Quadratic Loss Function Research Articles

Introduction to Neutrosophic Bayes Estimation Theory

This research presents the concept of neutrosophic Bayesian estimation defining the neutrosophic loss function, neutrosophic risk function, neutrosophic posterior risk function and neutrosophic maximum a posteriori estimator. Minimization of the neutrosophic posterior risk of the estimator is also discussed. An algebraic isomorphism is used to simplify equations solving. As an application of the presented theorems, a sample drawn from a neutrosophic gamma distribution with a conjugate prior is discussed and studied and the parameter of the formulated distribution is successfully estimated using neutrosophic quadratic loss function which results an estimator that equals the posterior mean.

ε-Insensitive Least Squares Support Vector Regression with Sequential Minimal Optimization

The [Formula: see text]-insensitive-Least Squares Support Vector Regression ([Formula: see text]-LSSVR) optimization problem with [Formula: see text]-insensitive quadratic loss function is a non-smooth, piecewise quadratic complex problem. Therefore, the optimal solution of [Formula: see text]-LSSVR takes longer than Classic LSSVR. Additionally, due to its unbounded influence function, [Formula: see text]-LSSVR, like classical LSSVR, shows poor generalization ability in the presence of outliers. To overcome the above-mentioned drawbacks, this paper presents an [Formula: see text]-insensitive Weighted Least Squares Support Vector Regression ([Formula: see text]-WLSSVR) with equality constraints to increase robustness against outliers. To reduce the computational complexity and time, the optimization problem of [Formula: see text]-WLSSVR is solved by the Sequential Minimal Optimization (SMO) algorithm. Extensive simulations have shown that the presented [Formula: see text]-WLSSVR model combined with the SMO algorithm is more effective and efficient than LSSVR and its known sparse variants.

Bayesian Estimation of a Scale Parameter of the Gumbel-Lomax Distribution Using Informative and Non Informative Priors

Estimating the scale parameter of the Gumbel-Lomax Distribution using the Bayesian method of estimation and evaluating the estimators by assuming two non-informative prior distributions and one informative prior distribution is very important for the general application of the Gumbel-Lomax distribution. These estimators are obtained using the squared error loss function (SELF), Quadratic loss function (QLF) and precautionary loss function (PLF). The posterior distributions of the scale parameter of the Gumbel-Lomax distribution are derived and the Estimators are also obtained using the above mentioned priors and loss functions. Furthermore, a simulation using a package in R software is carried out to assess the performance of the estimators by making use of the Mean Squared Errors of the Estimators under the Bayesian approach and Maximum likelihood method. Our results show that Bayesian Method using PLF under all priors produces the best estimators of the scale parameter compared to estimators using the Maximum Likelihood method, SELF and QLF under all the priors irrespective of the values of the parameters and the different sample sizes. It is also discovered that the other parameters have no effect on the estimators of the scale parameter.

Optimal Bayesian sampling plan for censored competing risks data

There is a substantial amount of literature in the area of acceptance sampling plan with censored lifetime data. However, the optimality of a Bayesian sampling plan in the presence of competing risks has not been considered so far. In this paper, first, the Bayesian sampling plans (BSP) for Type-II and Type-I hybrid censoring schemes are discussed in presence of competing risks when the lifetime distribution is exponential. The closed-form expression of the Bayes decision function is obtained analytically for a linear loss function. Then we consider the Weibull distribution with an unknown shape parameter under Type-I hybrid censoring scheme in presence of competing risks to obtain the BSP. However, the Bayes decision function cannot be obtained in closed-form for a general loss function, and in such cases, a numerical algorithm is proposed. As an illustration, in the exponential case, a quadratic loss function, and in Weibull case, a non-polynomial loss function, are considered for the application of the proposed numerical approach to obtain the optimum BSPs using the Bayes decision function.

Restricted Stein-rule estimation in ultrastructural linear measurement error models

We consider a multivariate ultrastructural measurement error model when some information on regression coefficients is available in the form of linear restriction. We propose Stein-rule estimation for the regression coefficients. Without any assumption about the distribution for the random components, the asymptotic risk under a specific quadratic loss function is studied and a sufficient condition for the dominance of the proposed estimator over a consistent estimator. We also carry out a simulation study to demonstrate the finite sample properties of the estimators.

Frequentist model averaging for zero‐inflated Poisson regression models

AbstractThis paper considers frequentist model averaging for estimating the unknown parameters of the zero‐inflated Poisson regression model. Our proposed weight choice procedure is based on the minimization of an unbiased estimator of a conditional quadratic loss function. We prove that the resulting model average estimator enjoys optimal asymptotic property and improves finite sample properties over the two commonly used information‐based model selection estimators and their model average estimators via simulation studies. The proposed method is illustrated by a real data example.

Quantile regression version of Hodrick–Prescott filter

Hodrick–Prescott (HP) filter is a popular trend filtering method for univariate macroeconomic time series such as real gross domestic product. This paper considers the quantile regression version of HP filter (qHP filter), which is a filtering method defined by replacing quadratic loss function of HP filter with quantile regression loss function. One of the essential properties of quantile regression is that if the regression includes intercept, then the ratio of negative residuals can be almost controlled. Does the suggested qHP filter also have the property? This paper answers this question. In addition to the main result, we provide an empirical illustration.

Forecast combination-based forecast reconciliation: Insights and extensions

In this paper, we build upon a recently proposed forecast combination-based approach to the reconciliation of a simple hierarchy (Hollyman R., Petropoulos F., Tipping M.E., Understanding forecast reconciliation, European Journal of Operational Research, 2021, 294, 149–160) and extend it in some new directions. In particular, we provide insights into the nature and mathematical derivation of the level-l conditional coherent (LlCC) point forecast reconciliation procedure for an elementary two-level hierarchy. We show that: (i) the LlCC procedure is the result of a linearly constrained minimization of a quadratic loss function, with an exogenous constraint given by the base forecast of the top level series of the hierarchy, which is not revised; and (ii) endogenous constraints may also be considered in the same framework, thereby resulting in level conditional reconciled forecasts where both the top and the bottom level time series forecasts are coherently revised. In addition, we show that the LlCC procedure (i.e., with exogenous constraints but the result also holds in the endogenous case) does not guarantee the non-negativity of the reconciled forecasts, which can be an issue in cases when non-negativity is a natural attribute of the variables that need to be forecast (e.g., sales and tourism flows). Finally, we consider two forecasting experiments to evaluate the performance of various cross-sectional forecast combination-based point forecast reconciliation procedures (vis-à-vis the state-of-the-art procedures) in a fair setting. In this framework, due to the crucial role played by the (possibly different) models used to compute the base forecasts, we re-interpret the combined conditional coherent reconciliation procedure (CCCH) of Hollyman et al. (2021) as a forecast pooling approach, and show that accuracy improvements may be obtained by adopting a simple forecast averaging strategy.

The role of asymmetric prediction losses in smart charging of electric vehicles

Climate change prompts humanity to look for decarbonisation opportunities, and a viable option is to supply electric vehicles with renewable energy. The stochastic nature of charging demand and renewable generation requires intelligent charging driven by predictions of charging behaviour. The conventional prediction models of charging behaviour usually minimise the quadratic loss function. Moreover, the adequacy of predictions is almost solely evaluated by accuracy measures, disregarding the consequences of prediction losses in an application context. Here, we study the role of asymmetric prediction losses which enable balancing the over- and under-predictions and adjust predictions to smart charging algorithms. Using the main classes of machine learning methods, we trained prediction models of the connection duration and compared their performance for various asymmetries of the loss function. In addition, we proposed a methodological approach to quantify the consequences of prediction losses on the performance of selected archetypal smart charging schemes. In concrete situations, we demonstrated that an appropriately selected degree of the loss function asymmetry is crucial as it almost doubles the price range where the smart charging is beneficial, and increases the extent to which the charging demand is satisfied up to 40%. Additionally, the proposed methods improve charging fairness since the distribution of unmet charging demand across vehicles becomes more homogeneous.

Progressive censoring schemes for marshall-olkin pareto distribution with applications: Estimation and prediction.

In this paper two prediction methods are used to predict the non-observed (censored) units under progressive Type-II censored samples. The lifetimes of the units follow Marshall-Olkin Pareto distribution. We observe the posterior predictive density of the non-observed units and construct predictive intervals as well. Furthermore, we provide inference on the unknown parameters of the Marshall-Olkin model, so we observe point and interval estimation by using maximum likelihood and Bayesian estimation methods. Bayes estimation methods are obtained under quadratic loss function. EM algorithm is used to obtain numerical values of the Maximum likelihood method and Gibbs and the Monte Carlo Markov chain techniques are utilized for Bayesian calculations. A simulation study is performed to evaluate the performance of the estimators with respect to the mean square errors and the biases. Finally, we find the best prediction method by implementing a real data example under progressive Type-II censoring schemes.

Optimum specification limit coefficient setting based on the smaller-the-better quality characteristics

In this work, the authors propose the application of type I continuous sampling plan (CSP-1 plan) with expected inspection cost, expected replacement cost, expected warranty cost, expected quality loss for conforming product, and expected quality loss for nonconforming product per item. The quality characteristics of product is considered as the normal distribution with smaller-the-better characteristics. The quality loss function of product addresses the quadratic quality loss function. The CSP-1 plan is applied in the process control. The optimal combination of parameters is obtained from the minimum expected total cost per item produced under the specified average outgoing quality limit (AOQL) protection. One needs to have the optimal clearance number, sampling frequency, and specification limit coefficient.

Universal mean-field upper bound for the generalization gap of deep neural networks.

Modern deep neural networks (DNNs) represent a formidable challenge for theorists: according to the commonly accepted probabilistic framework that describes their performance, these architectures should overfit due to the huge number of parameters to train, but in practice they do not. Here we employ results from replica mean field theory to compute the generalization gap of machine learning models with quenched features, in the teacher-student scenario and for regression problems with quadratic loss function. Notably, this framework includes the case of DNNs where the last layer is optimized given a specific realization of the remaining weights. We show how these results-combined with ideas from statistical learning theory-provide a stringent asymptotic upper bound on the generalization gap of fully trained DNN as a function of the size of the dataset P. In particular, in the limit of large P and N_{out} (where N_{out} is the size of the last layer) and N_{out}≪P, the generalization gap approaches zero faster than 2N_{out}/P, for any choice of both architecture and teacher function. Notably, this result greatly improves existing bounds from statistical learning theory. We test our predictions on a broad range of architectures, from toy fully connected neural networks with few hidden layers to state-of-the-art deep convolutional neural networks.

Generalized Bayes estimators with closed forms for the normal mean and covariance matrices

In the estimation of the mean matrix in a multivariate normal distribution, the generalized Bayes estimators with closed forms are provided, and the sufficient conditions for their minimaxity are derived relative to both matrix and scalar quadratic loss functions. The generalized Bayes estimators of the covariance matrix are also given with closed forms, and the dominance properties are discussed for the Stein loss function.

Estimation of order restricted standard deviations of normal populations with a common mean

The present paper deals with estimating standard deviations of two normal populations with a common mean and ordered variances. We propose a class of improved estimators of the standard deviations under the quadratic loss function. Considering k normal populations with a common mean and ordered variances, we establish inadmissibility results of the affine and scale equivariant estimators of the smallest standard deviation and derive its improved estimators. Simulation studies are performed to compare the risks of the improved estimators with the existing estimators. Two illustrative examples are presented as applications of the proposed estimators.

The integrated application of acceptance sampling in the quality control of manufacturing process and quality assurance of shipment product

In this work, the authors propose the integrated application of CSP-1 plan and Dodge-Romig rectifying inspection plan with expected inspection cost, expected replacement cost, expected warranty cost, expected quality loss for conforming product, and expected quality loss for non-conforming product per item. The quality characteristics of product is considered as the normal distribution with nominal-the-best and symmetric quadratic quality loss function. The CSP-1 plan is firstly applied in the process control. The optimal combination of parameters is obtained from the minimum expected total cost per item produced under the specified average outgoing quality limit (AOQL) protection. One needs to have the optimal clearance number, sampling frequency, and specification limit coefficient. Then Dodge-Romig rectifying inspection plan is applied in the product improvement. The optimal sample size, clearance number, and quality investment are obtained by minimizing the expected total cost per lot under the specified consumer’s risk.

Optimum specification limit setting under the larger-the-better quality characteristics

In this paper, the optimal clearance number and sampling frequency of type I continuous sampling plan (CSP-1 plan) and specification limit coefficient of product are determined with the minimum expected total cost per item under the specified average outgoing quality limit (AOQL) protection. The expected total cost per item includes the expected inspection cost, the expected replacement cost, the expected warranty cost, the expected quality loss for conforming product, and the expected quality loss for non-conforming product per item. The quality characteristic of product is considered as the normal distribution with larger-the-better characteristics. The quality loss function of product addresses the quadratic quality loss function. The sensitivity analysis of numerical example will be provided for illustration.

Two-sided strategic information transmission

We investigate a cheap talk model in which a decision maker and an expert are both privately informed. Both players observe independent signals that jointly determine ideal actions for the players, and the decision maker can send a cheap talk message to the expert, which is followed by the cheap talk of the expert and then the action of the decision maker. In equilibrium, the strategy of the decision maker is not monotonic, and the revelation of the decision maker concerning her information does not necessarily result in welfare improvement of the players. In particular, in models in which optimal actions are additively and/or multiplicatively separable in the information of two players and their preferences are represented by quadratic loss functions, the information revelation of the decision maker cannot facilitate information transmitted from the expert.

Estimating positive powers of the scale parameters under order restriction for two normal populations with a common mean

ABSTRACT The problem of estimating positive powers of scale parameters has been considered under order restriction when samples are available from two normal populations with a common mean μ. Maximum-likelihood estimators and several plug-in type estimators using some popular estimators of the common mean have been proposed. Sufficient conditions for improving equivariant estimators for the positive powers of the scale parameters under order restriction have been derived. Consequently, several improved estimators have been proposed. A numerical comparison among all the proposed estimators in the special cases (c = 1 and ) has been made in terms of risk values using the quadratic loss function, and recommendations are given for the use of the estimators. Finally, a real-life example has been considered to show the potential application of the model problem.

E-Bayesian Estimation under Loss Functions in Competing Risks

Using gamma prior distribution of which shape hyperparameter has beta distributio and rate parameter has three different distributions over a finite interval, we studied the E-Bayesian estimation of one scale parameter of Gompertz distribution based on progressively type I censored sample from the competing risks model subject to K independent causes. The estimators obtainedgeneralize those issued from the quadratic loss, entropy loss and DeGroot loss functions.

Optimal Bayesian Estimation of a Regression Curve, a Conditional Density, and a Conditional Distribution

In this paper, several related estimation problems are addressed from a Bayesian point of view, and optimal estimators are obtained for each of them when some natural loss functions are considered. The problems considered are the estimation of a regression curve, a conditional distribution function, a conditional density, and even the conditional distribution itself. These problems are posed in a sufficiently general framework to cover continuous and discrete, univariate and multivariate, and parametric and nonparametric cases, without the need to use a specific prior distribution. The loss functions considered come naturally from the quadratic error loss function commonly used in estimating a real function of the unknown parameter. The cornerstone of these Bayes estimators is the posterior predictive distribution. Some examples are provided to illustrate the results.