Quadratic Loss Function Research Articles

Non-minimaxity of debiased shrinkage estimators

We consider the estimation of the p-variate normal mean of X∼Np(θ,I)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X\\sim \\mathcal {N}_p(\ heta ,I)$$\\end{document} under the quadratic loss function. We investigate the decision theoretic properties of debiased shrinkage estimator, the estimator which shrinks towards the origin for smaller ‖x‖2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert x\\Vert ^2$$\\end{document} and which is exactly equal to the unbiased estimator X for larger ‖x‖2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert x\\Vert ^2$$\\end{document}. Such debiased shrinkage estimator seems superior to the unbiased estimator X, which implies minimaxity. However, we show that it is not minimax under mild conditions.

Investigation of half-normal model using informative priors under Bayesian structure

This paper considers properties of half-normal distribution using informative priors under the Bayesian criterion. It employs the squared root inverted gamma, Chi-square and Rayleigh distributions as the prior distribution to construct the Posterior distributions of the respective distributional parameters. Hyperparameters are elicited via prior predictive distribution. The properties of posterior distribution are studied, and their graphs are presented using a real data set. A comprehensive simulation scheme is conducted using informative priors. Bayes estimates are obtained using the loss functions (squared error loss function, modified loss function, quadratic loss function and Degroot loss function). Statistical inferences interval estimates and Bayesian hypothesis testing are presented to demonstrate the usefulness of the study.

Адаптивное формирование опорного изображения для бортовой корреляционно-экстремальной системы сопровождения движущихся объектов

To increase the efficiency of the correlation-extremal tracking system for moving objects, it is proposed to use Interactive Multiple Model (IMM), which includes Kalman filters of the 0th and 1st order and a Singer filter of the 0th order. The structure of the meter was obtained as a result of statistical synthesis according to the criterion of minimum a posteriori risk with a quadratic loss function. The statistical synthesis is based on the semi-Markov brightness model, which takes into account the random nature of the transition time from one state to another.

Process machining allowance for reliability analysis of mechanical parts based on hidden quality loss

The machining allowance variation is significant for the reliability of a part during the machining process. Usually, when the machining allowance of a part increases, the machining and production cost also increase. When the machining allowance decreases, the machining surface will have defects. The parts will produce many scraps and reliability will decrease. The machining allowance of a part consists of multiple process machining allowances. To analyze the impact caused by machining allowance variation, the hidden quality loss and process machining allowance are combined through the process capability index (PCI). Then the asymmetric quadratic quality loss function (AQF) and quadratic exponential function (QEF) are used to analyze them. A prediction model of hidden quality loss of process machining allowance is proposed. On the premise that the quality characteristic value obeys normal function distribution, a numerical model is given and used to obtain process machining allowance-inherent reliability of the product. The actual case is used to compare and verify the two models.

A COMPARISON OF BAYESIAN APPROACH WITH Maximum Likelihood Method To Estimate Parameter For Rayleigh Distribution

The main objective of this study is to obtain and compare the performance of "Maximum likelihood estimator (MLE) and Bayesian estimators of the scale parameter of the Rayleigh distribution. In order to get better understanding in our Bayesian analysis we consider informative prior as well as non-informative prior using Jeffery prior information under loss functions (modified squred error loss function, Quadratic error loss function) to find the best method for estimation, which used the samples size ( 10, 20, 30, 50, 100). The comparison of the estimators, based on their mean squared errors (MSE's), we obtain that, MinMSE is the best estimator, while the performance of Bayes estimator under modified squared error loss function with non-informative prior with a value of the scale parameter ( ) is the best estimator comparing to other for all simple size.

Bayesian analysis of single server Markovian queueing model with balking under asymmetric loss functions

The aim of this article is to obtain the Bayesian estimates of traffic intensity () in an M/M/1 queueing model with balking. Balking is a behavior of impatient customers in which upon arrival, customer decides not to enter the queue and leave the system without getting served. In this article, we considered balking probability as a function of the number of customers present in the system. We obtained the Bayesian estimates of for the model under study by using various asymmetric loss functions such as modified quadratic loss function (M/QLF), entropy loss function (ELF), De’groot loss function (DLF), and linex loss function (LLF). Also, we provide some simulation results for supporting our findings for different sample sizes and parametric values. Further, a numerical application, for the better understanding of the model, will also be analyzed.

The Influence of Narrowband Interference on DME System Operation

Within the new air traffic management concept, using the Global Navigation Satellite System (GNSS), it is assumed that distance measurement equipment (DME) will be retained. The results of research carried out by several authors have confirmed that global navigation satellite systems (GNSS) can be put out of operation in cases of interference. Therefore, it is very timely to investigate the accuracy and resistance to interference of DME systems. The presented work contains the results of research in the field of assessment of the accuracy and resistance of a DME system, which works in conditions of narrowband interference, using modeling and simulation. Based on the derived model of the DME measurement signal, its parameters, and narrowband interference, algorithms for processing the measurement signals of a DME system in conditions of narrowband interference were derived. A quasi-optimal nonlinear filtering method was used in the derivation of the measurement signal-processing algorithms. A quadratic loss function was used as an optimality criterion, which allows us to obtain the results of measuring the parameters of the DME measurement signal as a minimum of the a posteriori mean error. Within this method, Gaussian approximation and large and small parameter methods were used. Simulation of the operation of the DME system confirmed that the measurement accuracy of this system depends on the stability of the frequency of the DME support generator, and also depends on the signal-to-noise ratio and the signal-to-interference ratio of the DME receiver input. Comparing the results of the DME system receiver designed by us with the parameters listed in in published works that discuss the accuracy of this system, we can conclude that its accuracy is much better. The simulation results confirmed that the potential accuracy of the distance measurement is equal to 2.2 m. However, the mentioned algorithms require substantial simplification to be used for real-time signal processing. This will be our next research direction.

Improvement of Quadratic Exponential Quality Gain–Loss Function and Optimization of Engineering Specifications

In the past, the optimization of engineering specifications primarily included the influence of quality loss terms on product quality; however, in actual production practice, compensation quantity inevitably affects the optimization quality of engineering specifications. In this paper, the quadratic exponential quality loss and gain function was first supplemented, and the quadratic exponential quality loss and gain function was constructed under the larger-the-better characteristic and the smaller-the-better characteristic; in order to accurately represent the change in compensation amount in the process of quality control, the optimization method of engineering specification under the calculation of quality loss and compensation was given in combination with the hyperbolic tangent function. Finally, with the support of the dam construction quality acceptance assessment, the engineering specification optimization model was tested. The optimal coefficient was obtained as 0.162, and the specification range was reduced from 0.744 to 0.648, achieving the optimization goal.

Simple mandates, monetary rules, and trend-inflation

AbstractA mandate framework is proposed for delegating monetary policy to an instrument-independent, but goal-dependent central bank that emphasizes simplicity in both the objectives entering the welfare criterion and those in the instrument rule. It consists of: (i) a simple quadratic loss function penalizing deviations from target variables; (ii) a welfare-optimized, Taylor-type log-linear nominal interest-rate rule with targets that match those in the loss function; (iii) a zero-lower-bound (ZLB) constraint on the nominal interest rate imposing a low unconditional probability of ZLB episodes; and (iv) a long-run inflation target. In an estimated New Keynesian model with these features, we find that for a quarterly probability of 5%, an optimal annual inflation target is close to 2%, weights for real variables in the loss function are small compared with inflation except for the real wage growth mandate and the optimized rules mimic a price-level rule.

Implicit Stochastic Gradient Descent for Training Physics-Informed Neural Networks

Physics-informed neural networks (PINNs) have effectively been demonstrated in solving forward and inverse differential equation problems, but they are still trapped in training failures when the target functions to be approximated exhibit high-frequency or multi-scale features. In this paper, we propose to employ implicit stochastic gradient descent (ISGD) method to train PINNs for improving the stability of training process. We heuristically analyze how ISGD overcome stiffness in the gradient flow dynamics of PINNs, especially for problems with multi-scale solutions. We theoretically prove that for two-layer fully connected neural networks with large hidden nodes, randomly initialized ISGD converges to a globally optimal solution for the quadratic loss function. Empirical results demonstrate that ISGD works well in practice and compares favorably to other gradient-based optimization methods such as SGD and Adam, while can also effectively address the numerical stiffness in training dynamics via gradient descent.

Using Different Loss Function to Estimate the Parameters of Birnbaum-Saunders Distribution by Bayesian Method with Application

The Birnbaum-Saunders distribution is one of the most important distributions that explain fatigue time in general, as it has many engineering and industrial applications. In our Paper we introduce three estimation methods to estimate the parameters of Birnbaum-Saunders parameters, first is the Maximum Likelihood Estimator MLE and second, is Bayes estimation with quadratic loss function BQ and last is the Bayes estimator with weighted loss function BW. simulated data were used as well as real data used which represented by the fatigue time of a concrete block under pressure before its final collapse. It was concluded that the Bayes estimator with squared loss function BQ is the best.

Bayesian estimation of a geometric distribution using informative priors based on a Type-I censoring scheme

In this paper, the geometric distribution parameter is estimated under a type-I censoring scheme by means of the Bayesian estimation approach. The Beta and Kumaraswamy informative priors, as well as five loss functions are used for this purpose. Expressions of Bayes estimators and Bayes risks are derived under the Squared Error Loss Function (SELF), the Quadratic Loss Function (QLF), the Precautionary Loss Function (PLF), the Simple Asymmetric Precautionary Loss Function (SAPLF), and the DeGroot Loss Function (DLF) using the two aforementioned priors. The prior densities are obtained through prior predictive distributions. Simulation studies are carried out to make comparisons using Bayes risks. Finally, a real-life data example is used to verify the model’s efficiency.

Overall error analysis for the training of deep neural networks via stochastic gradient descent with random initialisation

In spite of the accomplishments of deep learning based algorithms in numerous applications and very broad corresponding research interest, at the moment there is still no rigorous understanding of the reasons why such algorithms produce useful results in certain situations. A thorough mathematical analysis of deep learning based algorithms seems to be crucial in order to improve our understanding and to make their implementation more effective and efficient. In this article we provide a mathematically rigorous full error analysis of deep learning based empirical risk minimisation with quadratic loss function in the probabilistically strong sense, where the underlying deep neural networks are trained using stochastic gradient descent with random initialisation. The convergence speed we obtain suffers under the curse of dimensionality. However, it is presumably close to optimal in the generality of the framework we consider and, to the best of our knowledge, we establish the first full error analysis in the scientific literature for a deep learning based algorithm in the probabilistically strong sense as well as the first full error analysis in the scientific literature for a deep learning based algorithm where stochastic gradient descent with random initialisation is the employed optimisation method.

Estimating reciprocals of scale parameters of two exponential populations with common location and ordered scales using censored samples

The problem of estimating reciprocals of scale parameters (hazard rates) from two exponential populations with a common location and ordered scale parameters have been considered under the progressive type-II censoring scheme from a decision-theoretic viewpoint. The loss function is considered as quadratic. The maximum-likelihood estimators (MLEs) and the uniformly minimum variance unbiased estimators (UMVUEs) are derived. Sufficient conditions are derived for improving estimators in affine and scale equivariant classes. Consequently, improved estimators over the MLEs and the UMVUEs are derived. A numerical comparison among all the proposed estimators is made, and conclusions are drawn regarding their performances with respect to the quadratic loss function.

Doubly Type II Censoring of Two Stress-Strength System Reliability Estimation for Generalized Exponential-Poisson Distribution

In this paper, a Bayesian analysis is made to estimate the Reliability of two stress-strength model systems. First: the reliability of a one component strengths X under stress Y. Second, reliability of one component strength under three stresses. Where X and Y are independent generalized exponential-Poison random variables with parameters (α,λ,θ) and (β,λ,θ) . The analysis is concerned with and based on doubly type II censored samples using gamma prior under four different loss functions, namely quadratic loss function, weighted loss functions, linear and non-linear exponential loss function. The estimators are compared by mean squared error criteria due to a simulation study. We also find that the mean square error is the best performance of the estimator from that found in quadratic, weighted, linear and non-linear exponential loss functions.

Quantile regression for massive data set

Traditional statistical analysis is challenged by modern massive data sets, which have huge sample size and dimension. Quantile regression has become a popular alternative to least squares method for providing comprehensive description of the response distribution and robustness against heavy-tailed error distributions. On the other hand, non-smooth quantile loss poses a new challenge to massive data sets. To address the problem, we transform the non-differentiable quantile loss function into a convex quadratic loss function based on Expectation-maximization (EM) algorithm using an asymmetric Laplace distribution. Both simulations and real data application are conducted to illustrate the performance of the proposed methods.

An adaptive EWMA control chart based on Hampel function to monitor the process location parameter

AbstractThe classical memory Exponential Weighted Moving Average (EWMA) and Cumulative Sum (CUSUM) control charts (CCs) are important to detect small‐to‐moderate sizes of shift in the process location and/or dispersion parameters. To make the memory control chart (CC) more efficient for broad range of shift, the score functions in particular Huber and Bi‐square functions are incorporated in the classical memory CCs. These score functions help to formulate the parameters of memory CCs and are called Adaptive EWMA (AEWMA) and Adaptive CUSUM (ACUSUM). The AEWMA and ACUSUM are capable to handle various sizes (i.e., small, moderate, and large) of shift. The aim of this study is to utilize Hampel function as the alternative of the mentioned score functions, and propose a new AEWMA CC, symbolized as AEWMAHample. To develop the proposed AEWMAHampleCC, the standard EWMA statistic and Hampel function are incorporated in the classical EWMA CC structure. The proposed AEWMAHampleCC is also efficient to handle certain sizes of shift in the process location parameter. Performance evaluation procedures, like average run length is considered for a particular shift. Similarly, the performance comparison index, relative average run length, and extra quadratic loss function are estimated over a range of shift. All these performance measures are calculated from run length (RL) characteristic of the proposed control chart. An algorithm in MATLAB is developed based on Monte Carlo simulation procedure to generate the RLs. Some existing control charts including EWMA, CUSUM, MEC, MCE, , , IACCUSUM, , and are taken for comparison reason. For specific sizes of shift, the proposed AEWMAHampleCC reveals the superiority. Besides, to address the practical significance for practitioners and quality engineers, the real‐life and simulated data examples are considered.

On Statistical Modeling Using a New Version of the Flexible Weibull Model: Bayesian, Maximum Likelihood Estimates, and Distributional Properties with Applications in the Actuarial and Engineering Fields

In this article, we present a new statistical modification of the Weibull model for updating the flexibility, called the generalized Weibull-Weibull distribution. The new modification of the Weibull model is defined and studied in detail. Some mathematical and statistical functions are studied, such as the quantile function, moments, the information generating measure, the Shannon entropy and the information energy. The joint distribution functions of the two marginal univariate models via the Copula model are provided. The unknown parameters are estimated using the maximum likelihood method and Bayesian method via Monte Carlo simulations. The Bayesian approach is discussed using three different loss functions: the quadratic error loss function, the LINEX loss function, and the general entropy loss function. We perform some numerical simulations to show how interesting the theoretical results are. Finally, the practical application of the proposed model is illustrated by analyzing two applications in the actuarial and engineering fields using corporate data to show the elasticity and advantage of the proposed generalized Weibull-Weibull model. The practical applications show that proposed model is very suitable for modeling actuarial and technical data sets and other related fields.

A Bayesian look at classical estimation: The shape of the pareto distribution and its functions

The Pareto Distribution has been found to be useful in describing socio-economic, physical, and biological phenomena. This paper explores different methods of estimating the shape parameter of the Pareto distribution and two of its functions (reliability and hazard functions), assuming the scale parameter is known. In particular, this paper shows that the Bayesian approach, using both the quadratic and entropy loss functions, can provide a general method for estimating the shape parameter and these two functions. These Bayesian estimators generalize the maximum likelihood estimator (MLE) and the uniformly most powerful unbiased estimator (UMVUE). Simulation results show that the general Bayesian method can perform better than the MLE and UMVUE, especially when the sample size is small.

Optimal capital allocation for individual risk model using a Mean-Variance Principle

<p style='text-indent:20px;'>In this article, we study the problem of optimal capital allocation for the individual risk model using the Mean-Variance principle with quadratic loss function when a total amount of initial capital is granted for allocation. The determination of the optimal capital allocation strategy proceeds in two phases. First, explicit formulas for the optimal allocations are presented based on the assumption that the claim occurrence indicators are independent of the claim severities when the allocated total capital is fixed. Second, an approximating algorithm is proposed to find out the optimal value of capital used for allocation through minimizing the mean-variance loss function. As a result, the exact allocation policy can be provided by following the first phase again. Numerical examples and applications are provided to illustrate the main results and some discussions on the effect of neglecting the dependence between the claim occurrence indicators and claim severities on the optimal allocations are also given to shed light on future research.</p>