Some results on quadratic credibility premium using the balanced loss function

PurposeThis paper generalizes the quadratic framework introduced by Le Courtois (2016) and Sumpf (2018), to obtain new credibility premiums in the balanced case, i.e. under the balanced squared error loss function. More precisely, the authors construct a quadratic credibility framework under the net quadratic loss function where premiums are estimated based on the values of past observations and of past squared observations under the parametric and the non-parametric approaches, this framework is useful for the practitioner who wants to explicitly take into account higher order (cross) moments of past data.Design/methodology/approachIn the actuarial field, credibility theory is an empirical model used to calculate the premium. One of the crucial tasks of the actuary in the insurance company is to design a tariff structure that will fairly distribute the burden of claims among insureds. In this work, the authors use the weighted balanced loss function (WBLF, henceforth) to obtain new credibility premiums, and WBLF is a generalized loss function introduced by Zellner (1994) (see Gupta and Berger (1994), pp. 371-390) which appears also in Dey et al. (1999) and Farsipour and Asgharzadhe (2004).FindingsThe authors declare that there is no conflict of interest and the funding information is not applicable.Research limitations/implicationsThis work is motivated by the following: quadratic credibility premium under the balanced loss function is useful for the practitioner who wants to explicitly take into account higher order (cross) moments and new effects such as the clustering effect to finding a premium more credible and more precise, which arranges both parts: the insurer and the insured. Also, it is easy to apply for parametric and non-parametric approaches. In addition, the formulas of the parametric (Poisson–gamma case) and the non-parametric approach are simple in form and may be used to find a more flexible premium in many special cases. On the other hand, this work neglects the semi-parametric approach because it is rarely used by practitioners.Practical implicationsThere are several examples of actuarial science (credibility).Originality/valueIn this paper, the authors used the WBLF and a quadratic adjustment to obtain new credibility premiums. More precisely, the authors construct a quadratic credibility framework under the net quadratic loss function where premiums are estimated based on the values of past observations and of past squared observations under the parametric and the non-parametric approaches, this framework is useful for the practitioner who wants to explicitly take into account higher order (cross) moments of past data.

Bayesian Approaches for Poisson Distribution Parameter Estimation

The Bayesian approach, a non-classical estimation technique, is very widely used in statistical inference for real world situations. The parameter is considered to be a random variable, and knowledge of the prior distribution is used to update the parameter estimation. Herein, two Bayesian approaches for Poisson parameter estimation by deriving the posterior distribution under the squared error loss or quadratic loss functions are proposed. Their performances were compared with frequentist (maximum likelihood estimator) and Empirical Bayes approaches through Monte Carlo simulations. The mean square error was used as the test criterion for comparing the methods for point estimation; the smallest value indicates the best performing method with the estimated parameter value closest to the true parameter value. Coverage Probabilities (CPs) and average lengths (ALs) were obtained to evaluate the performances of the methods for constructing confidence intervals. The results reveal that the Bayesian approaches were excellent for point estimation when the true parameter value was small (0.5, 1 and 2). In the credible interval comparison, these methods obtained CP values close to the nominal 0.95 confidence level and the smallest ALs for large sample sizes (50 and 100), when the true parameter value was small (0.5, 1 and 2). Doi: 10.28991/esj-2021-01310 Full Text: PDF

Robust classification with feature selection using an application of the Douglas-Rachford splitting algorithm

This paper deals with supervised classification and feature selection with application in the context of high dimensional features. A classical approach leads to an optimization problem minimizing the within sum of squares in the clusters (I2 norm) with an I1 penalty in order to promote sparsity. It has been known for decades that I1 norm is more robust than I2 norm to outliers. In this paper, we deal with this issue using a new proximal splitting method for the minimization of a criterion using I2 norm both for the constraint and the loss function. Since the I1 criterion is only convex and not gradient Lipschitz, we advocate the use of a Douglas-Rachford minimization solution. We take advantage of the particular form of the cost and, using a change of variable, we provide a new efficient tailored primal Douglas-Rachford splitting algorithm which is very effective on high dimensional dataset. We also provide an efficient classifier in the projected space based on medoid modeling. Experiments on two biological datasets and a computer vision dataset show that our method significantly improves the results compared to those obtained using a quadratic loss function.

Acute Myeloid Leukemia (AML) Detection Using AlexNet Model

Acute Myeloid Leukemia (AML) is a kind of fatal blood cancer with a high death rate caused by abnormal cells’ rapid growth in the human body. The usual method to detect AML is the manual microscopic examination of the blood sample, which is tedious and time-consuming and requires a skilled medical operator for accurate detection. In this work, we proposed an AlexNet-based classification model to detect Acute Myeloid Leukemia (AML) in microscopic blood images and compared its performance with LeNet-5-based model in Precision, Recall, Accuracy, and Quadratic Loss. The experiments are conducted on a dataset of four thousand blood smear samples. The results show that AlexNet was able to identify 88.9% of images correctly with 87.4% precision and 98.58% accuracy, whereas LeNet-5 correctly identified 85.3% of images with 83.6% precision and 96.25% accuracy.

Deep learning: a statistical viewpoint

The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture that specific principles underlie these phenomena: that overparametrization allows gradient methods to find interpolating solutions, that these methods implicitly impose regularization, and that overparametrization leads to benign overfitting, that is, accurate predictions despite overfitting training data. In this article, we survey recent progress in statistical learning theory that provides examples illustrating these principles in simpler settings. We first review classical uniform convergence results and why they fall short of explaining aspects of the behaviour of deep learning methods. We give examples of implicit regularization in simple settings, where gradient methods lead to minimal norm functions that perfectly fit the training data. Then we review prediction methods that exhibit benign overfitting, focusing on regression problems with quadratic loss. For these methods, we can decompose the prediction rule into a simple component that is useful for prediction and a spiky component that is useful for overfitting but, in a favourable setting, does not harm prediction accuracy. We focus specifically on the linear regime for neural networks, where the network can be approximated by a linear model. In this regime, we demonstrate the success of gradient flow, and we consider benign overfitting with two-layer networks, giving an exact asymptotic analysis that precisely demonstrates the impact of overparametrization. We conclude by highlighting the key challenges that arise in extending these insights to realistic deep learning settings.

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.

A generalized quadratic loss for SVM and Deep Learning

We consider some supervised binary classification tasks and a regression task, whereas SVM and Deep Learning, at present, exhibitthe best generalization performances. We extend the work [3] on a gen-eralized quadratic loss for learning problems that examines pattern cor-relations in order to concentrate the learning problem into input spaceregions where patterns are more densely distributed. From a shallowmethods point of view (e.g.: SVM), since the following mathematicalderivation of problem (9) in [3] is incorrect, we restart from problem (8)in [3] and we try to solve it with one procedure that iterates over the dualvariables until the primal and dual objective functions converge. In ad-dition we propose another algorithm that tries to solve the classificationproblem directly from the primal problem formulation. We make alsouse of Multiple Kernel Learning to improve generalization performances.Moreover, we introduce for the first time a custom loss that takes in con-sideration pattern correlation for a shallow and a Deep Learning task.We propose some pattern selection criteria and the results on 4 UCIdata-sets for the SVM method. We also report the results on a largerbinary classification data-set based on Twitter, again drawn from UCI,combined with shallow Learning Neural Networks, with and without thegeneralized quadratic loss. At last, we test our loss with a Deep NeuralNetwork within a larger regression task taken from UCI. We comparethe results of our optimizers with the well known solver SVMlightandwith Keras Multi-Layers Neural Networks with standard losses and witha parameterized generalized quadratic loss, and we obtain comparable results.

OUTLIER-ROBUST KERNEL HIERARCHICAL-OPTIMIZATION RLS ON A BUDGET WITH AFFINE CONSTRAINTS

<div>This paper introduces a non-parametric learning framework to combat outliers in online, multi-output, and nonlinear regression tasks. A hierarchical-optimization problem underpins the learning task: Search in a reproducing kernel Hilbert space (RKHS) for a function that minimizes a sample average $\ell_p$-norm ($1 \leq p \leq 2$) error loss on data contaminated by noise and outliers, subject to side information that takes the form of affine constraints defined as the set of minimizers of a quadratic loss on a finite number of faithful data devoid of noise and outliers. To surmount the computational obstacles inflicted by the choice of loss and the potentially infinite dimensional RKHS, approximations of the $\ell_p$-norm loss, as well as a novel twist of the criterion of approximate linear dependency are devised to keep the computational-complexity footprint of the proposed algorithm bounded over time. Numerical tests on datasets showcase the robust behavior of the advocated framework against different types of outliers, under a low computational load, while satisfying at the same time the affine constraints, in contrast to the state-of-the-art methods which are constraint agnostic.</div><div><br></div><div>-------</div><div><br></div><div>© 20XX IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works.<br></div>