparing the powers of different tests. The theory developed is illustrated by using the famous eye vision data from Stuart (1953).     

Extreme returns and intensity of trading

Asymmetric information models of market microstructure claim that variables such as trading intensity are proxies for latent information on the value of financial assets. We consider the interval‐valued time series (ITS) of low/high returns and explore the relationship between these extreme returns and the intensity of trading. We assume that the returns (or prices) are generated by a latent process with some unknown conditional density. At each period of time, from this density, we have some random draws (trades) and the lowest and highest returns are the realized extreme observations of the latent process over the sample of draws. In this context, we propose a semiparametric model of extreme returns that exploits the results provided by extreme value theory. If properly centered and standardized extremes have well‐defined limiting distributions, the conditional mean of extreme returns is a nonlinear function of the conditional moments of the latent process and of the conditional intensity of the process that governs the number of draws. We implement a two‐step estimation procedure. First, we estimate parametrically the regressors that will enter into the nonlinear function, and in a second step we estimate nonparametrically the conditional mean of extreme returns as a function of the generated regressors. Unlike current models for ITS, the proposed semiparametric model is robust to misspecification of the conditional density of the latent process. We fit several nonlinear and linear models to the 5‐minute and 1‐minute low/high returns to seven major banks and technology stocks, and find that the nonlinear specification is superior to the current linear models and that the conditional volatility of the latent process and the conditional intensity of the trading process are major drivers of the dynamics of extreme returns.     

Characterizations of logarithmic series distrimtrans

The distributions of X, Y and (X. Y ), where X and Y are random variables with probability functions of a logarithmic series law, are characterized by the regression function of X on Y and the conditional distribution of Y given X. Moreover, characterizations are given for binomial or Pascal conditional distributions in terms of the regression function of X on Y and the marginal distribution of X.     

Expected utility theory with non-commutative probability theory

In this paper, we extend von Neumann and Morgenstern’s expected utility approach to a non-commutative probability theory. We introduce a new representation of the decision maker’s set of events which extends the canonical representation. We reformulate von Neumann and Morgenstern’s approach to modeling decision maker behavior by non-commutative probability theory. We introduce a set of preference axioms similar to von Neumann and Morgenstern’s axioms, and show that they lead to a generalization of the expected utility theorem. Our generalization allows for decision makers to make an intuitive distinction between representations of a set of events. We find that this methodology enables several paradoxes and inconsistencies in traditional expected utility theory (e.g., Allais paradox, etc.) to be solved or better understood.     

On truncated bivariate discrete distributions: A unified treatment

A unified treatment of three types of zero class truncation for bivariate discrete distributions is presented. Using the probability generating function approach, various properties of the truncated distributions are examined in association with the corresponding properties of the initial complete form of the distribution. Expressions for moments and conditional distributions are also obtained. Bivariate versions of the Thomas and the Intervened Poisson distributions are introduced and used as illustrative examples. Received November 2000/Revised March 2002     

A characterization using the conditional variance

A class of probability distributions is characterized assuming that the conditional variance of a functionh (X), givenX>x, is constant.     

Compound gamma,beta and F distributions

Summary In this paper a compound gamma distribution has been derived by compounding a gamma distribution with another gamma distribution. The resulting compound gamma distribution has been reduced to the Beta distributions of the first kind and the second kind and to theF distribution by suitable transformations. This includes theLomax distribution as a special case which enjoys a useful property. Moment estimators for two of its parameters are explicitly obtained, which tend to a bivariate normal distribution. The paper contains expressions for a bivariate probability density function, its conditional expectation, conditional variance and the product moment correlation coefficient. Finally, all the parameters of the compound gamma distribution are explicitly expressed in terms of the functions of the moments of the functions of random variables in two different ways. This note is based on a technical report prepared by the author while he was with the Procter and Gamble Company.