Class Of Loss Functions Research Articles

Estimation of two ordered bivariate mean residual life functions

Situations occur frequently in which the mean residual life (mrl) functions of two populations must be ordered. For example, if a mechanical device is improved, the mrl function for the improved device should not be less than that of the original device. Also, mrl functions for medical patients should often be ordered depending on the status of concomitant variables. This paper proposes nonparametric estimators of the bivariate mrl function under a mrl ordering. The estimators are shown to be asymptotically unbiased, strongly uniformly consistent and weakly convergent to a bivariate Gaussian process. The estimators are shown to be the projections, in a sense to be made precise, of the empirical mrl function onto an appropriate convex set of mrl functions. In the one-sample problem, the new estimators dominate the empirical mrl function in terms of risk with respect to a wide class of loss functions.

Asymptotics in Bayesian decision theory with applications to global robustness

We provide the rate of convergence of the Bayes action derived from non smooth loss functions involved in Bayesian robustness. Such loss functions are typically not twice differentiable but admit right and left second derivatives. The asymptotic limit of three measures of global robustness is given. These measures are the range of the Bayes actions set associated with a class of loss functions, the maximum regret of using a particular loss when the subjective loss belongs to a given class and the range of the posterior expected loss when the loss ranges over a given class. An application to prior robustness with density ratio classes is provided.

Asymptotic global robustness in bayesian decision theory

In Bayesian decision theory, it is known that robustness with respect to the loss and the prior can be improved by adding new observations. In this article we study the rate of robustness improvement with respect to the number of observations n. Three usual measures of posterior global robustness are considered: the (range of the) Bayes actions set derived from a class of loss functions, the maximum regret of using a particular loss when the subjective loss belongs to a given class and the range of the posterior expected loss when the loss function ranges over a class. We show that the rate of convergence of the first measure of robustness is $\sqrt{n}$ , while it is n for the other measures under reasonable assumptions on the class of loss functions. We begin with the study of two particular cases to illustrate our results.

On Measuring Loss Robustness Using Maximum A Posteriori Estimate

This paper investigates robustness of the loss function in a Bayesian framework. We incorporate maximum a posteriori (MAP) estimate, posterior expected loss (PEL) and introduce MAP range, in order to measure loss robustness under certain classes of loss functions, including Hellinger and LINEX loss functions. These classes of loss functions are considered for the estimation of the natural parameters of the continuous exponential family. The methods introduced are illustrated with some standard examples, including Normal and Gamma distributions. Finally, a comparison between MAP and PEL based loss robustness measures is presented to illustrate the importance of MAP estimation in measuring loss robustness.

Adaptive Estimators and Tests of Stationary and Nonstationary Short- and Long-Memory ARFIMA–GARCH Models

This article considers the fractionally autoregressive integrated moving average [ARFIMA(p,d,q)] models with GARCH errors. The process generated by this model is short memory, long memory, stationary, and nonstationary, respectively, when d ∈ (-1/2, 0), d ∈ (0, 1/2), d ∈ (-1/2, 1/2), and d ∈ (1/2, ∞). Using a unified approach, the local asymptotic normality of the model is established for d ∈ ∪∞j = 0(J - 1/2, J + 1/2). The adaptivity and efficiency of the estimating parameters are discussed. In a class of loss functions, the asymptotic minimax bound of the estimators for the model is given when the density f of rescaled residuals is unknown. An adaptive estimator is constructed for the parameters in the ARFIMA part when f is symmetric, and a general form of the efficient estimator is also constructed for all the parameters in the ARFIMA and GARCH parts. When the density f is unknown, Wald tests are constructed for testing the unit root + 1 against the class of fractional unit roots. It is shown that these tests asymptotically follow the chi-squared distribution and are locally most powerful. These results are new contributions to the literature, even for the ARFIMA model with iid errors, except for the adaptive estimator in this case with d ∈ (0, 1/2). The performance of the asymptotic results in finite samples is examined through Monte Carlo experiments. An application to the U.S. Consumer Price Index inflation series is given, and a clear conclusion from this is that the series is neither an I(0) nor an I(1), but rather than an I(d) process with d ≈ 0.288.

Optimal stopping in software testing

AbstractIn this paper we address the problem of how to decide when to terminate the testing/modification process and to release the software during the development phase. We present a Bayesian decision theoretic approach by formulating the optimal release problem as a sequential decision problem. By using a non‐Gaussian Kalman filter type model, proposed by Chen and Singpurwalla (1994), to track software reliability, we are able to obtain tractable expressions for inference and determine a one‐stage look ahead stopping rule under reasonable conditions and a class of loss functions. © 2002 Wiley Periodicals, Inc. Naval Research Logistics, 2003

The inverted beta loss function: properties and applications

In this paper, a general class of loss functions based on the inversion of the standard beta probability density function (pdf) is examined. The extension of this loss function from a standard beta pdf ranging from (0, 1) to the general beta pdf ranging from (p, q) is examined through the scale invariance property under a linear transformation. An industrial application in quality assurance is used to demonstrate this general class of loss functions. Mathematical derivations are attached in the Appendices.

Uncertainty in Beliefs and Preferences: Conditions for Optimal Alternatives

Bayes decision problems require subjective elicitation of inputs: beliefs and preferences. Elicitation methods are far from simple, and elicited quantities cannot perfectly represent the judgements of the Decision Maker. Several foundations suggest overlaying this problem using robust approaches. In these models, beliefs are modelled by a class of probability distributions, and preferences by a class of loss functions. Then we have a Pareto order. So, the solution concept is an efficient set. In this paper, we deal with the problem of existence of a solution. Moreover, we study the relations between Bayes actions and nondominated ones.

Asymptotic Limit of the Bayes Actions Set Derived from a Class of Loss Functions

Similarly to the determination of a prior in Bayesian Decision theory, an arbitrarily precise determination of the loss function is unrealistic. Thus, analogously to global robustness with respect to the prior, one can consider a set of loss functions to describe the imprecise preferences of the decision maker. In this paper, we investigate the asymptotic behavior of the Bayes actions set derived from a class of loss functions. When the collection of additional observations induces a decrease in the range of the Bayes actions, robustness is improved. We give sufficient conditions for the convergence of the Bayes actions set with respect to the Hausdorff metric and we also give the limit set. Finally, we show that these conditions are satisfied when the set of decisions and the set of states of nature are subsets of Rp.

VARIANCE ESTIMATION FOR THE NORMAL DISTRIBUTION UNDER LOG SYMMETRIC LOSS

For a normal sample with unknown mean, the almost universally used estimator of the variance, σ2, is “the sample variance.” This estimator is the minumum variance unbiased estimator of σ2, but it is inadmissible under square error loss. It is dominated by the maximum likelihood estimator, which is also inadmissible. We consider a class of estimators and compare these estimators under a class of loss functions which we call “log symmetric.”

Dealing With Label Switching in Mixture Models

Summary In a Bayesian analysis of finite mixture models, parameter estimation and clustering are sometimes less straightforward than might be expected. In particular, the common practice of estimating parameters by their posterior mean, and summarizing joint posterior distributions by marginal distributions, often leads to nonsensical answers. This is due to the so-called ‘label switching’ problem, which is caused by symmetry in the likelihood of the model parameters. A frequent response to this problem is to remove the symmetry by using artificial identifiability constraints. We demonstrate that this fails in general to solve the problem, and we describe an alternative class of approaches, relabelling algorithms, which arise from attempting to minimize the posterior expected loss under a class of loss functions. We describe in detail one particularly simple and general relabelling algorithm and illustrate its success in dealing with the label switching problem on two examples.

A note on sequential estimation of the size of a population under a general loss function

In estimating the size of a finite population under a sequential sampling scheme where the stopping rule is to stop sampling when a fixed number of marked items are observed, it has been shown that the maximum likelihood estimator (MLE) does not have an explicit expression and is inadmissible under weighted-squared-error loss. This note shows that the MLE is inadmissible under a very general class of loss functions. Also, a class of estimators which dominate the MLE is constructed and given in the article. Finally, an optimal class of estimators for some commonly used loss functions will be derived.

Analytic approximation of the interval of Bayes actions derived from a class of loss functions

Due to a lack of time or limited information, a precise determination of the loss function, and the computation of the associated Bayes action is not always possible. For these reasons, statisticians use mathematically tractable losses instead of subjective ones. We study the legitimacy of such lossesl0 with specific properties, by first constructing a class aroundl0 which contains all plausible subjective losses, and then computing the setAπ of Bayes actions associated with this class. We derive an analytic approximation toAπ. We apply these results whenlo is the quadratic, LIXEX and entropy loss. Finally, we investigate robustness characteristics using the analytic expression ofAπ.

Minimax detection of a signal for nondegenerate loss functions, and extremal convex problems

We study a wide class of minimax problems of signal detection under nonparametric alternatives and a modification of these problems for a special class of loss functions. Under rather general assumptions, we obtain the exact asymptotics (of Gaussian type) for the minimax error probabilities and the structure of asymptotically minimax tests. The methods are based on a reduction of the problems under consideration to extremal problems of minimization of a certain Hilbert norm on a convex set of sequences of probability measures on the real line. These extremal problems are investigated in a paper by I. A. Suslina for alternatives having the type of lq-ellipsoids with lp-balls removed. Bibliography: 16 titles.

Application of the Limit of Truncated Isotonic Regression in Optimization Subject to Isotonic and Bounding Constraints

An isotonic regression truncated by confining its domain to a union of its level sets is the isotonic regression in the reduced function space. When some of the weights with which the inner product system is defined go to infinity, the truncated isotonic regression converges. This limit can be used in discribing the projection onto the set of vectors which satisfy an order restriction and have one or more of its coordinates bounded above and/or below. Through this characterization, two inequalities associated with the projection are established and found useful in order and bound restricted statistical inference. The results obtained show that for an exponential distribution family the inequalities lead to the linkage of the order and bound restricted MLE with the projection of the unrestricted MLE, and the dominance of the order and bound restricted MLE over the unrestricted MLE with respect to two classes of loss functions and risks as well as Bayes risks.

On the estimation of a survival function under a dispersive order constraint

The concept of dispersion is of great relevance in experimental situations. For example, it is well-known in corrosion engineering that lifetimes of metals under corrosive experimental conditions are affected by the concentration in solution of the corrosive material. Thus, the data shows more dispersion the weaker the corrosivity of the solution. Here, the problem of estimating F subject to the constraint that F is less dispersed than G is considered. Under suitable regularity conditions, it is shown that the proposed estimators dominate the empirical distribution function for a wide class of loss functions. Results for censored data are also discussed any asymptotic confidence intervals and bands are derived.

A General Class of Loss Functions with Industrial Applications

Over the last few years, loss functions have become increasingly important in quality assurance settings. In this paper, we develop a class of loss functions based on inversions of probability density functions and present industrial examples to illustrate process monitoring and control using economic loss instead of measurements.

Single-sample Bayes and empirical Bayes rules for ranking and estimating multinomial probabilities

Abstract The problem of selecting the most probable cell, more generally the problem of partitioning k multinomial cells according to the values of the cell probabilities is considered. Bayes rules are obtained for the general ranking problem under a general class of loss functions. A sequence of parametric empirical Bayes selection rules is proposed and shown to be asymptotically optimal of order O(e-cn) for some positive constant c. Also the problem of selecting the most (least) probable cell and simultaneously estimating the associated probability of the selected cell is considered. For this problem also a sequence of parametric empirical Bayes rules is proposed and shown to be asymptotically optimal of order O(n-1).

ON Optimality the Feosd-Accuracy Estimate Op Theparameter In Ah Explosive Autoregressive Process Op The First Order

The unif orm asymptotic normality of the fixed accuracy estimate for the parameter in explosive autoregressive process of the first order is established. The asymptotic optimality of the estimate inminimax sense for sufficiently broad class of loss functions is shown. The f ixed-accuracy estimate is constructed as a sequential modification of the least squares estimate.

Minimax invariant estimator of a continuous distribution function

Consider the problems of the continuous invariant estimation of a distribution function with a wide class of loss functions. It has been conjectured for long that the best invariant estimator is minimax for all sample sizes n≥1. This conjecture is proved in this short note.