Detecting Extreme Values with Order Statistics in Samples from Continuous Distributions

Timelike Circular Surfaces and Singularities in Minkowski 3-Space

The present paper is focused on time-like circular surfaces and singularities in Minkowski 3-space. The timelike circular surface with a constant radius could be swept out by moving a Lorentzian circle with its center while following a non-lightlike curve called the spine curve. In the present study, we have parameterized timelike circular surfaces and examined their geometric properties, such as singularities and striction curves, corresponding with those of ruled surfaces. After that, a different kind of timelike circular surface was determined and named the timelike roller coaster surface. Meanwhile, we support the results of this work with some examples.

Bivariate Step-Stress Accelerated Life Tests for the Kavya–Manoharan Exponentiated Weibull Model under Progressive Censoring with Applications

A new three-parameter survival model is proposed using the Kavya–Manoharan (KM) transformation family and the exponentiated Weibull (EW) distribution. The shapes of the pdf for the new model can be asymmetric and symmetric shapes, such as unimodal, decreasing, right-skewed and symmetric. In addition, the shapes of the hrf for the suggested model can be increasing, decreasing, constant and J-shaped. Statistical properties are obtained: quantile function, mode, moments, incomplete moments, residual life time, reversed residual life time, probability weighted moments, order statistics and entropy. We discuss the maximum likelihood estimation for the model. The relevance and flexibility of the model are demonstrated using two real datasets. The distribution is very flexible, and it outperforms many known distributions, such as the three-parameter exponentiated Weibull, the modified Weibull model, the Kavya–Manoharan Weibull, the extended Weibull, the odd Weibull inverse Topp–Leone and the extended odd Weibull inverse Nadarajah–Haghigh model. A bivariate step-stress accelerated life test based on progressive type-I censoring (PTIC) using the model is presented. This pattern is noticed when a particular number of lifetime test units are routinely eliminated from the test at the conclusion of each post-test period of time. Minimizing the asymptotic variance of the MLE of the log of the scale parameter at design stress under PTIC yields an expression for the ideal test plan under PTIC.

Robust Sparse Reduced-Rank Regression with Response Dependency

In multiple response regression, the reduced rank regression model is an effective method to reduce the number of model parameters and it takes advantage of interrelation among the response variables. To improve the prediction performance of the multiple response regression, a method for the sparse robust reduced rank regression with covariance estimation(Cov-SR4) is proposed, which can carry out variable selection, outlier detection, and covariance estimation simultaneously. The random error term of this model follows a multivariate normal distribution which is a symmetric distribution and the covariance matrix or precision matrix must be a symmetric matrix that reduces the number of parameters. Both the element-wise penalty function and row-wise penalty function can be used to handle different types of outliers. A numerical algorithm with a covariance estimation method is proposed to solve the robust sparse reduced rank regression. We compare our method with three recent reduced rank regression methods in a simulation study and real data analysis. Our method exhibits competitive performance both in prediction error and variable selection accuracy.

Binomial Distributed Data Confidence Interval Calculation: Formulas, Algorithms and Examples

When collecting experimental data, the observable may be dichotomous. Sampling (eventually with replacement) thus emulates a Bernoulli trial leading to a binomial proportion. Because the binomial distribution is discrete, the analytical evaluation of the exact confidence interval of the sampled outcome is a mathematical challenge. This paper proposes three alternative confidence interval calculation methods that are characterized and exemplified.

Experimental Design for the Lifetime Performance Index of Weibull Products Based on the Progressive Type I Interval Censored Sample

In this study, the experimental design is developed based on the testing procedure for the lifetime performance index of products following Weibull lifetime distribution under progressive type I interval censoring. This research topic is related to asymmetrical probability distributions and applications across disciplines. The asymptotic distribution of the maximum likelihood estimator of the lifetime performance index is utilized to develop the testing procedure. In order to reach the given power level, the minimum sample size is determined and tabulated. In order to minimize the total cost that occurred under progressive type I interval censoring, the sampling design is investigated to determine the minimum number of inspection intervals and equal interval lengths when the termination time of experiment is fixed or not fixed. For illustrative aims, one practical example is given for the implementation of our proposed sampling design to collect the progressive type I interval censored sample so that the users can use this sample to test if the lifetime performance index exceeds the desired target level.

An Overview of Non-Destructive Testing of Goss Texture in Grain-Oriented Magnetic Steels

Grain oriented steels are widely used for electrical machines and components, such as transformers and reactors, due to their high magnetic permeability and low power losses. These outstanding properties are due to the crystalline structure known as Goss texture, obtained by a suitable process that is well-known and in widespread use among industrial producers of ferromagnetic steel sheets. One of the most interesting research areas in this field has been the development of non-destructive methods for the quality assessment of Goss texture. In particular, the study of techniques that can be implemented in industrial processes is very interesting. Here, we provide an overview of techniques developed in the past, novel approaches recently introduced, and new perspectives. The reliability and accuracy of several methods and equipment are presented and discussed.

The Censored Beta-Skew Alpha-Power Distribution

This paper introduces a new family of distributions for modelling censored multimodal data. The model extends the widely known tobit model by introducing two parameters that control the shape and the asymmetry of the distribution. Basic properties of this new family of distributions are studied in detail and a model for censored positive data is also studied. The problem of estimating parameters is addressed by considering the maximum likelihood method. The score functions and the elements of the observed information matrix are given. Finally, three applications to real data sets are reported to illustrate the developed methodology.

Principal Component Wavelet Networks for Solving Linear Inverse Problems

In this paper we propose a novel learning-based wavelet transform and demonstrate its utility as a representation in solving a number of linear inverse problems—these are asymmetric problems, where the forward problem is easy to solve, but the inverse is difficult and often ill-posed. The wavelet decomposition is comprised of the application of an invertible 2D wavelet filter-bank comprising symmetric and anti-symmetric filters, in combination with a set of 1×1 convolution filters learnt from Principal Component Analysis (PCA). The 1×1 filters are needed to control the size of the decomposition. We show that the application of PCA across wavelet subbands in this way produces an architecture equivalent to a separable Convolutional Neural Network (CNN), with the principal components forming the 1×1 filters and the subtraction of the mean forming the bias terms. The use of an invertible filter bank and (approximately) invertible PCA allows us to create a deep autoencoder very simply, and avoids issues of overfitting. We investigate the construction and learning of such networks, and their application to linear inverse problems via the Alternating Direction of Multipliers Method (ADMM). We use our network as a drop-in replacement for traditional discrete wavelet transform, using wavelet shrinkage as the projection operator. The results show good potential on a number of inverse problems such as compressive sensing, in-painting, denoising and super-resolution, and significantly close the performance gap with Generative Adversarial Network (GAN)-based methods.

Monte Carlo Simulation of a Modified Chi Distribution Considering Asymmetry in the Generating Functions: Application to the Study of Health-Related Variables

Random variables in biology, social and health sciences commonly follow skewed distributions. Many of these variables can be represented by exGaussian functions; however, in practice, they are sometimes considered as Gaussian functions when statistical analysis is carried out. The asymmetry can play a fundamental role which can not be captured by central tendency estimators such as the mean. By means of Monte Carlo simulations, the effect of a small asymmetry in the generating functions of the chi distribution is studied. To this end, the k generating functions are taken as exGaussian functions. The limits of this approximation are tested numerically for the practical case of three health-related variables: one physical (body mass index) and two cognitive (verbal fluency and short-term memory). This work is in line with our previous works on a physics-inspired mathematical model to represent the reaction times of a group of individuals.

Bounds on the Lifetime Expectations of Series Systems with IFR Component Lifetimes

We consider series systems built of components which have independent identically distributed (iid) lifetimes with an increasing failure rate (IFR). We determine sharp upper bounds for the expectations of the system lifetimes expressed in terms of the mean, and various scale units based on absolute central moments of component lifetimes. We further establish analogous bounds under a more stringent assumption that the component lifetimes have an increasing density (ID) function. We also indicate the relationship between the IFR property of the components and the generalized cumulative residual entropy of the series system lifetime.

A new goodness of fit test in the presence of uncertain parameters

The Weibull distribution has been widely used in the areas of quality and reliability. The Anderson–Darling test has been popularly used either the data in hand follow the Weibull distribution or not. The existing Anderson–Darling test under classical statistics is applied when all the observations in quality and reliability work are determined, précised, and exact. In the areas of reliability and quality, the data may indeterminate, in-interval and fuzzy. In this case, the existing Anderson–Darling test cannot be applied for testing the assumption of the Weibull distribution. In this paper, we present the Anderson–Darling test under neutrosophic statistics. We present the methodology to fit the neutrosophic Weibull distribution on the data. We discuss the testing procedure with the help of reliability data. We present the comparisons of the proposed test with the existing Anderson–Darling the goodness of fit test under classical statistics. From the comparison, it is concluded that the proposed test is more informative than the existing Anderson–Darling test under an indeterminate environment. In addition, the proposed test gives information about the measure of indeterminacy.

Asymptotic Approximations of Ratio Moments Based on Dependent Sequences

The widely orthant dependent (WOD) sequences are very weak dependent sequences of random variables. For the weighted sums of non-negative m-WOD random variables, we provide asymptotic expressions for their appropriate inverse moments which are easy to calculate. As applications, we also obtain asymptotic expressions for the moments of random ratios. It is pointed out that our random ratios can include some models such as change-point detection. Last, some simulations are illustrated to test our results.