Optimal selection from distributions with unknown parameters: Robustness of Bayesian models

We consider the problem of making one choice from a known number of i.i.d. alternatives. It is assumed that the distribution of the alternatives has some unknown parameter. We follow a Bayesian approach to maximize the discounted expected value of the chosen alternative minus the costs for the observations. For the case of gamma and normal distribution we investigate the sensitivity of the solution with respect to the prior distributions. Our main objective is to derive monotonicity and continuity results for the dependence on parameters of the prior distributions. Thus we prove some sort of Bayesian robustness of the model.     

A preliminary test procedure for the scale parameter of exponential distribution when the selection parameter is unknown

Summary A preliminary test estimator is considered for the scale parameter of the two-parameter exponential distribution with unknown selection parameter, where the distribution does not satisfy the regularity condition of Wilks' theorem—the density is not differentiable. A method of specifying the level of significance of the preliminary test based on is proposed AIC. This work was partly supported by Scientific Research Fund No. 58450058 from the Ministry of Education of Japan. The Institute of Statistical Mathematics     

Optimal parameter selection for nonlinear multistage systems with time-delays

In this paper, we consider a novel dynamic optimization problem for nonlinear multistage systems with time-delays. Such systems evolve over multiple stages, with the dynamics in each stage depending on both the current state of the system and the state at delayed times. The optimization problem involves choosing the values of the time-delays, as well as the values of additional parameters that influence the system dynamics, to minimize a given cost functional. We first show that the partial derivatives of the system state with respect to the time-delays and system parameters can be computed by solving a set of auxiliary dynamic systems in conjunction with the governing multistage system. On this basis, a gradient-based optimization algorithm is proposed to determine the optimal values of the delays and system parameters. Finally, two example problems, one of which involves parameter identification for a realistic fed-batch fermentation process, are solved to demonstrate the algorithm’s effectiveness.     

Estimation of a non-negative location parameter with unknown scale

For a vast array of general spherically symmetric location-scale models with a residual vector, we consider estimating the (univariate) location parameter when it is lower bounded. We provide conditions for estimators to dominate the benchmark minimax MRE estimator, and thus be minimax under scale invariant loss. These minimax estimators include the generalized Bayes estimator with respect to the truncation of the common non-informative prior onto the restricted parameter space for normal models under general convex symmetric loss, as well as non-normal models under scale invariant \(L^p\) loss with \(p>0\) . We cover many other situations when the loss is asymmetric, and where other generalized Bayes estimators, obtained with different powers of the scale parameter in the prior measure, are proven to be minimax. We rely on various novel representations, sharp sign change analyses, as well as capitalize on Kubokawa’s integral expression for risk difference technique. Several properties such as robustness of the generalized Bayes estimators under various loss functions are obtained.     

Deconvolution from panel data with unknown error distribution

We devise a new method of estimating a distribution in a deconvolution model with panel data and an unknown distribution of the additive errors. We prove strong consistency under a minimal condition concerning the zero sets of the involved characteristic functions.     

Computing the noncentral gamma distribution, its inverse and the noncentrality parameter

The noncentral gamma distribution can be viewed as a generalization of the noncentral chi-squared distribution and it can be expressed as a mixture of a Poisson density function with a incomplete gamma function. The noncentral gamma distribution is not available in free conventional statistical programs. This paper aimed to propose an algorithm for the noncentral gamma by combining the method originally proposed by Benton and Krishnamoorthy (Comput Stat Data Anal 43(2):249–267, 2003) for the noncentral distributions with the method of inversion of the distribution function with respect to the noncentrality parameter using Newton–Raphson. The algorithms are available in pseudocode and implemented as R functions. To evaluate the accuracy and speed of computation of the algorithms implemented in R, results of the distribution function, density function, quantiles and noncentrality parameter of the noncentral incomplete gamma and its particular case, the noncentral chi-squared, were obtained for the arguments settings used by Benton and Krishnamoorthy (Comput Stat Data Anal 43(2):249–267, 2003) and Chen (J Stat Comput Simul 75(10):813–829, 2005). The implemented routines performed well and, in general, were as accurate than other approximations. The R package denoted ncg is available to download on the CRAN-R package repository http://cran.r-project.org/.     

The newsvendor problem with unknown distribution

Newsvendor theory assumes that the decision-maker faces a knowndistribution. But in real-life situations, demand distribution is not alwaysknown. In the experimental study which this paper presents, half of theparticipants assuming the newsvendor role were unaware of the underlying demanddistribution, while the other half knew the demand distribution. Participantshad to decide how many papers to order each day (for 100 days). The experimentalfindings indicate that subjects who know the demand distribution behavedifferently to those who do not. However, interestingly enough, knowing thedemand distribution does not necessarily lead the subject closer to the optimalsolution or improve profits. It was found that supply surplus at a certainperiod strongly affects the order quantity towards the following period, despitethe knowledge of the demand distribution.     

Pointwise deconvolution with unknown error distribution

This Note presents rates of convergence for the pointwise mean squared error in the deconvolution problem with estimated characteristic function of the errors.     

Optimal damping of perturbations in a system with unknown initial conditions

We consider the perturbation damping problem for a system in which, along with an external perturbation bounded in the L ₂-norm, there is an initial perturbation caused by unknown nonzero initial conditions. We state necessary and sufficient conditions for the existence of an optimal control law minimizing the maximum L ₂-norm of the system output for all L ₂-bounded external perturbations and bounded initial states and synthesize this control law.     

Optimal stopping of a Brownian bridge with an unknown pinning point

The problem of stopping a Brownian bridge with an unknown pinning point to maximise the expected value at the stopping time is studied. A few general properties, such as continuity and various bounds of the value function, are established. However, structural properties of the optimal stopping region are shown to crucially depend on the prior, and we provide a general condition for a one-sided stopping region. Moreover, a detailed analysis is conducted in the cases of the two-point and the mixed Gaussian priors, revealing a rich structure present in the problem.     

Linearization conditions for regression models with unknown variance parameter

In the case of the nonlinear regression model, methods and procedures have been developed to obtain estimates of the parameters. These methods are much more complicated than the procedures used if the model considered is linear. Moreover, unlike the linear case, the properties of the resulting estimators are unknown and usually depend on the true values of the estimated parameters. It is sometimes possible to approximate the nonlinear model by a linear one and use the much more developed linear methods, but some procedure is needed to recognize such situations. One attempt to find such a procedure, taking into account the requirements of the user, is given in [4], [5], [3], where the existence of an a priori information on the parameters is assumed. Here some linearization criteria are proposed and the linearization domains, i.e. domains in the parameter space where these criteria are fulfilled, are defined. The aim of the present paper is to use a similar approach to find simple conditions for linearization of the model in the case of a locally quadratic model with unknown variance parameter ². Also a test of intrinsic nonlinearity of the model and an unbiased estimator of this parameter are derived.     

A simulation based approach to the parameter estimation for the three-parameter gamma distribution

The gamma distribution is one of the commonly used statistical distribution in reliability. While maximum likelihood has traditionally been the main method for estimation of gamma parameters, Hirose has proposed a continuation method to parameter estimation for the three-parameter gamma distribution. In this paper, we propose to apply Markov chain Monte Carlo techniques to carry out a Bayesian estimation procedure using Hirose’s simulated data as well as two real data sets. The method is indeed flexible and inference for any quantity of interest is readily available.     

Nonparametric estimation for censored lifetimes suffering from unknown selection bias

In a population of individuals, where the random variable (r.v.) σ denotes the birth time and X the lifetime, we consider the case, where an individual can be observed only if its life-line (σ, X) = {(σ + y, y), 0 ≤ y ≤ X} intersects a given Borel set S in ℝ × ℝ₊. Denoting by σ _S and X _S the birth time and lifetime for the observed individuals, we point out that the distribution function (d.f.) F _S of the r.v. X _S suffers from a selection bias in the sense that F _S = ∝ w d F/μ _S, where w and μ _S depend only on the distribution of σ and on F, the d.f. of X. Assuming in addition that the r.v. X _S is randomly right-censored as soon as the individual is selected, we construct a productlimit estimator for the d.f. F _S and a nonparametric estimator ŵ for the weight function w. We prove a consistency result for ŵ and a weak convergence result for . We establish in addition an exponential bound for .      

An adaptive discretization for Tikhonov-Phillips regularization with a posteriori parameter selection

Summary. The aim of this paper is to describe an efficient adaptive strategy for discretizing ill-posed linear operator equations of the first kind: we consider Tikhonov-Phillips regularization with a finite dimensional approximation instead of A. We propose a sparse matrix structure which still leads to optimal convergences rates but requires substantially less scalar products for computing compared with standard methods. Received September 16, 1998 / Revised version received August 4, 1999 / Published online August 2, 2000     

On the estimation of the shape parameter of the gamma distribution in second-order asymptotics

In the two-parameter case with parameter orthogonality, we propose a method of constructing an estimator with second-order admissibility under any loss function with a given loss coefficient. Furthermore, we give a sufficient condition for any estimator to be second-order admissible or inadmissible. On the basis of these results, the problem of estimating the shape parameter of the gamma distribution is discussed at the level of second-order asymptotics.     

Optimal adaptive solution of initial-value problems with unknown singularities

The optimal solution of initial-value problems in ODEs is well studied for smooth right-hand side functions. Much less is known about the optimality of algorithms for singular problems. In this paper, we study the (worst case) solution of scalar problems with a right-hand side function having r   continuous bounded derivatives in R$\mathbf{R}$, except for an unknown singular point. We establish the minimal worst case error for such problems (which depends on r similarly as in the smooth case), and define optimal adaptive algorithms. The crucial point is locating an unknown singularity of the solution by properly adapting the grid. We also study lower bounds on the error of an algorithm for classes of singular problems. In the case of a single singularity with nonadaptive information, or in the case of two or more singularities, the error of any algorithm is shown to be independent of r.     

Optimal selection with randomly fading memory

In this paper, we will consider optimal selection problems in which the recall of a random number of observations and uncertainty of selection are both allowed at each stage of the selection process. General rules as well as closed form solutions for specific examples with Markov memory system and finite memory are obtained.