Unobserved Random Variables Research Articles

On empirical Bayes estimation of multivariate regression coefficient

We investigate the empirical Bayes estimation problem of multivariate regression coefficients under squared error loss function. In particular, we consider the regression modelY=Xβ+ε, whereYis anm-vector of observations,Xis a knownm×kmatrix,βis an unknownk-vector, andεis anm-vector of unobservable random variables. The problem is squared error loss estimation ofβbased on some “previous” dataY1,…,Ynas well as the “current” data vectorYwhenβis distributed according to some unknown distributionG, whereYisatisfiesYi=Xβi+εi,i=1,…,n. We construct a new empirical Bayes estimator ofβwhenεi∼N(0,σ2Im),i=1,…,n. The performance of the proposed empirical Bayes estimator is measured using the mean squared error. The rates of convergence of the mean squared error are obtained.

Bayesian Predictive Probability Functions for Count Data that are Subject to Misclassification

AbstractWe develop three Bayesian predictive probability functions based on data in the form of a double sample. One Bayesian predictive probability function is for predicting the true unobservable count of interest in a future sample for a Poisson model with data subject to misclassification and two Bayesian predictive probability functions for predicting the number of misclassified counts in a current observable fallible count for an event of interest. We formulate a Gibbs sampler to calculate prediction intervals for these three unobservable random variables and apply our new predictive models to calculate prediction intervals for a real‐data example. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Optimal Portfolio in Partially Observed Stochastic Volatility Models

We address the maximization problem of expected utility from terminal wealth. The special feature of this paper is that we consider a financial market where price process of risky assets follows a stochastic volatility model and we require that investors observe just the vector of stock prices. Using stochastic filtering techniques and adapting martingale duality methods in this partially observed incomplete model, we characterize the value function and the optimal portfolio policies. We study in detail the Bayesian case, when risk premia of the stochastic volatility model are unobservable random variables with known prior distribution. We also consider the case of unobservable risk premia modelled by linear Gaussian processes.

Adjusting for measurement error to assess health effects of variability in biomarkers

Longitudinal studies of health effects often relate individuals' biomarker levels to disease progression. Repeated measurements also provide an opportunity to assess within-individual biomarker variability, and it is reasonable to postulate that this measure might provide additional information about a particular outcome variable. Given the existing precedent for application of adjustment methods to account for measurement error in subject-specific average levels of a covariate, this concept motivates the application of such methods to incorporate variability as well. In this paper, we investigate the nature of the relationship between the decline of CD4 cell count induced by infection with human immunodeficiency virus, and CD4 level and variability prior to infection. We first describe the distribution of repeated CD4 measurements prior to infection using a model that accounts both for random average levels and random subject-specific variance components. Based on this model, we define true unobservable random variables that correspond to prior level and stability. We perform a linear regression analysis, using these latent variables as covariates, by means of a full maximum likelihood approach. We compare the resulting parameter estimates with those based on regressions employing sample-based estimates of pre-infection levels and variances, and empirical Bayes estimates of these quantities. Although the final inferences are similar to those based on the unadjusted analysis, we find that the magnitude of association with prior level decreases, while that with prior stability increases. Stratified analyses indicate that smoking status affects the relationship between prior CD4 level and initial CD4 decline. We point out advantages associated with the maximum likelihood approach in this particular application.

Estimating mixing density from a system of observations

Let Y11,…Y1m,Y21…,Yn1,…Ynm be observations such that Yij=Xi+εij, where xi are i.i.d. unobservable random variables with density f and εij are independent measurement errors, with density fε. First, this paper study the estimation of f from the system of observations {Yij}n,m i=1,j=1 when fε is known. Upper bounds on the rate of quadratic mean convergence are obtained. The upper bounds depend on the geometry of the problem and on the rate of growing of m as function of n. Then, we study two special situations inside the system of observations. The first case is when a parameter of the characteristic function of ε is unknown. The second case corresponds to εij non identically distributed. A simulation study has been conducted to show the sampling behavior of the estimators for distinct values of n and m.

Portfolio choice and the Bayesian Kelly criterion

We derive optimal gambling and investment policies for cases in which the underlying stochastic process has parameter values that are unobserved random variables. For the objective of maximizing logarithmic utility when the underlying stochastic process is a simple random walk in a random environment, we show that a state-dependent control is optimal, which is a generalization of the celebrated Kelly strategy: the optimal strategy is to bet a fraction of current wealth equal to a linear function of the posterior mean increment. To approximate more general stochastic processes, we consider a continuous-time analog involving Brownian motion. To analyze the continuous-time problem, we study the diffusion limit of random walks in a random environment. We prove that they converge weakly to a Kiefer process, or tied-down Brownian sheet. We then find conditions under which the discrete-time process converges to a diffusion, and analyze the resulting process. We analyze in detail the case of the natural conjugate prior, where the success probability has a beta distribution, and show that the resulting limit diffusion can be viewed as a rescaled Brownian motion. These results allow explicit computation of the optimal control policies for the continuous-time gambling and investment problems without resorting to continuous-time stochastic-control procedures. Moreover they also allow an explicit quantitative evaluation of the financial value of randomness, the financial gain of perfect information and the financial cost of learning in the Bayesian problem.

Prediction and Asymptotics

Prediction of an unobserved random variable is considered from a frequentist viewpoint. After a brief review of previous work, a number of examples in which an exact solution is possible are given, partly for their intrinsic interest and partly to illustrate general results. A new form of predictive density is derived accurate to the third order of asymptotic theory under ordinary repeated sampling. The formula is invariant under transformation of the observed and unobserved random variables and under reparametrization. It respects the conditionality principle and may be based on the minimal prediction sufficient statistic. Some open problems are noted.

Portfolio choice and the Bayesian Kelly criterion

We derive optimal gambling and investment policies for cases in which the underlying stochastic process has parameter values that are unobserved random variables. For the objective of maximizing logarithmic utility when the underlying stochastic process is a simple random walk in a random environment, we show that a state-dependent control is optimal, which is a generalization of the celebrated Kelly strategy: the optimal strategy is to bet a fraction of current wealth equal to a linear function of the posterior mean increment. To approximate more general stochastic processes, we consider a continuous-time analog involving Brownian motion. To analyze the continuous-time problem, we study the diffusion limit of random walks in a random environment. We prove that they converge weakly to a Kiefer process, or tied-down Brownian sheet. We then find conditions under which the discrete-time process converges to a diffusion, and analyze the resulting process. We analyze in detail the case of the natural conjugate prior, where the success probability has a beta distribution, and show that the resulting limit diffusion can be viewed as a rescaled Brownian motion. These results allow explicit computation of the optimal control policies for the continuous-time gambling and investment problems without resorting to continuous-time stochastic-control procedures. Moreover they also allow an explicit quantitative evaluation of the financial value of randomness, the financial gain of perfect information and the financial cost of learning in the Bayesian problem.

Prediction of Outstanding Claims: A Hierarchical Credibility Approach

We consider the problem of predicting unpaid losses in respect of incurred, but not yet reported claims in the case where the distribution of delays until notification may depend on the type of claim. A hierarchical credibility model is presented, where the distribution of delays corresponding to different types of claims, as well as other characteristics affecting the claims experience, are viewed as realizations of unobservable random variables. The proposed method is illustrated by a numerical example.

Approximate confidence bounds for ratios of variance components in two-way unbalanced crossed models with interactions

Consider the general unbalanced two-factor crossed components-of-variance model with interaction given by Yijk: = μ+Ai: +Bj: + Cij: +Eijk: (i = 1,2, … a; j = 1,…,b; k = 1,…,.nij:=0) Ai:,Bj:, Cij: and Eijk: are independent unobservable random variables. Also Ai:sim; N(0,σ2 A),Bj: ∼ N(0,σ2 B), Cij:∼N(0,s2 C:) and Eijk:∼N(0,s2 E:). In this paper approximate confidence bounds are obtained for ρA: = ρ2 A/2 and ρB: = ρ2 B:/ρ2 (where σ2 = σ2 A:+ σ2 B+σ2 Cσ2 E) for special cases of the above model. The balanced incomplete block model is studied as a special case.

Factor Indeterminacy in Generalizability Theory

Generalizability theory and common factor analysis are based upon the random effects model of the analy sis of variance, and both are subject to the factor inde terminacy problem: The unobserved random variables (common factor scores or universe scores) are indeter minate. In the one-facet (repeated measures) design, the extent to which true or universe scores and com mon factor scores are not uniquely defined is shown to be a function of the dependability (reliability) of the data. The minimum possible correlation between equivalent common factor scores is a lower bound es timate of reliability.

The Correspondence of Efficiency Frontier as a Generalization of the Cost Function

As THE TITLE SUGGESTS, this paper generalizes the concept of a cost function, first developed by Shephard [5], refined and extended by Uzawa [10], McFadden f 3], Diewart [1] and Hanoch [2]. The nature of the generalization is introduced -below in a somewhat general context. Empirical analyses of behavioral functions, such as product supply or factor ,demand, based on cross-section data often lead to the conclusion that prices are relatively unimportant explanatory variables. Such a result is simply an indication of the fact that there is a wide spread in the output and input composition of different firms operating under similar economic conditions. This spread can be attributed to unobserved random variables which, among other things, indicate that producers fail to locate their optimal points. Such an explanation is unsatisfactory and leaves room for a broader framework which can account for the empirical evidence. An attempt in this direction is undertaken in this paper. The starting point is the assumption that the utility function of the firm is a function of several arguments, rather than of profit alone (cf. [4]). From this assumption it immediately follows that differences among firms may result from ,differences in the components that enter the utility functions and the weights given to them. Technically, the problem can be stated as follows: Let u(c) be a quasi concave utility function monotonic in the components of the m-vector c. Let x be a k-vector of outputs and inputs that come from a production set X possessing some desirable properties. Let P be a m X k matrix (m < k) of coefficients to be referred to as prices. The problem of maximization is: