An adaptive rule for stopping a discovery process under time censorship

Consider a finite population of large but unknown size of hidden objects. Consider searching for these objects for a period of time, at a certain cost, and receiving a reward depending on the sizes of the objects found. Suppose that the size and discovery time of the objects both have unknown distributions, but the conditional distribution of time given size is exponential with an unknown non-negative and non-decreasing function of the size as failure rate. The goal is to find an optimal way to stop the discovery process. Assuming that the above parameters are known, an optimal stopping time is derived and its asymptotic properties are studied. Then, an adaptive rule based on order restricted estimates of the distributions from truncated data is presented. This adaptive rule is shown to perform nearly as well as the optimal stopping time for large population size.     

Bounded risk estimation of linear combinations of the location and scale parameters in exponential distributions under two-stage sampling

Two-stage sampling is proposed for estimating linear combinations of the location and scale parameters of exponential distributions with bounded quadratic risk functions. Exact formulae for the expected values and risks of the estimators are derived, and the performance of estimators is studied. Illustrations with real data are included.     

Robust designs for one-point extrapolation

We consider the construction of designs for the extrapolation of a regression response to one point outside of the design space. The response function is an only approximately known function of a specified linear function. As well, we allow for variance heterogeneity. We find minimax designs and corresponding optimal regression weights in the context of the following problems: (P1) for nonlinear least squares estimation with homoscedasticity, determine a design to minimize the maximum value of the mean squared extrapolation error (MSEE), with the maximum being evaluated over the possible departures from the response function; (P2) for nonlinear least squares estimation with heteroscedasticity, determine a design to minimize the maximum value of MSEE, with the maximum being evaluated over both types of departures; (P3) for nonlinear weighted least squares estimation, determine both weights and a design to minimize the maximum MSEE; (P4) choose weights and design points to minimize the maximum MSEE, subject to a side condition of unbiasedness. Solutions to (P1)–(P4) are given in complete generality. Numerical comparisons indicate that our designs and weights perform well in combining robustness and efficiency. Applications to accelerated life testing are highlighted.     

Optimal crossover designs when carryover effects are proportional to direct effects

Crossover designs are used for a variety of different applications. While these designs have a number of attractive features, they also induce a number of special problems and concerns. One of these is the possible presence of carryover effects. Even with the use of washout periods, which are for many applications widely accepted as an indispensable component, the effect of a treatment from a previous period may not be completely eliminated. A model that has recently received renewed attention in the literature is the model in which first-order carryover effects are assumed to be proportional to direct treatment effects. Under this model, assuming that the constant of proportionality is known, we identify optimal and efficient designs for the direct effects for different values of the constant of proportionality. We also consider the implication of these results for the case that the constant of proportionality is not known.     

ON EMPIRICAL BAYES STOPPING TIMES

We consider the empirical Bayes decision theory where the component problems are the optimal fixed sample size decision problem and a sequential decision problem. With these components, an empirical Bayes decision procedure selects both a stopping rule function and a terminal decision rule function. Empirical Bayes stopping rules are constructed for each case and the asymptotic behaviours are investigated.     

Bayesian mixture modeling for spatial Poisson process intensities,with applications to extreme value analysis

We propose a method for the analysis of a spatial point pattern, which is assumed to arise as a set of observations from a spatial nonhomogeneous Poisson process. The spatial point pattern is observed in a bounded region, which, for most applications, is taken to be a rectangle in the space where the process is defined. The method is based on modeling a density function, defined on this bounded region, that is directly related with the intensity function of the Poisson process. We develop a flexible nonparametric mixture model for this density using a bivariate Beta distribution for the mixture kernel and a Dirichlet process prior for the mixing distribution. Using posterior simulation methods, we obtain full inference for the intensity function and any other functional of the process that might be of interest. We discuss applications to problems where inference for clustering in the spatial point pattern is of interest. Moreover, we consider applications of the methodology to extreme value analysis problems. We illustrate the modeling approach with three previously published data sets. Two of the data sets are from forestry and consist of locations of trees. The third data set consists of extremes from the Dow Jones index over a period of 1303 days.     

Verallgemeinerte bayes-verfahren bei sequentiellen testproblemen

In this paper, problems of sequential decision theory are taken into consideration by extending the definition of the BAYES rule and treating BAYES rules. This generalisation is quite useful for practice. In many cases only BAYES rules can be calculated. The conditions under which such sequential decision procedures exist are demonstrated, as well as how to construct them on a scheme of backward induction resulting in the conclusion that the existence of BAYES rules needs essentially weaker assumptions than the existence of BAYES rules.Futhermore, methods are searched to simplify the construction of optimal stopping rules. Some illustrative examples are given.     

Some inferential procedures in randomized repeated measurement design for binary response

We consider a two-treatment two-period crossover design in the presence of possible carryover effects, where the treatment responses are binary. We provide some simple probability models incorporating the possible carryover effects. Asymptotic distributions of the estimates of the parameters under the proposed models are derived. We carry out tests for treatment difference and carryover effects. Finally we use a data set to illustrate the applicability of the proposed procedures.     

Considerations on group-wise identical designs for linear mixed models

We consider a general class of mixed models, where the individual parameter vector is composed of a linear function of the population parameter vector plus an individual random effects vector. The linear function can vary for the different individuals. We show that the search for optimal designs for the estimation of the population parameter vector can be restricted to the class of group-wise identical designs, i.e., for each of the groups defined by the different linear functions only one individual elementary design has to be optimized. A way to apply the result to non-linear mixed models is described.     

Robust Linear Calibration

We regard the simple linear calibration problem where only the response y of the regression line y = β₀ + β₁ t is observed with errors. The experimental conditions t are observed without error. For the errors of the observations y we assume that there may be some gross errors providing outlying observations. This situation can be modeled by a conditionally contaminated regression model. In this model the classical calibration estimator based on the least squares estimator has an unbounded asymptotic bias. Therefore we introduce calibration estimators based on robust one-step-M-estimators which have a bounded asymptotic bias. For this class of estimators we discuss two problems: The optimal estimators and their corresponding optimal designs. We derive the locally optimal solutions and show that the maximin efficient designs for non-robust estimation and robust estimation coincide.     

Bias-reduced extreme quantile estimators of Weibull tail-distributions

In this paper, we consider the problem of estimating an extreme quantile of a Weibull tail-distribution. The new extreme quantile estimator has a reduced bias compared to the more classical ones proposed in the literature. It is based on an exponential regression model that was introduced in Diebolt et al. [2007. Bias-reduced estimators of the Weibull-tail coefficient. Test, to appear]. The asymptotic normality of the extreme quantile estimator is established. We also introduce an adaptive selection procedure to determine the number of upper order statistics to be used. A simulation study as well as an application to a real data set is provided in order to prove the efficiency of the above-mentioned methods.     

Joint modeling of degradation and failure time data

This paper surveys some approaches to model the relationship between failure time data and covariate data like internal degradation and external environmental processes. These models which reflect the dependency between system state and system reliability include threshold models and hazard-based models. In particular, we consider the class of degradation–threshold–shock models (DTS models) in which failure is due to the competing causes of degradation and trauma. For this class of reliability models we express the failure time in terms of degradation and covariates. We compute the survival function of the resulting failure time and derive the likelihood function for the joint observation of failure times and degradation data at discrete times. We consider a special class of DTS models where degradation is modeled by a process with stationary independent increments and related to external covariates through a random time scale and extend this model class to repairable items by a marked point process approach. The proposed model class provides a rich conceptual framework for the study of degradation–failure issues.     

A Note on Dirichlet One-Armed Bandits

One of the two independent stochastic processes (or ‘arms’) is selected and observed sequentially at each of n(≤ ∝) stages. Arm 1 yields observations identically distributed with unknown probability measure P with a Dirichlet process prior whereas observations from arm 2 have known probability measure Q. Future observations are discounted and at stage m, the payoff is a _m(≥0) times the observation Z _m at that stage. The objective is to maximize the total expected payoff. Clayton and Berry (1985) consider this problem when a _m equals 1 for m ≤ n and 0 for m > n(< ∝) In this paper, the Clayton and Berry (1985) results are extended to the case of regular discount sequences of horizon n, which may also be infinite. The results are illustrated with numerical examples. In case of geometric discounting, the results apply to a bandit with many independent unknown Dirichlet arms.     

Sharp upper bounds for the expected values of non-extreme order statistics from discrete distributions

We consider the problem of determining sharp upper bounds on the expected values of non-extreme order statistics based on i.i.d. random variables taking on N values at most. We show that the bound problem is equivalent to the problem of establishing the best approximation of the projection of the density function of the respective order statistic based on the standard uniform i.i.d. sample onto the family of non-decreasing functions by arbitrary N  -valued functions in the norm of L²(0,1)$L^2(0,1)$ space. We also present an algorithm converging to the local minima of the approximation problems.     

Empirical likelihood for a partially linear model with covariate data missing at random

In this paper, we consider the partial linear model with the covariables missing at random. Empirical likelihood ratios for the regression coefficients and the baseline function are investigated, the empirical log-likelihood ratios are proven to be asymptotically chi-squared and the corresponding confidence regions for the parameters of interest are then constructed. The finite sample behavior of the proposed method is evaluated with simulation and illustrated with an AIDS clinical trial dataset.     

Binary-image restoration

We consider the problem of binary-image restoration. The image being restored is not random, and we make no assumption about the nature of its contents. The estimate of the colour at each site is a fixed (the same for all sites) function of the data available in a neighbourhood of that site. Under this restriction, the estimate minimizing the overall mean squared error of prediction is the conditional expectation of the true colour given the observations in the neighbourhood of a site. The computation of this conditional expectation leads to the formal definition of the local characteristics of an image, namely, the frequency with which each pattern appears in the true unobserved image. When the “true” distribution of the patterns is unknown, it can be estimated from the records. The conditional expectation described above can then be evaluated using the estimated distribution of the patterns, and this procedure leads to a very natural estimate of the colour at each site. We propose two unbiased and consistent estimates for the distribution of patterns when the noise is a Gaussian white noise. Since the size of realistic images is very large, the estimated pattern distribution is usually close to the true one. This suggests that the estimated conditional expectation can be expected to be nearly optimal. An interesting feature of the proposed restoration methods is that they do not require prior knowledge of the local or global properties of the true underlying image. Several examples based on synthetic images show that the new methods perform fairly well for a variety of images with different degrees of colour continuity or textures.     

Small-sample performance of Bernoulli two-armed bandit Bayesian strategies

In this paper we examine the small-sample performance of a number of strategies for Bernoulli two-armed bandit problems with independent arms. We first investigate strategies based on a one-armed bandit threshold value (an index analogous to the ‘Gittins index’) and on upper confidence bounds for θ_i. Using backward induction and the Bayesian viewpoint, we observe that these strategies improve on the myopic strategy and get much closer to optimal in terms of total expected reward, even though for very small samples, the myopic worth itself is already close to optimal. Second, we find that the myopic strategy and the strategy based on the one-armed threshold value dominate the Bayesian optimal strategy over a region in the parameter space that can have large probability under the assumed prior. Finally, through examples we show how this has an impact on robustness: small specifications of the prior can lead to the myopic strategy performing better than the optimal strategy in terms of Bayes worth.     

Optimal two-stage group-sequential designs

We derive optimal two-stage adaptive group-sequential designs for normally distributed data which achieve the minimum of a mixture of expected sample sizes at the range of plausible values of a normal mean. Unlike standard group-sequential tests, our method is adaptive in that it allows the group size at the second look to be a function of the observed test statistic at the first look. Using optimality criteria, we construct two-stage designs which we show have advantage over other popular adaptive methods. The employed computational method is a modification of the backward induction algorithm applied to a Bayesian decision problem.     

A probablistic analysis for an optimal screening problem

Summary We consider a lotL formed byN apparently similar unitsW ₁,…,W _N, where each of theW _i may come from one of two different populationsP ₁ andP ₂;T ₁,…,T _N denote the corresponding lifetimes. The units fromP _i undergo a failure of kindi and their survival function isS _i (t). We assume that the failure rate function are known and that the units fromP ₁ are ?substandard?: λ ₁ (t)≥λ ₂ (t), ∀t≥0. We want to putW ₁,…,W _N under a pre-operational test (burn-in test) in order to eliminate at least a great part of the substandard units and we face the problem of obtaining a rule for stopping the test under the assumption that, with the failure of a unit, it is possible to recognize the population from which the unit comes. Such a problem will be formalized as an optimal stopping problem for a suitably defined Markov process. Our study shall evidentiate some fundamental aspects of the problem and the role of the prior distribution of the (random) numberM ₀ of those units inL coming fromP ₁ (substandard). The latter distribution has a great influence on the form of the solution. This research was supported by the C.N.R. Project ?Statistica Bayesiana e Simulazione in Affidalità e Modellistica Biologica?.     

Robust designs for nearly linear regression

In this paper we seek designs and estimators which are optimal in some sense for multivariate linear regression on cubes and simplexes when the true regression function is unknown. More precisely, we assume that the unknown true regression function is the sum of a linear part plus some contamination orthogonal to the set of all linear functions in the L₂ norm with respect to Lebesgue measure. The contamination is assumed bounded in absolute value and it is shown that the usual designs for multivariate linear regression on cubes and simplices and the usual least squares estimators minimize the supremum over all possible contaminations of the expected mean square error. Additional results for extrapolation and interpolation, among other things, are discussed. For suitable loss functions optimal designs are found to have support on the extreme points of our design space.