Exponential Loss Function Research Articles

Assessing the effect of E-Bayesian inference for Poisson inverse exponential distribution parameters under different loss functions and its application

This paper present E-Bayesian and Bayesian estimators of parameters of Poisson inverse exponential distribution (PIED) under Squared error loss function (SELF), General entropy loss function (GELF) and Linear Exponential loss function (LINEX) for progressive type-II censored data with binomial removals (PT-II CBRs). The E-Bayesian and corresponding Bayesian estimators are compared in terms of their risks based on simulated samples from PIED. The proposed methodology is applied to survival time of multiple myeloma patients data.

Approximate Bayes Estimators of the Parameters of the Inverse Gaussian Distribution Under Different Loss Functions

Inverse Gaussian is a popular distribution especially in the reliability and life time modelling, and thus the estimation of its unknown parameters has received considerable interest. This paper aims to obtain the Bayes estimators for the two parameters of the inverse Gaussian distribution under varied loss functions (squared error, general entropy and linear exponential). In Bayesian procedure, we consider commonly used non-informative priors such as the vague and Jeffrey’s priors, and also propose using the extension of Jeffrey’s prior. In the case where the two parameters are unknown, the Bayes estimators cannot be obtained in the closed-form. Hence, we employ two approximation methods, namely Lindley and Tierney Kadane (TK) approximations, to attain the Bayes estimates of the parameters. In this paper. the effects of considered loss functions, priors and approximation methods on Bayesian parameter estimation are also presented. The performance of Bayes estimates is compared with the corresponding classical estimates in terms of the bias and the relative efficiency throughout an extensive simulation study. The results of the comparison show that Bayes estimators obtained by TK method under linear exponential loss function using the proposed prior outperform the other estimators for estimating the parameters of inverse Gaussian distribution most of the time. Finally, a real data set is provided to illustrate the results.

Bayesian and non-Bayesian estimation of the Lomax model based on upper record values under weighted LINEX loss function

In this article, we developed a new loss function, as the simplification of linear exponential loss function (LINEX) by weighting LINEX function. We derive a scale parameter, reliability and the hazard functions in accordance with upper record values of the Lomax distribution (LD). To study a small sample behavior performance of the proposed loss function using a Monte Carlo simulation, we make a comparison among maximum likelihood estimator, Bayesian estimator by means of LINEX loss function and Bayesian estimator using square error loss (SE) function. The consequences have shown that a modified method is the finest for valuing a scale parameter, reliability and hazard functions.

Bayesian Estimation and Prediction Based on Progressively First Failure Censored Scheme from a Mixture of Weibull and Lomax Distributions

This paper develops Bayesian estimation and prediction, for a mixture of Weibull and Lomax distributions, in the context of the new life test plan called progressive first failure censored samples. Maximum likelihood estimation and Bayes estimation, under informative and non-informative priors, are obtained using Markov Chain Monte Carlo methods, based on the symmetric square error Loss function and the asymmetric linear exponential (LINEX) and general entropy loss functions. The maximum likelihood estimates and the different Bayes estimates are compared via a Monte Carlo simulation study. Finally, Bayesian prediction intervals for future observations are obtained using a numerical example

Bayesian Method for the Generalized Exponential Model Using Fuzzy Data

This paper focuses on Bayesian inference for the parameters of the generalized exponential model under asymmetric and symmetric loss functions when the observations are described in terms of fuzzy numbers. First, a generalized likelihood function based on fuzzy data is derived. Then, considering general entropy, linear exponential and squared error loss functions, the Bayes estimates of the parameters are obtained. Since Bayes estimates could not be expressed in closed forms, Metropolis–Hasting samplers are used to compute the approximate Bayes estimates. For comparison purposes, the maximum likelihood estimates of the parameters are also computed. The proposed inferences are illustrated using three real-world examples. The numerical simulation results demonstrate the superiority of the Bayesian method over the maximum likelihood procedure.

A robust two-stage procedure for the Poisson process under the linear exponential loss function

Within the framework of Bayesian model, we consider the problem of sequentially estimating the intensity parameter of a homogeneous Poisson process with linear exponential (LINEX) loss function and fixed cost per unit time. Instead of fully sequential sampling, a two-stage sampling technique is introduced to solve the problem in this paper. The proposed two-stage procedure is robust in the sense that it does not depend on the prior. It is shown that the two-stage procedure shares the asymptotic properties with the asymptotically pointwise optimal procedures for a large class of the priors. A simulation study is conducted to compare the performances of the proposed two-stage procedure and the asymptotically pointwise optimal procedures.

Fuzzy C-Means Clustering Using Asymmetric Loss Function

In this work, a fuzzy clustering algorithm is proposed based on the asymmetric loss function instead of the usual symmetric dissimilarities. Linear Exponential (LINEX) loss function is a commonly used asymmetric loss function, which is considered in this paper. We prove that the negative likelihood of an extreme value distribution is equal to LINEX loss function and clarify some of its advantages. Using such a loss function, the so-called LINEX Fuzzy C-Means algorithm is introduced. The introduced clustering method is compared with its crisp version and Fuzzy C-Means algorithms through a few real datasets as well as some simulated datasets.

Bayesian Estimates Based On Record Values Under Weighted LINEX Loss Function

In this paper, we developed linear exponential (LINEX) loss function by emerging weights to produce weighted linear exponential (WLINEX) loss function. Then we utilized WLINEX to derive scale parameter and reliability function of the Weibull distribution based on record values when the shape parameter is known. After, we estimated scale parameter and reliability function of Weibull distribution by using maximum likelihood (ML) estimation and by several Bayes estimations. The Bayes estimates were obtained with respect to symmetric loss function (squared error loss (SEL)), asymmetric loss function (LINEX) and asymmetric loss function (WLINEX). The ML and the different Bayes estimates are compared via a Monte Carlo simulation study. The result of simulation mentioned that the proposed WLINEX loss function is promising and can be used in real environment especially at the case of underestimate where it revealed better performance than LINEX loss function for estimating scale parameter.

Bayesian inference of dynamic cumulative residual entropy from Pareto Ⅱ distribution with application to COVID-19

<abstract> <p>Dynamic cumulative residual entropy is a recent measure of uncertainty which plays a substantial role in reliability and survival studies. This article comes up with Bayesian estimation of the dynamic cumulative residual entropy of Pareto Ⅱ distribution in case of non-informative and informative priors. The Bayesian estimator and the corresponding credible interval are obtained under squared error, linear exponential (LINEX) and precautionary loss functions. The Metropolis-Hastings algorithm is employed to generate Markov chain Monte Carlo samples from the posterior distribution. A simulation study is done to implement and compare the accuracy of considered estimates in terms of their relative absolute bias, estimated risk and the width of credible intervals. Regarding the outputs of simulation study, Bayesian estimate of dynamic cumulative residual entropy under LINEX loss function is preferable than the other estimates in most of situations. Further, the estimated risks of dynamic cumulative residual entropy decrease as the value of estimated entropy decreases. Eventually, inferential procedure developed in this paper is illustrated via a real data.</p> </abstract>

Estimation of multicomponent stress-strength reliability following Weibull distribution based on upper record values

ABSTRACT Stress-strength models are of special importance in reliability literature and engineering applications. This paper deals with the estimation problem of a stress-strength model incorporating multi-component system. The system is regarded as alive only if at least out of strength components exceed the stress. The reliability of such system is obtained when strength and stress variables have Weibull distributions. Maximum likelihood estimator of and asymptotic confidence intervals are obtained based on upper record values. Bayesian estimator under squared error and linear exponential loss functions using gamma prior distributions and the corresponding credible intervals are obtained. Due to the lack of explicit forms for the Bayes estimates, the Markov Chain Monte Carlo (MCMC) method is employed. A simulation study is implemented to assess the performance of estimates. A real-life example is presented to show how the proposed model may be utilized in breaking strength of jute fibre data.

A Novel Fault Identification Method for Photovoltaic Array via Convolutional Neural Network and Residual Gated Recurrent Unit

Under the background of the large-scale construction of photovoltaic (PV) power stations, it is crucial to discover and solve module failures in time for improving the service life and maintaining the normal operation efficiency of modules. Based on analyzing the difference of I-V curves of PV arrays under different fault states, the I-V curves, temperatures and irradiances are taken as input data, and a fusion model of convolutional neural network (CNN) and residual-gated recurrent unit (Res-GRU) is proposed to identify the PV array fault. This model consists of a 1-dimensional CNN module with a 4-layer structure and a Res-GRU module. It has the advantages of end-to-end fault diagnosis, no manual feature extraction, strong anti-interference ability, and usable in the absence of irradiances and temperatures. Moreover, it can not only identify a single fault (e.g., short circuit, partial shading, abnormal aging, etc.), but also can effectively identify hybrid faults. Experimental results show that the classification accuracy of the proposed method is 98.61%, which is better than the ones of the artificial neural network (ANN), the extreme learning machine with kernel function (KELM), the fuzzy C-mean (FCM) clustering, the residual neural network (ResNet), and the stage-wise additive modeling using multi-class exponential loss function based on the classification and regression tree (SAMME-CART). In addition, in the absence of temperatures and irradiances, the classification accuracy still reaches 95.23%, which has a broad application prospect in the online fault diagnoses of PV arrays.

A Study on LINEX Loss Function with Different Estimating Methods

LINEX means linear exponential loss function which used in the analysis of statistical estimation and prediction problem which rises exponentially on one side of zero and almost linearly on the other side of zero. It is used in both overestimation and underestimation problems. Ali Shadrokh and Hassan Pazira [1] presented Shrinkage estimator in Gamma Type-II Censored Data under LINEX loss function. In that paper, they have explained how the LINEX loss function works however no practical or detail explanations were given in terms of changing the shape parameter and the error function. In this study we have explained how the LINEX loss function works through practical or detail explanations in terms of changing the shape parameter and the error function, also see how the loss function works with the data generated from gamma distribution through resampling methods to compare the performance of LINEX loss function considering the relative estimation error and usual estimation error through generating random numbers from gamma distribution like randomization method and by using bootstrapping samples. The very intention is to find out which resampling method performs well in using the LINEX loss function. Using Monte Carlo Simulations these estimators are compared. It is doing draw random number from the gamma distribution and finds the maximum likelihood estimate of θ is and using this estimator to explain the LINEX loss function ; , or , where c is the shape parameter and is any estimate of the parameter . The shape of this loss function is determined by the value of c. In the analysis we use the values of shape parameter c = -0.25, -0.50, -0.75, -1 and c = 0.25, 0.50, 0.75, 1. The same procedure is done by using bootstrapping method, and finally compared between this two methods. The relative estimation error should be used instead of the estimation error where the LINEX loss function works better in both of the cases. Between the two estimators, bootstrap method is better work because although the characteristics are same, bootstrap method is more dispersed than others.

A paradoxical argument about domination

We consider the Bayes estimator of the mean vector θ of a multivariate normal distribution under uncertain prior information when the covariance matrix Σ is unknown. We use the plug-in estimator S to obtain the approximate Bayes as well as the empirical Bayes estimators under a multiparameter linear exponential loss function. Then the risks of the proposed estimators are compared and dominating the unbiased estimator of θ with respect to Bayes risk is discussed. The empirical Bayes estimator is compared with the best of the estimators in Srivastava and Ehsanes Saleh (2005) by means of a simulation study. The empirical Bayes estimator is shown to have smaller mean absolute LINEX errors for a wide range of parameter values. The biases do not appear to differ much.

Bayesian inference on power Lindley distribution based on different loss functions

This paper focuses on Bayesian estimation of the parameters and reliability function of the power Lindley distribution by using various symmetric and asymmetric loss functions. Assuming suitable priors on the parameters, Bayes estimates are derived by using squared error, linear exponential (linex) and general entropy loss functions. Since, under these loss functions, Bayes estimates of the parameters do not have closed forms we use lindley’s approximation technique to calculate the Bayes estimates. Moreover, we obtain the Bayes estimates of the parameters using a Markov Chain Monte Carlo (MCMC) method. Simulation studies are conducted in order to evaluate the performances of the proposed estimators under the considered loss functions. Finally, analysis of a real data set is presented for illustrative purposes.

BAYESIAN ESTIMATION OF PARETO SURVIVAL MODEL WITH INFORMATIVE PRIOR ON CENSORED DATA

This research conducts a case of the cancer patients in censored data using Bayesian methodology. There are three types of loss function in Bayesian estimation method such as squared error loss function (self), linear exponential loss function (lelf) and general entropy loss function (gelf). Pareto survival model is selected as presentation data. To construct a posterior distribution, framing a likelihood function of Pareto and a prior, requires the prior distribution. An exponential distribution is chosen as a prior that describes parameter character of the Pareto. The posterior distribution is used to discover estimators in three loss functions of Bayesian methods. There are estimators held down by Bayesian self , Bayesian lelf and Bayesian gelf which substance 3.79, 3.78 and 3.90 correspondingly. After getting those estimators, the hazard functions , and and survival functions , and can be determined. The result shows that all of survival values under Bayesian approaches are lower than the real survival value. It means the result is more trusted because as a prior, the parameter is defined more precisely than before. The hazard function confirmations a same shape in all approaches. The rates of hazard are decreasing along with survival values which show the same behavior. The curves are strictly dropping after first data. This occurrence because due to a heavy-tailed character of Pareto. The result indicates that MSE of parameter estimation under the Bayesian self, lelf and gelf are 1.3x10-2, 1.2x10-2 and 0 respectively. The mse of survival estimation under the Bayesian self, lelf and gelf are 10-4, 1.1x10-4 and 3x10-5 individually. It concludes that the Bayesian gelf is the best approximation.

Optimal investment strategies and risk-sharing arrangements for a hybrid pension plan

A continuous time stochastic model is used to study a hybrid pension plan, where both the contribution and benefit levels are adjusted depending on the performance of the plan, with risk sharing between different generations. The pension fund is invested in a risk-free asset and multiple risky assets. The objective is to seek an optimal investment strategy and optimal risk-sharing arrangements for plan trustees and participants so that this proposed hybrid pension system provides adequate and stable income to retirees while adjusting contributions effectively, as well as keeping its sustainability in the long run. These goals are achieved by minimizing the expected discount disutility of intermediate adjustment for both benefits and contributions and that of terminal wealth in finite time horizon. Using the stochastic optimal control approach, closed-form solutions are derived under quadratic loss function and exponential loss function. Numerical analysis is presented to illustrate the sensitivity of the optimal strategies to parameters of the financial market and how the optimal benefit changes with respect to different risk aversions. Through numerical analysis, we find that the optimal strategies do adjust the contributions and retirement benefits according to fund performance and model objectives so the intergenerational risk sharing seem effectively achieved for this collective hybrid pension plan.

Bayesian Inference of δ = P(X < Y) for Burr Type XII Distribution Based on Progressively First Failure-Censored Samples

Let X and Y follow two independent Burr type XII distributions and δ = P ( X < Y ) . If X is the stress that is applied to a certain component and Y is the strength to sustain the stress, then δ is called the stress–strength parameter. In this study, The Bayes estimator of δ is investigated based on a progressively first failure-censored sample. Because of computation complexity and no closed form for the estimator as well as posterior distributions, the Markov Chain Monte Carlo procedure using the Metropolis–Hastings algorithm via Gibbs sampling is built to collect a random sample of δ via the joint distribution of the progressively first failure-censored sample and random parameters and the empirical distribution of this collected sample is used to estimate the posterior distribution of δ . Then, the Bayes estimates of δ using the square error, absolute error, and linear exponential error loss functions are obtained and the credible interval of δ is constructed using the empirical distribution. An intensive simulation study is conducted to investigate the performance of these three types of Bayes estimates and the coverage probabilities and average lengths of the credible interval of δ . Moreover, the performance of the Bayes estimates is compared with the maximum likelihood estimates. The Internet of Things and a numerical example about the miles-to-failure of vehicle components for reliability evaluation are provided for application purposes.

Robust AdaBoost based ensemble of one-class support vector machines

One-class support vector machine (OCSVM) is a commonly used one-class classification method for tackling novelty detection problems. Unfortunately, employing the traditional AdaBoost on it may not produce satisfying performance when there are outliers within training samples. In this paper, the conventional loss function of AdaBoost is redesigned, i.e., substituting the exponential loss function by a more robust one to enhance the robustness of the traditional AdaBoost based ensemble of OCSVMs. The proposed loss function is defined as the weighted combination of the modified exponential loss function and the squared loss function. The robust AdaBoost based on the proposed loss function is introduced by redesigning the update formulae for the weights of base classifiers and the probability distribution of training samples. The upper bounds of empirical error and generalization error for the robust AdaBoost based ensemble of OCSVMs are derived. Experimental results on the synthetic and benchmark data sets demonstrate that the proposed ensemble method is superior to its related approaches.

Instance-Based Cost-Sensitive Boosting

Many classification algorithms aim to minimize just their training error count; however, it is often desirable to minimize a more general cost metric, where distinct instances have different costs. In this paper, an instance-based cost-sensitive Bayesian consistent version of exponential loss function is proposed. Using the modified loss function, the derivation of instance-based cost-sensitive extensions of AdaBoost, RealBoost and GentleBoost are developed which are termed as ICSAdaBoost, ICSRealBoost and ICSGentleBoost, respectively. In this research, a new instance-based cost generation method is proposed instead of doing this expensive process by experts. Thus, each sample takes two cost values; a class cost and a sample cost. The first cost is equally assigned to all samples of each class while the second cost is generated according to the probability of each sample within its class probability density function. Experimental results of the proposed schemes imply 12% enhancement in terms of [Formula: see text]-measure and 13% on cost-per-sample over a variety of UCI datasets, compared to the state-of-the-art methods. The significant priority of the proposed method is supported by applying the pair of [Formula: see text]-tests to the results.

Empirical Bayes estimator of parameter, reliability and hazard rate for Kumaraswamy distribution

This paper proposes empirical Bayes estimators of parameter, reliability and hazard function for Kumaraswamy distribution under the linear exponential loss function for progressively type II censored samples with binomial removal and type II censored samples. The proposed estimators have been compared with the corresponding Bayes estimators for their simulated risks. The applicability of the proposed estimators have been illustrated through ulcer patient data.