Squared Error Loss Function Research Articles

Fuzzy Estimators for Parameters and Hazard Rate Function of Lindley Distribution: Simulation Study

The quantitative description of data is very essential for statistics. In standard statistics data are assumed to be numbers, vectors or classical functions. But in applications, real data are frequently not precise numbers or vectors, but often more or less imprecise. All kinds of data which cannot be presented as precise numbers or cannot be precisely classified are called non-precise or fuzzy. The present paper focused on obtaining the maximum likelihood and approximate Bayesian estimators of the parameters and hazard rate function for Lindley distribution when the data are available in fuzzy form. Since there are no exact forms, maximum likelihood estimators have been derived according to the algorithm of Newton- Raphson as an iterative technique and Bayes estimators have been derived based on informative gamma priors relative to squared error and precautionary loss functions according to a numerical approximate Lindley‘s technique. The obtained estimators have been compared through Monte-Carlo simulation study.

A 3-Component Mixture of Exponential Distribution Assuming Doubly Censored Data: Properties and Bayesian Estimation

The output of an engineering process is the result of several inputs, which may be homogeneous or heterogeneous and to study them, we need a model which should be flexible enough to summarize efficiently the nature of such processes. As compared to simple models, mixture models of underlying lifetime distributions are intuitively more appropriate and appealing to model the heterogeneous nature of a process in survival analysis and reliability studies. Moreover, due to time and cost constraints, in the most lifetime testing experiments, censoring is an unavoidable feature. This article focuses on studying a mixture of exponential distributions, and we considered this particular distribution for three reasons. The first reason is its application in reliability modeling of electronic components and the second important reason is its skewed behavior. Similarly, the third and the most important reason is that exponential distribution has the memory-less property. In particular, we deal with the problem of estimating the parameters of a 3-component mixture of exponential distributions using type-II doubly censoring sampling scheme. The elegant closed-form expressions for the Bayes estimators and their posterior risks are derived under squared error loss function, precautionary loss function and DeGroot loss function assuming the noninformative (uniform and Jeffreys’) and the informative priors. A detailed Monte Carlo simulation and real data studies are carried out to investigate the performance (in terms of posterior risks) of the Bayes estimators. From results, it is observed that the Bayes estimates assuming the informative prior perform better than the noninformative priors.

Reliability estimation in a multicomponent stress-strength model for Burr XII distribution under progressive censoring

We consider estimation of the multicomponent stress-strength reliability under progressive Type II censoring under the assumption that stress and strength variables follow Burr XII distributions with a common shape parameter. Maximum likelihood estimates of the reliability are obtained along with asymptotic intervals when common shape parameter may be known or unknown. Bayes estimates are also derived under the squared error loss function using different approximation methods. Further, we obtain exact Bayes and uniformly minimum variance unbiased estimates of the reliability for the case common shape parameter is known. The highest posterior density intervals are also obtained. We perform Monte Carlo simulations to compare the performance of proposed estimates and present a discussion based on this study. Finally, two real data sets are analyzed for illustration purposes.

Entropy Estimation for Exponential Distribution under Multiply Type II Censored Competing Risks Data

In lifetime data analysis, generally, it is known that more than one or several risk factors (RiF) may be present at the same time. A failure item of experiment is resulted by one of the RiFs. Generally, it is also known that the lifetimes of test items may not be recorded exactly. Therefore, analysis of censored competing risks data (CRD) has been studied by many researchers. In this article, we derive the Bayes estimators (BEs) of the entropy under the ExpD (ExpD) with multiply type II censored competing risks data (Mu2CeCr). Approximate confidence interval (CI) and bootstrap CIs are also provided. The BEs of entropy for the ExpD with Mu2CeCr under the squared error loss function (SqL), precautionary loss function (PrL) and DeGroot loss function (DeL) are provided. Tiernery and Kadane (TiKa) approximate method is used to compute these estimators. To know the performance of proposed estimators of entropy based on ExpD with CRD under multiply type II censoring scheme (Mu2Ce), a numerical study is conducted.

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.

Bayesian Estimators of the parameter and Reliability Function of Inverse Rayleigh Distribution" A comparison study "

In this paper, Bayesian estimator for the parameter and reliability function of inverse Rayleigh distribution (IRD) were obtained Under three types of loss function, namely, square error loss function (SELF), Modified Square error loss function (MSELF) and Precautionary loss function (PLF),taking into consideration the informative and non- informative prior. The performance of such estimators was assessed on the basis of mean square error (MSE) criterion by performing a Monte Carlo simulation technique.

Bayesian Estimation of Marshall Olkin Extended Inverse Weibull Distribution Using MCMC Approach

In this paper, we invoke a new prospective to discuss the estimation of a three-parameter Marshall Olkin extended inverse Weibull distribution based on Markov Chain Monte Carlo (MCMC) approach. The Bayes estimators under the squared error loss and LINEX loss functions are derived for three parameters. MCMC approach is applied to compute the Bayesian estimation of the unknown parameters. Using a real data application, it is shown that the superior performance of Bayesian estimation.

Measuring Uncertainty Under Prior Information

There are various situations where prior information is available on parameters in the form of order relations. One needs to take into account this information to obtain superior estimators. Shannon modeled uncertainty mathematically and called it entropy. In this paper, we study the problem of estimating entropy of two exponential populations associated with a common scale parameter, and unknown but ordered location parameters. In particular, the unrestricted best affine equivariant estimator is shown to be inadmissible under order restricted location parameters with respect to a class of location invariant loss functions. Under some specific location invariant loss functions, various improved estimators are obtained. Applications of this problem are developed for various sampling schemes: $(i)$ i.i.d. sampling, $(ii)$ record values, $(iii)$ Type-II censoring and $(iv)$ progressive Type-II censoring. Numerically, we have compared the risk performance of the proposed estimators for the squared error and linex loss functions.

Reliability estimation for the bathtub-shaped distribution based on progressively first-failure censoring sampling

In this article, we consider estimating the parameters, reliability function R(t) and failure rate function H(t) of the two-parameter bathtub-shaped distribution introduced by Chen (2000) based on the progressive first-failure censored sample. The maximum likelihood estimators and Bayes estimators under squared error loss function are derived. We obtain the asymptotic confidence intervals for the parameters using the observed Fisher information matrix. The parametric bootstrap confidence intervals of reliability characteristics are also proposed. Lindley approximation procedure is adopted to establish Bayes estimates. Furthermore, we conduct Monte Carlo simulation to compare the behaviors of different methods. A real data set is analyzed to illustrate the proposed methods.

Robust Forecasting of River-Flow Based on Convolutional Neural Network

In this paper, a novel method is developed for day-ahead daily river-flow forecasting based on convolutional neural network (CNN). The proposed method incorporates both spatial and temporal information to improve the forecasting performance. A CNN model is usually trained by minimizing the mean squared error which is, however, sensitive to few particularly large errors. This character of squared error loss function will result in a poor estimator. To tackle the problem, a robust loss function is proposed to train the CNN. To facilitate the training of CNNs for multiple sites forecasting, transfer learning is also applied in this study. With transfer learning, a new CNN inherits the structure and partial learnable parameters from a well-trained CNN to reduce the training complexity. The forecasting performance of the proposed method is validated with real data of four rivers by comparing with widely used benchmarking models including the autoregressive model, multilayer perception network, kernel ridge regression, radial basis function neural network, and generic CNN. Numerical results show that the proposed method performs best in terms of the root mean squared error, mean absolute error, and mean absolute percentage error. The two-sample Kolmogorov-Smirnov test is further applied to assess the confidence on the conclusion.

Performance of Three-Stage Sequential Estimation of the Normal Inverse Coefficient of Variation Under Type II Error Probability: A Monte Carlo Simulation Study

This paper sheds light on the performance of the three-stage sequential estimation of the population inverse coefficient of variation of the normal distribution under a moderate sample size. We estimate the final sample size generated by the three-stage procedure, and the population mean, the population variance, the population inverse coefficient of variation, the asymptotic coverage probability, and the asymptotic regret incurred by estimating the population inverse coefficient of variation by its sample statistics under squared-error loss function plus linear sampling cost. Besides, we address the sensitivity of the constructed confidence interval to detect a potential shift that may occur in the population inverse coefficient of variation under uncontrolled and controlled optimal sample size against type II error probability. We do so by computing the characteristic operating function. Besides, we address the sensitivity of the three-stage procedure as the underlying distribution departs away from normality. We consider two classes of distributions: Student’s t distribution and beta distribution. We use Monte Carlo simulations for this study. We write FORTRAN codes and use Microsoft developer studio software. The simulation results revealed that the controlled confidence intervals provide coverage probabilities that exceed the prescribed nominal value even for small optimal sample size contrary to the uncontrolled case that attains the nominal value only asymptotically. Moreover, under the controlled case, the sensitivity of the procedure to depict a potential shift in the parameter of concern becomes more sensitive than the uncontrolled case. Finally, the three-stage procedure is non-sensitive to departure from normality for normal likewise distributions.

Estimation and Prediction for the Generalized Half Normal Distribution under Hybrid Censoring

Abstract In this article, we make estimation and prediction inferences for the generalized half normal distribution. The maximum likelihood and Bayes estimators of unknown parameters are obtained based on hybrid Type I censored samples. We obtain asymptotic intervals using the observed Fisher information matrix and also construct bootstrap intervals of unknown parameters. Bayes estimators are obtained under the squared error loss function using different approximation methods. We also construct the highest posterior density intervals of unknown parameters. Further one- and two-sample predictors and prediction intervals of censored observations are discussed. A Monte Carlo simulation study is conducted to compare the performance of the proposed methods. We further analyze a real data set for illustrative purposes. Finally, conclusions are presented.

Shifted Exponential Distribution: Bayesian Estimation, Prediction and Expected Test Time under Progressive Censoring

Abstract In this article, we present Bayesian estimation of a shifted exponential distribution assuming progressive type-II censoring with random removals. Bayes estimators and their respective posterior risks are derived under the squared error loss function and the precautionary loss function. Moreover, the credible intervals and expected experiment time along with the Bayesian predictive intervals for one- and two-sample cases are also discussed. A comparison of the expected experiment time under complete and type-II progressive censoring is also a part of this study. Finally, a real-life example is provided to illustrate the proposed methodology.

On the Bayesian Analysis of Constant-Stress Life Test Model Under Type-Ii Censoring

In this article, a constant-stress partially accelerated life test model of high reliability products and materials with type-II censored data from the linear failure rate distribution is considered. Two different methods of estimation likelihood and Bayesian are used to estimate the unknown parameters of the model. The maximum likelihood estimates of the model parameters are obtained using the Newton–Raphson technique. The posterior means and posterior variances are derived under the squared error loss function using Lindley’s approximation procedure. The advantages of this approximation are discussed. Monte Carlo simulations are made for comparing and evaluating the performance of the proposed methods of estimation.

Bayesian Estimation of Parameters of Weibull Distribution Using Linex Error Loss Function

This paper develops a Bayesian analysis of the scale parameter in the Weibull distribution with a scale parameter  θ  and shape parameter  β (known). For the prior distribution of the parameter involved, inverted Gamma distribution has been examined. Bayes estimates of the scale parameter, θ  , relative to LINEX loss function are obtained. Comparisons in terms of risk functions of those under LINEX loss and squared error loss functions with their respective alternate estimators, viz: Uniformly Minimum Variance Unbiased Estimator (U.M.V.U.E) and Bayes estimators relative to squared error loss function are made. It is found that Bayes estimators relative to squared error loss function dominate the alternative estimators in terms of risk function.

Two-Parameter Logistic-Exponential Distribution: Some New Properties and Estimation Methods

The logistic exponential (LE) distribution is the only two parameter distribution that exhibits five hazard rate shapes such as constant, increasing, decreasing, bathtub, and upside-down bathtub. Due to its practical utility, this article considers ten frequentist methods of estimation, namely, maximum likelihood, least square, weighted least square, percentiles, maximum and minimum spacing distance, and variant of the method of the minimum distances for the LE parameters. A Monte Carlo simulation study is carried out to compare the performance of these estimation methods. Furthermore, Bayesian estimation under the squared error loss function assuming gamma priors for both parameters of the LE distribution is also discussed. The posterior summaries are obtained by using a Markov chain Monte Carlo algorithm. To show the practical superiority, a real-life data is analyzed to show that the LE model performs better than the well-known two-parameter models, like, exponentiated-exponential, Nadarajah–Haghighi, Birnbaum–Saunders, Weibull, Gamma, inverse Gaussian, and log-normal.

Estimation of Parameters and Reliability Characteristics in Lindley Distribution Using Randomly Censored Data

This article deals with the estimation of parameters and reliability characteristics of Lindley distribution underrandom censoring. Expected time on test based on randomly censored data is obtained. The maximum likelihood estimators of the unknown parameters and reliability characteristics are derived. The asymptotic, bootstrap p and bootstrap t confidence intervals of the parameters are constructed. The Bayes estimators of the parameters and reliability characteristics under squared error loss function using non-informative and gamma informative priors are obtained. For computing of Bayes estimates, Lindley approximation and MCMC methods are considered. Highest posterior density (HPD) credible intervals of the parameters are obtained using MCMC method. Various estimation procedures are compared using a Monte Carlo simulation study. Finally, a real data set is analyzed for illustration purposes.

A study on the effect of the loss function on Bayesian estimation and posterior risk of binomial distribution

The effect of different loss functions on Bayesian estimation and its posterior risk is studied in this paper. The definition of quasi-modified squared error loss function is proposed based on the modified squared error loss function, and the formulas of Bayesian estimation and its posterior risk under the quasi-modified squared error loss function are developed, respectively. Moreover, the Bayesian estimations and its posterior risks of binomial distribution parameter under different loss functions are introduced. Monte Carlo simulation and application examples are provided for illustrative purpose, and the results are compared on the basis of posterior risk and mean square error (MSE). Finally, the Bayesian estimations and the Bayesian credible intervals are obtained, respectively, by MCMC method (also using OpenBUGS).

Estimation of the exponentiated half-logistic distribution based on multiply Type-I hybrid censoring

In this paper, we derive some estimators of the scale parameter of the exponentiated half-logistic distribution based on the multiply Type-I hybrid censoring scheme. We assume that the shape parameter λ is known. We obtain the maximum likelihood estimator of the scale parameter σ. The scale parameter is estimated by approximating the given likelihood function using two different Taylor series expansions since the likelihood equation is not explicitly solved. We also obtain Bayes estimators using prior distribution. To obtain the Bayes estimators, we use the squared error loss function and general entropy loss function (shape parameter q = -0.5, 1.0). We also derive interval estimation such as the asymptotic confidence interval, the credible interval, and the highest posterior density interval. Finally, we compare the proposed estimators in the sense of the mean squared error through Monte Carlo simulation. The average length of 95% intervals and the corresponding coverage probability are also obtained.

On A Shape Parameter of Gompertz Inverse Exponential Distribution Using Classical and Non Classical Methods of Estimation

The Gompertz inverse exponential distribution is a three-parameter lifetime model with greater flexibility and performance for analyzing real life data. It has one scale parameter and two shape parameters responsible for the flexibility of the distribution. Despite the importance and necessity of parameter estimation in model fitting and application, it has not been established that a particular estimation method is better for any of these three parameters of the Gompertz inverse exponential distribution. This article focuses on the development of Bayesian estimators for a shape of the Gompertz inverse exponential distribution using two non-informative prior distributions (Jeffery and Uniform) and one informative prior distribution (Gamma prior) under Square error loss function (SELF), Quadratic loss function (QLF) and Precautionary loss function (PLF). These results are compared with the maximum likelihood counterpart using Monte Carlo simulations. Our results indicate that Bayesian estimators under Quadratic loss function (QLF) with any of the three prior distributions provide the smallest mean square error for all sample sizes and different values of parameters.