Minimax regret treatment choice with finite samples

This paper applies the minimax regret criterion to choice between two treatments conditional on observation of a finite sample. The analysis is based on exact small sample regret and does not use asymptotic approximations or finite-sample bounds. Core results are: (i) Minimax regret treatment rules are well approximated by empirical success rules in many cases, but differ from them significantly–both in terms of how the rules look and in terms of maximal regret incurred–for small sample sizes and certain sample designs. (ii) Absent prior cross-covariate restrictions on treatment outcomes, they prescribe inference that is completely separate across covariates, leading to no-data rules as the support of a covariate grows. I conclude by offering an assessment of these results.     

Statistical treatment choice based on asymmetric minimax regret criteria

This paper studies the problem of treatment choice between a status quo treatment with a known outcome distribution and an innovation whose outcomes are observed only in a finite sample. I evaluate statistical decision rules, which are functions that map sample outcomes into the planner’s treatment choice for the population, based on regret, which is the expected welfare loss due to assigning inferior treatments. I extend previous work started by Manski (2004) that applied the minimax regret criterion to treatment choice problems by considering decision criteria that asymmetrically treat Type I regret (due to mistakenly choosing an inferior new treatment) and Type II regret (due to mistakenly rejecting a superior innovation) and derive exact finite sample solutions to these problems for experiments with normal, Bernoulli and bounded distributions of outcomes. The paper also evaluates the properties of treatment choice and sample size selection based on classical hypothesis tests and power calculations in terms of regret.     

Minimax-regret treatment choice with missing outcome data

I use the minimax-regret criterion to study choice between two treatments when some outcomes in the study population are unobservable and the distribution of missing data is unknown. I first assume that observable features of the study population are known and derive the treatment rule that minimizes maximum regret over all possible distributions of missing data. When no treatment is dominant, this rule allocates positive fractions of persons to both treatments. I then assume that the data are a random sample of the study population and show that in some instances, treatment rules that estimate certain point-identified population means by sample averages are finite-sample minimax regret.     

Zum Minimax-Prinzip in der statistischen Qualitätskontrolle

Summary To decide about acception or rejection of a lot of items by sampling the sample size and the critical value of the test statistic are to be determined suitably. In this paper the consequences of the minimax principle are studied when it is applied to the expectation of the total loss. Under the assumption that the loss for acceptance and the loss for rejection without control are linear functions of the portion of defective items, it is proved that the minimax principle implies the decision rule do not control and accept the lot with a certain probability 0. In other words: In this case and in this form the minimax principle is not adequate.     

Approximately optimum two-stage sampling plans

Summary In order to compare two-stage sampling plans we use the minimax regret principle i.e. the minimax principle applied to the corresponding regret functions. In this paper approximation formulas for optimum two-stage sampling plans are derived in the case of sampling by attributes as well as in the case of sampling by variables; furthermore a method is presented how to improve the approximate solution in the first case.     

Deriving utilities using the analytic hierarchy process

The analytic hierarchy process is used to derive the utilities of outcomes in a decision problem. Such utilities may be verifiable or adjusted by the familiar standard gamble procedure. For multiattribute outcomes, the methodology may be easier to employ than scoring rule procedures.     

Procedures to determine optimum two-stage sampling plans by attributes

Summary In order to compare two sampling plans we use the minimax regret principle, i.e. the minimax principle applied to regret functions. It is shown that among all two-stage sampling plans there exists an optimum sampling plan which can be computed with the aid of a procedure presented in this paper; furthermore another procedure is described how to obtain an approximately optimum two-stage sampling plan in a more direct way. Finally only those two-stage sampling plans are regarded which satisfy an additional condition; among these sampling plans an optimum one exists and is to be determined, too.     

On the natural selection rule in general linear models

We consider a problem of selecting the best treatment in a general linear model. We look at the properties of the natural selection rule. It is shown that the natural selection rule is minimax under to “0–1” loss function and it is a Bayes rule under a monotone permutation invariant loss function with respect to a permutation invariant prior for every variance balanced design. Some other condition on the design matrix is given so that a Bayes rule with respect to a normal prior will be of simple structure.     

Ungünstige Verteilungen bei einparametrigen Parameterschätzproblemen

Summary One of the commonly used methods for determining minimax point estimators is based on least favorable distributions, because Bayes estimators with respect to a least favorable distribution are frequently minimax point estimators. Therefore it is worthwhile to investigate the structure of least favorable distributions. In the present paper it will be proved that, under certain conditions, each least favorable distribution is finite discrete. We give sufficient conditions for point estimation problems with the property that each least favorable distribution is finite discrete. Some examples are presented.     

Kostenoptimale Prüfpläne für ein quantitatives Merkmal

Summary A sample inspection plan is said to be optimal in the sense of the minimax regret principle, if it minimizes the difference between the expected total costs and the unavoidable costs. The results of this article can be used to calculate such sample inspection plans for a quantitative quality control with one-sided tolerance limits and known or unknown variance of the test variate. As an example of practical importance the case of a normal variate with unknown variance is considered. Formulae are given to estimate the error that arises if the assumed distribution of the test variate differs from the actual distribution.     

Optimal rules and robust Bayes estimation of a Gamma scale parameter

For estimating an unknown scale parameter of Gamma distribution, we introduce the use of an asymmetric scale invariant loss function reflecting precision of estimation. This loss belongs to the class of precautionary loss functions. The problem of estimation of scale parameter of a Gamma distribution arises in several theoretical and applied problems. Explicit form of risk-unbiased, minimum risk scale-invariant, Bayes, generalized Bayes and minimax estimators are derived. We characterized the admissibility and inadmissibility of a class of linear estimators of the form $cX\,{+}\,d$ , when $X\sim \varGamma (\alpha ,\eta )$ . In the context of Bayesian statistical inference any statistical problem should be treated under a given loss function by specifying a prior distribution over the parameter space. Hence, arbitrariness of a unique prior distribution is a critical and permanent question. To overcome with this issue, we consider robust Bayesian analysis and deal with Gamma minimax, conditional Gamma minimax, the stable and characterize posterior regret Gamma minimax estimation of the unknown scale parameter under the asymmetric scale invariant loss function in detail.     

Robust optimal monetary policy in a forward‐looking model with parameter and shock uncertainty

This paper characterizes a robust optimal policy rule in a simple forward‐looking model, when the policymaker faces uncertainty about model parameters and shock processes. We show that the robust optimal policy rule is likely to involve a stronger response of the interest rate to fluctuations in inflation and the output gap than is the case in the absence of uncertainty. Thus parameter uncertainty alone does not necessarily justify a small response of monetary policy to perturbations. However, uncertainty may amplify the degree of ‘super‐inertia’ required by optimal monetary policy. We finally discuss the sensitivity of the results to alternative assumptions. Copyright © 2007 John Wiley & Sons, Ltd.     

Decision rules for the choice of structural equations

In practical econometric analysis we are faced with the problem of how to specify structural equations. The conventional t-test of coefficients is apparently inappropriate. The smallest root, say λ, of a certain determinantal equation provides us with basis for the test of overidentifying restrictions. The preliminary test, based on λ, may give us a possible decision rule for choosing a structural equation from nested alternatives. However, ambiguity remains in specifying the significance level. We propose a decision method called the unbiased decision rule; unbiased in the sense that we attain a correct decision with probability of more than a half. The critical points are found as the medians of non-central F-distributions. The degrees of freedom and the non-centrality parameter of non-central F-distributions are determined by the properties of contending models. We also discuss the implications of the unbiased decision rule in the context of the conventional pre-test.     

Stochastic Route Choice Equilibrium Assignment for Travelers with Heterogeneous Regret Aversions

Most route choice models assume that people are completely rational. Recently, regret theory has attracted researchers’ attentions because of its power to depict real travel behavior. This paper proposes a multiclass stochastic user equilibrium assignment model by using regret theory. All users are differentiated by their own regret aversion. The route travel disutility for users of each class is defined as a linear combination of the travel time and anticipated regret. The proposed model is formulated as a variational inequality problem and solved by using the self-regulated averaging method. The numerical results show that users’ regret aversion indeed influences their route choice behavior and that users with high regret aversion are more inclined to change route choice when the traffic congestion degree varies.     

Kosten-optimale Prüfpläne für die Gut-Schlecht-Prüfung

Summary A lot is accepted if the number of defective units in a sample of sizen does not exceed the acceptance numberc. The usefulness of the sampling plan (n, c) is described by the regret function. This regret functionR(p), depending on the proportionp of defective units in the lot, is the expectation of the avoidable costs. There always exists an optimum sampling plan which minimizes the maximum ofR(p). The dependence of the maxima ofR(p) onn andc is studied and some theorems are given which are useful for calculating the minimax solution, that is the optimum sampling plan.      

Treatment effect estimation with covariate measurement error

This paper investigates the effect that covariate measurement error has on a treatment effect analysis built on an unconfoundedness restriction in which there is conditioning on error free covariates. The approach uses small parameter asymptotic methods to obtain the approximate effects of measurement error for estimators of average treatment effects. The approximations can be estimated using data on observed outcomes, the treatment indicator and error contaminated covariates without employing additional information from validation data or instrumental variables. The results can be used in a sensitivity analysis to probe the potential effects of measurement error on the evaluation of treatment effects.     

A conditional minimax approach in survey sampling

Summary Applying the usual minimax criterion in finite sampling theory yields complicated solutions except the parameter space has certain invariance properties. A conditional minimax criterion is suggested. After a sample is selected it is reasonable to seek an estimator that has good properties (e.g. minimaxity) for that sample. Explicit solutions are given in the case where the parameter space is described by quadratic forms.     

Regression discontinuity inference with specification error

A regression discontinuity (RD) research design is appropriate for program evaluation problems in which treatment status (or the probability of treatment) depends on whether an observed covariate exceeds a fixed threshold. In many applications the treatment-determining covariate is discrete. This makes it impossible to compare outcomes for observations “just above” and “just below” the treatment threshold, and requires the researcher to choose a functional form for the relationship between the treatment variable and the outcomes of interest. We propose a simple econometric procedure to account for uncertainty in the choice of functional form for RD designs with discrete support. In particular, we model deviations of the true regression function from a given approximating function—the specification errors—as random. Conventional standard errors ignore the group structure induced by specification errors and tend to overstate the precision of the estimated program impacts. The proposed inference procedure that allows for specification error also has a natural interpretation within a Bayesian framework.     

Identificeren op grond van waarnemingen

This paper is an exposition of some elements of Wald's decision theory. Concepts like a priori distribution, decision function, loss function, risk, Bayes procedures, admissible procedures, minimax procedures, least favourable distribution are introduced all in connection with the problem of classification of observations into two given populations. The exact treatment of the procedures and their mutual relations are illustrated by numerical examples concerning univariate and multivariate normal populations. The extension towards the classification into more than two given populations concludes the paper.     

Minimax estimation of a cumulative distribution function by converting to a parametric problem

Let X = (X ₁,...,X _n ) be a sample from an unknown cumulative distribution function F defined on the real line . The problem of estimating the cumulative distribution function F is considered using a decision theoretic approach. No assumptions are imposed on the unknown function F. A general method of finding a minimax estimator d(t;X) of F under the loss function of a general form is presented. The method of solution is based on converting the nonparametric problem of searching for minimax estimators of a distribution function to the parametric problem of searching for minimax estimators of the probability of success for a binomial distribution. The solution uses also the completeness property of the class of monotone decision procedures in a monotone decision problem. Some special cases of the underlying problem are considered in the situation when the loss function in the nonparametric problem is defined by a weighted squared, LINEX or a weighted absolute error.