Quadratic Loss Function Research Articles

Bayesian Estimation of Stress-Strength P(T < X < Z) for Dagum Distribution

In this paper, the reliability formula is derived for the stress-strength model of the probability P(T < X < Z) for a component’s strength X falling between two stresses T and Z, based on Dagum Distribution with unknown parameter β and known and common parameters λ and δ. Bayesian estimation is discussed to estimate the reliability under complete data by using Gamma prior based on two loss function (weighted and quadratic loss functions), and the comparison between these estimators based on a simulation study using mean square error criteria (MSE) for each of the small, medium and large samples. The most important conclusion is that this comparison confirms that the performance of the estimator according to the weighted loss function works better for the most experiments studied.

Preference of Prior for Two-Component Mixture of Lomax Distribution

Recently, El-Sherpieny et al., (2020), suggested Type-II hybrid censoring method for parametric estimation of Lomax distribution (LD) without due regard being given to the choice of priors and posterior risk associated with the model. This paper fills this gap and derived the new LD model with minimum posterior risk for the selection of priors. It derives a closed form expression for Bayes estimates and posterior risks using square error loss function (SELF), weighted loss function (WLF), quadratic loss function (QLF) and DeGroot loss function (DLF). Prior predictive approach is used to elicit the hyperparameters of mixture model. Analysis of Bayes estimates and posterior risks is presented in terms of sample size (n), mixing proportion (p) and censoring rate (t0), with the help of simulation study. Usefulness of the model is demonstrated on applying it to simulated and real-life data which show promising results in terms of better estimation and risk reduction.

Efficient adaptive CUSUM control charts based on generalized likelihood ratio test to monitor process dispersion shift

AbstractIn practice, a shift in the process parameters (location and/or dispersion) is unknown in prior and cannot be diagnosed precisely with the classical cumulative sum (CUSUM) control chart. To overcome this issue, this study proposed two adaptive CUSUM (ACUSUM) control charts. The proposed control charts utilized linear weighted function that is inspired by generalized likelihood ration test (GLRT) to monitor small and certain range of shift in the process dispersion. In more details, the proposed control charts methodologies are based on GLRT, exponentially weight moving average statistic, and score functions. To obtain the run length of the proposed control charts for performance assessment, algorithms are designed in MATLAB based on Monte Carlo simulation technique. Further, average run length (ARL) is used as a performance measure tool to compare the control charts performance for a single shift. For certain range of shift, extra quadratic loss function, relative ARL, and performance comparison index performance measures based on ARL are calculated. Some existing control charts are used for comparison purpose. The proposed control charts show outstanding capability to detect out‐of‐control signal against these control charts. Moreover, real‐life data of inside diameter of cylinder bore in an engine block are used to reveal the practicality and worth of the proposed control charts relative to other control charts.

The virtual loss function in the summary perception of motion and its limited adjustability

Humans can grasp the “average” feature of a visual ensemble quickly and effortlessly. However, it is largely unknown what is the exact form of the summary statistic humans perceive and it is even less known whether this form can be changed by feedback. Here we borrow the concept of loss function to characterize how the summary perception is related to the distribution of feature values in the ensemble, assuming that the summary statistic minimizes a virtual expected loss associated with its deviation from individual feature values. In two experiments, we investigated a random-dot motion estimation task to infer the virtual loss function implicit in ensemble perception and see whether it can be changed by feedback. On each trial, participants reported the average moving direction of an ensemble of moving dots whose distribution of moving directions was skewed. In Experiment 1, where no feedback was available, participants’ estimates fell between the mean and the mode of the distribution and were closer to the mean. In particular, the deviation from the mean and toward the mode increased almost linearly with the mode-to-mean distance. The pattern was best modeled by an inverse Gaussian loss function, which punishes large errors less heavily than the quadratic loss function does. In Experiment 2, we tested whether this virtual loss function can be altered by feedback. Two groups of participants either received the mode or the mean as the correct answer. After extensive training up to five days, both groups’ estimates moved slightly towards the mode. That is, feedback had no specific influence on participants’ virtual loss function. To conclude, the virtual loss function in the summary perception of motion is close to inverse Gaussian, and it can hardly be changed by feedback.

Classification Performance of Thresholding Methods in the Mahalanobis–Taguchi System

The Mahalanobis–Taguchi System (MTS) is a pattern recognition tool employing Mahalanobis Distance (MD) and Taguchi Robust Engineering philosophy to explore and exploit data in multidimensional systems. The MD metric provides a measurement scale to classify classes of samples (Abnormal vs. Normal) and gives an approach to measuring the level of severity between classes. An accurate classification result depends on a threshold value or a cut-off MD value that can effectively separate the two classes. Obtaining a reliable threshold value is very crucial. An inaccurate threshold value could lead to misclassification and eventually resulting in a misjudgment decision which in some cases caused fatal consequences. Thus, this paper compares the performance of the four most common thresholding methods reported in the literature in minimizing the misclassification problem of the MTS namely the Type I–Type II error method, the Probabilistic thresholding method, Receiver Operating Characteristics (ROC) curve method and the Box–Cox transformation method. The motivation of this work is to find the most appropriate thresholding method to be utilized in MTS methodology among the four common methods. The traditional way to obtain a threshold value in MTS is using Taguchi’s Quadratic Loss Function in which the threshold is obtained by minimizing the costs associated with misclassification decision. However, obtaining cost-related data is not easy since monetary related information is considered confidential in many cases. In this study, a total of 20 different datasets were used to evaluate the classification performances of the four different thresholding methods based on classification accuracy. The result indicates that none of the four thresholding methods outperformed one over the others in (if it is not for all) most of the datasets. Nevertheless, the study recommends the use of the Type I–Type II error method due to its less computational complexity as compared to the other three thresholding methods.

COVID-19 mortality analysis from soft-data multivariate curve regression and machine learning.

A multiple objective space-time forecasting approach is presented involving cyclical curve log-regression, and multivariate time series spatial residual correlation analysis. Specifically, the mean quadratic loss function is minimized in the framework of trigonometric regression. While, in our subsequent spatial residual correlation analysis, maximization of the likelihood allows us to compute the posterior mode in a Bayesian multivariate time series soft-data framework. The presented approach is applied to the analysis of COVID-19 mortality in the first wave affecting the Spanish Communities, since March 8, 2020 until May 13, 2020. An empirical comparative study with Machine Learning (ML) regression, based on random k-fold cross-validation, and bootstrapping confidence interval and probability density estimation, is carried out. This empirical analysis also investigates the performance of ML regression models in a hard- and soft-data frameworks. The results could be extrapolated to other counts, countries, and posterior COVID-19 waves.Supplementary InformationThe online version contains supplementary material available at 10.1007/s00477-021-02021-0.

Robust Synthetic Control Charting

charting is based on the normality assumption. This assumption, however, is hard to attain in practice. Therefore, it is important to examine how the chart would response under some types of non-normal data. The focus of this article is to monitor location shifts using a synthetic chart and to validate its performance under the g-and-h distributions. This study shows that the effect of non-normality on the standard synthetic chart is not trivial, especially when the underlying distributions are heavy-tailed. With these types of distribution, Phase II monitoring of location using median-based estimators is advisable. In doing so, the synthetic chart is more robust to departure in the normality assumption with little effect on its out-of-control performance. This paper shows how the synthetic parameters should be attained to reflect the use of the modified one-step M-estimator (MOM) in its Winsorized version, and the median for Phase II. The assessment is based on the average run length and supported by the extra quadratic loss function. Finally, the practical application of the proposed synthetic charts is illustrated using real data.

Sustainability of pension schemes

We build a &#8220;smooth&#8221; automatic balancing mecanism (S-ABM) which would result from an optimal tradeoff between increasing the receipts and reducing the expenditures of a pension scheme. The S-ABM obtains from minimizing a sum of discounted quadratic loss function under the constraint of an intertemporal budget balance. One advantage of this model of &#8220;optimal&#8221; adjustment is its ability to analyse various configurations in terms of ABMs by controlling the adjustment pace. Notably, this S-ABM permits to specify two particular cases we respectively denote &#8220;flat Swedish-type ABM&#8221; and &#8220;fiscal-cliff US-type ABM&#8221;. They are obtained by assuming very high adjustment costs on revenue (implying only pension benefit adjustment) and by choosing specific sequences of social time preference rate. We apply this ABM to the case of the United States Social Security to evaluate the potential adjustments necessary to ensure financial sustainability. These assessments are made under various assumptions about forecast time horizon, social time preference and weighting of social costs associated with increased receipts and/or lower expenditures.

Optimal Policy with Occasionally Binding Constraints: Piecewise Linear Solution Methods

This paper develops a piecewise linear toolkit for optimal policy analysis of linear rational expectations models, subject to occasionally binding constraints on (multiple) policy instruments and other variables. Optimal policy minimises a quadratic loss function under either commitment or discretion. The toolkit accounts for the presence of ‘anticipated disturbances’ to the model equations, allowing optimal policy analysis around scenarios or forecasts that are not produced using the model itself (for example, judgement-based forecasts such as those often produced by central banks). The flexibility and applicability of the toolkit to very large models is demonstrated in a variety of applications, including optimal policy experiments using a version of the Federal Reserve Board’s FRB/US model.

SURVFIT: Doubly sparse rule learning for survival data

Survival data analysis has been leveraged in medical research to study disease morbidity and mortality, and to discover significant bio-markers affecting them. A crucial objective in studying high dimensional medical data is the development of inherently interpretable models that can efficiently capture sparse underlying signals while retaining a high predictive accuracy. Recently developed rule ensemble models have been shown to effectively accomplish this objective; however, they are computationally expensive when applied to survival data and do not account for sparsity in the number of variables included in the generated rules. To address these gaps, we present SURVFIT, a “doubly sparse” rule extraction formulation for survival data. This doubly sparse method can induce sparsity both in the number of rules and in the number of variables involved in the rules. Our method has the computational efficiency needed to realistically solve the problem of rule-extraction from survival data if we consider both rule sparsity and variable sparsity, by adopting a quadratic loss function with an overlapping group regularization. Further, a systematic rule evaluation framework that includes statistical testing, decomposition analysis and sensitivity analysis is provided. We demonstrate the utility of SURVFIT via experiments carried out on a synthetic dataset and a sepsis survival dataset from MIMIC-III.

Optimum profit model based on the average outgoing quality limit protection

The optimal setting of process mean is an important issue in operations management and a variable sampling inspection plan is an acceptance sampling technique for quality assurance. In the present study, the newsvendor problem (i.e., the mixed procurement system) is modified by applying a variable sampling inspection plan with specified average outgoing quality limit (AOQL) with maximization of the expected total profit including the manufacturer and the retailer. The optimal retailer’s order quantity and manufacturer’s process mean and the parameters of sampling plan are jointly determined by maximizing the expected total profit. In the model, the single sampling rectifying inspection plan is adopted to determine the quality of lot product and the used cost of customer for product is measured by Taguchi’s quadratic loss function. A numerical example is given and the sensitivity analysis of model parameters are provided for illustration. Based on the results of sensitivity analysis, it may be seen that a higher processing cost per unit generally leads to a lower expected profit. In the meantime, a higher retailer’s selling price always results in a higher expected profit.

Bayesian selector of adaptive bandwidth for multivariate gamma kernel estimator on [0,∞ ) d

ABSTRACT Bayesian bandwidth selections in multivariate associated kernel estimation of probability density functions are known to improve classical methods such as cross-validation techniques in terms of execution time and smoothing quality. The paper focuses on a basic multivariate gamma kernel which is appropriated to estimate densities with support . For this purpose, we consider a Bayesian adaptive estimation of the bandwidths vector under the usual quadratic loss function. The exact expression of the posterior distribution and the vector of bandwidths are obtained. Simulations studies highlight the excellent performance of the proposed approach, comparing to the global cross-validation bandwidth selection, and under integrated squared errors. Two bivariate and trivariate applications to the Old Faithful geyser data and new ones on drinking water pumps in the Sahel, respectively, are made.

Integrated economic design of quality control and maintenance management: Implications for managing manufacturing process

This paper develops an integrated economic model for the joint optimization of quality control parameters and a preventive maintenance policy using the cumulative sum (CUSUM) control chart and variable sampling interval fixed time sampling policy. To determine the in-control and out-of-control cost for both mean and variance, a Taguchi quadratic loss function and modified linear loss function are used, respectively. Imperfect preventive maintenance and minimal corrective maintenance policies were considered in developing the model, which determines the optimal values for significant process parameters to minimize the total expected cost per unit of time. A numerical example is used to test the model, which is followed by a sensitivity analysis. The integration of CUSUM mean and variance charts with the maintenance actions are proven successful to detect the slightest shift of the process. The findings reveal that among all the cost components, process failure due to external causes and equipment breakdown has a noteworthy attribute to the total costs of the optimized model. It is expected that top managers can utilize the suggested combined model to minimize the costs related to quality loss and maintenance policy and achieve economical advantages as well.

A note on estimation of a shape parameter in a Pareto distribution

Considering an estimation problem of a shape parameter in a Pareto distribution under a restriction of a scale parameter, an estimator dominating all the previous estimators is proposed. Especially, under the quadratic loss function, the proposed estimator is shown to be second-order admissible in the class of all modified maximum likelihood estimators.

Model averaging prediction for nonparametric varying-coefficient models with B-spline smoothing

Model averaging has been demonstrated as a powerful tool in statistical prediction over the past decade. However, a majority of related works focus on the parametric model averaging. In this paper, we propose a model averaging estimation under nonparametric varying-coefficient models. Differing from existing works, our proposal concentrates on the development of the B-spline approximation to nonparametric varying coefficient functions for model average estimator, rendering the computational burden more cheaply than the kernel smoothing based estimator. Furthermore, our procedure is asymptotically optimal under mild conditions. The asymptotic optimality established in current paper is in terms of conditional quadratic loss function when the variance of model error is known or unknown, respectively. Three different cases of candidate models are considered. Extensive simulations are carried out to evaluate the finite-sample performance of our estimator. A real data is analyzed for illustration as well.

Deep learning-based detection and prediction of trending topics from streaming data

Detecting and predicting trending topics from steaming social data has always been the point of active research area in business and research firms to take quick decisions, change marketing strategies and set new goals. Topic modelling is one of the excellent methods to analyse the contents from large collection of documents in an unsupervised manner and it is a popular method used in natural language processing, information retrieval, text processing and many other research domains. In this paper, deep learning-based topic modelling technique has been proposed to detect and predict the trending topics from streaming data. The online version of latent semantic analysis with regularisation constraints has been designed using long short-term memory network. Specifically, a problem of detecting the topics from streaming media is handled as the minimisation of quadratic loss function constrained by l1 and l2 regularisation. The online learning mechanism supports scalable topic modelling. For topic prediction, sequence-to-sequence long short-term memory network has been designed. Experimentally, significant results have been achieved in terms of query retrieval performance and topic relevance metrics for topic detection on our published dataset. For topic prediction, the results obtained in terms of root mean squared error are also significant.

Bayesian Estimation of Transmuted Weibull Distribution under Different Loss Functions

In this article, we aim to estimate the parameters of the transmuted Weibull distribution (TWD) using Bayesian approach, as the Weibull distribution plays an important role in reliability engineering and life testing problems. Informative and non-informative priors under squared error loss function (SELF), precautionary loss function (PLF) and quadratic loss function (QLF) are assumed to estimate the scale, the shape and the transmuted parameter of the TWD. In addition to this, we also compute the Bayesian credible intervals (BCIs). To estimate parameters, we adopt Markov Chain Monte Carlo (MCMC) technique assuming uncensored and censored environments in terms of different sample sizes and censoring rates. The posterior risks, associated with each estimator are used to compare the performance of different estimators. Two real data sets are analyzed to illustrate the flexibility of the proposed distribution.

Estimating the scale parameter of an exponential distribution under progressive type II censoring

We consider estimation of the scale parameter of an exponential distribution with unknown location under an arbitrary strictly convex loss function when samples are progressive type II censored. Stein-type procedures, improving upon the minimum risk equivariant (MRE) estimator, are obtained and illustrated upon using quadratic and entropy loss functions. We also study a class of improving estimators using the integral expression of risk difference (IERD) method and useful consequences are discussed. Some known procedures are shown to belong to the proposed class of estimators.

Bayesian Analysis of the Weibull Paired Comparison Model Using Numerical Approximation

The method of paired comparisons (PC) is widely used to rank items using sensory evaluations. The PC models are developed to provide basis for such comparisons. In this study, the Weibull PC model is analyzed under the Bayesian paradigm using noninformative priors and different loss functions, namely, Squared Error Loss Function (SELF), Quadratic Loss Function (QLF), DeGroot Loss Function (DLF), and Precautionary Loss Function (PLF). Numerical approximation is used to illustrate the entire estimation procedure. A real dataset showing usage preferences for different cellphone brands, Huawei (HW), Samsung (SS), Oppo (OP), QMobile (QM), and Nokia (NK), is used. Quadrature method is used to evaluate the Bayes estimates, their posterior risks, preference probabilities, predictive probabilities, and posterior probabilities to establish and verify ranking order of the competing cellphone brands under study. The results show that the paired comparison model under the study using Bayesian approach involving various loss functions can offer mathematical approach to evaluate cellphone brand preferences. The ranking provided by the model is justifiable according to the usage preference for these cellphone brands. The ranking given by the model indicates that cellphone brand Samsung is preferred the most and QMobile is the least preferred. The plausibility of the model is also assessed using the Chi square test of goodness of fit.

Mathematical Model of Object Classifier based on Bayesian Approach

The paper claims that the primary importance in solving the classification problem is to find the conditions for dividing the General complexity into classes, determine the quality of such a bundle, and verify the classifier model. We consider a mathematical model of a non-randomized classifier of features obtained without a teacher, when the number of classes is not set a priori, but only its upper bound is set. The mathematical model is presented in the form of a statement of a minimax conditional extreme task, and it is a problem of searching for the matrix of belonging of objects to a class, and representative (reference) elements within each class. The development of the feature classifier is based on the synthesis of two-dimensional probability density in the coordinate space: classes-objects. Using generalized functions, the probabilistic problem of finding the minimum Bayesian risk is reduced to a deterministic problem on a set of non-randomized classifiers. At the same time, the use of specially introduced constraints fixes non-randomized decision rules and plunges the integer problem of nonlinear programming into a General continuous nonlinear problem. For correct synthesis of the classifier, the dispersion curve of the isotropic sample is necessary. It is necessary to use the total intra-class and inter-class variance to characterize the quality of classification. The classification problem can be interpreted as a particular problem of the theory of catastrophes. Under the conditions of limited initial data, a minimax functional was found that reflects the quality of classification for a quadratic loss function. The developed mathematical model is classified as an integer nonlinear programming problem. The model is given using polynomial constraints to the form of a General problem of nonlinear continuous programming. The necessary conditions for the bundle into classes are found. These conditions can be used as sufficient when testing the hypothesis about the existence of classes.