Sufficiency and unbiased estimation

Bayesian decision problems require subjective elicitation of the inputs: beliefs and preferences. Sometimes, elicitation methods may not represent perfectly the judgements of the decision maker. Several foundations propose to overlay 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 are in the conditions of a Pareto order. Hence the solution concept is the set of non dominated alternatives. In this article we focus on the computation of the efficient set when the preferences are modeled by a class of convex loss functions.

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.

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.

For a multivariate normal set-up, it is well known that themaximumlikelihood estimator (MLE) of covariance matrix is neither admissible nor minimax under the Stein loss function. In this paper, we reveal that the MLE based on the Iwasawa parameterization leads to minimaxity with respect to the Stein loss function. Furthermore, a novel class of loss functions is proposed so that the minimum risks of the MLEs are identical in different coordinate systems, Cholesky parameterization and full Iwasawa parameterization. In other words, the MLEs based on these two different parameterizations are characterized by the property of minimaxity, without a Stein paradox. The application of our novel method to the high-dimensional covariance matrix problem is also discussed.

The idea to exploit label dependencies for better prediction is at the core of methods for multi-label classification (MLC), and performance improvements are normally explained in this way. Surprisingly, however, there is no established methodology that allows to analyze the dependence-awareness of MLC algorithms. With that goal in mind, we introduce a class of loss functions that are able to capture the important aspect of label dependence. To this end, we leverage the mathematical framework of non-additive measures and integrals. Roughly speaking, a non-additive measure allows for modeling the importance of correct predictions of label subsets (instead of single labels), and thereby their impact on the overall evaluation, in a flexible way. The well-known Hamming and subset 0/1 losses are rather extreme special cases of this function class, which give full importance to single label sets or the entire label set, respectively. We present concrete instantiations of this class, which appear to be especially appealing from a modeling perspective. The assessment of multi-label classifiers in terms of these losses is illustrated in an empirical study, clearly showing their aptness at capturing label dependencies. Finally, while not being the main goal of this study, we also show some preliminary results on the minimization of this parametrized family of losses.

4 - Recursive Enumerability

In stochastic modeling, probability distributions often involve unknown parameters, and this chapter deals with some principles of estimation of such parameters. That involves various classes of loss functions, and the study concentrates on certain desirable properties of estimators which are (known) functions of random variables. These include a detailed mathematical analysis of Bayes and maximum likelihood estimation as well as (nonlinear) prediction. Also treated are the asymptotics of the methodology to be explored for particular types of processes later on, and a relatively short account of sequential estimation together with some important complements.

We study the learnability of the edge cost functions in routing games from observations of the equilibrium flows, where the learnability is measured as the minimum number of samples needed to maintain a high consistency of the empirical risk, which is a statistical estimator for the quality of the learned model. To provide an upper bound on the minimum sample size, we motivate the analysis of the uniform laws of large numbers on a class of loss functions indexed over the space of parameters we want to estimate. On one hand, leveraging results on the complexity of function classes and in approximation theory, we investigate how the behavior of the empirical risk relates to a notion of complexity of the parameter space. On the other hand, using sensitivity analysis in optimization, we study how variations in the parameter space translate into variations in the class of loss functions. This allows us to show, independently of the sample distribution and of the edge costs parametrization, that the sample size required to maintain a high consistency grows quadratically with the number of edges in the network (or linearly if all the edge cost functions have the same shape), and grows linearly with the Lipschitz constant of the edge cost functions, and quadratically with the inverse of the strong monotonicity constant of the edge cost functions.

Admissibility and inadmissibility of a generalized Bayes unbiased estimator in a multivariate linear model

Sensitivity of a general class of shape functionals to topological changes

This study examines the estimation of the scale parameter of the Weibull-Power Function Distribution (WPFD) using both maximum likelihood and Bayesian approaches. Bayesian estimation is conducted under one informative Gamma prior with hyperparameters , as well as two non-informative priors: the uniform prior and Jeffreys’ prior. For each prior specification, Bayes estimators are derived under squared error, quadratic, and precautionary loss functions. Closed-form expressions for the posterior distributions and corresponding Bayes estimators are obtained. The finite-sample performance of the competing estimators is evaluated through a Monte Carlo simulation study based on 1000 replications. Estimator performance is assessed using mean squared error (MSE), bias, and coverage probability. The results indicate that all estimators are consistent, with bias and MSE decreasing as sample size increases. Across different prior specifications and parameter settings, the Bayesian estimator under the quadratic loss function consistently attains the lowest MSE, yielding reductions of approximately 10-20% relative to the maximum likelihood estimator in small and moderate samples. These findings suggest that Bayesian estimation under quadratic loss provides improved finite-sample efficiency for estimating the WPFD scale parameter, while maintaining asymptotic comparability with the maximum likelihood approach.

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Suppose Y <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> is obtained by observing a uniform Bernoulli random vector X <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> through a binary symmetric channel. Courtade and Kumar asked how large the mutual information between Yn and a Boolean function b(X <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> ) could be, and conjectured that the maximum is attained by a dictator function. An equivalent formulation of this conjecture is that dictator minimizes the prediction cost in a sequential prediction of Y <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> under logarithmic loss, given b(X <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> ). In this paper, we study the question of minimizing the sequential prediction cost under a different (proper) loss function-the quadratic loss. In the noiseless case, we show that majority asymptotically minimizes this prediction cost among all Boolean functions. We further show that for weak noise, majority is better than a dictator, and that for a strong noise dictator outperforms majority. We conjecture that for quadratic loss, there is no single sequence of Boolean functions that is simultaneously (asymptotically) optimal at all noise levels.

Our starting point are several general classes of real functions defined on the unit square satisfying some basic properties such as a boundary condition or several types of monotonicity and continuity. Applying to these functions some parameterized transformations and other constructions such as the transpose and flipping (which describe different aspects of symmetry) and truncation, we ask for conditions yielding (again) a bivariate copula. Some of these transformations are involutive (on one or more classes of functions), others are not even injective, and occasionally they induce additional properties, yielding, e.g., a (quasi-)copula. For several typical scenarios we identify the (not necessarily convex) sets of parameters leading to a copula and conditions imposing a minimal set of parameters.

In this paper, we introduce notable Jensen–Mercer inequality for a general class of convex functions, namely uniformly convex functions. We explore some interesting properties of such a class of functions along with some examples. As a result, we establish Hermite–Jensen–Mercer inequalities pertaining uniformly convex functions by considering the class of fractional integral operators. Moreover, we establish Mercer–Ostrowski inequalities for conformable integral operator via differentiable uniformly convex functions. Finally, we apply our inequalities to get estimations for normal probability distributions (Gaussian distributions).