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The common method for estimating nonlinear regression coefficients with Gaussian errors is nonlinear least squares. It is not robust for regression with heavy-tailed errors. The nonlinear least trimmed squares method is an efficient procedure for regression with heavy-tailed errors. We are introducing nonlinear least trimmed squares for the nonlinear regression with non-Gaussian stable errors and estimate the parameters of the stable distributions based on the order statistics. The traditional least trimmed squares regression algorithm is adopted for errors with infinite variance by modifying the trimming procedure. This paper estimated the nonlinear regression coefficients with stable errors through the proposed nonlinear least trimmed squares algorithm based on a new trimming procedure based on the properties of stable order statistics moments. The proposed algorithm was applied to real and simulated datasets and evaluated using a bootstrap standard deviation estimator. [ABSTRACT FROM AUTHOR]

When assessing the reliability of a system or the viability of a material, it is necessary to consider the stress conditions of the operating system. Let X be the strength of the system and Y be the stress of the system caused by the environment. In a stress–strength model, the system’s reliability is defined as P(X>Y), and due to ignoring environmental shocks or human failure, it is usually overestimated. To underestimate reliability, we assume that when X goes to the environmental conditions, the system’s strength reduces to ZX, where Z is a random variable on (0,1). We propose a modified reliability model with Ru=P(ZX>Y) for the gamma distribution. In this model, stress Y and the scale mixture of the strength of the gamma distribution, ZX, are modeled as independent random variables. The proposed model is compared with the traditional model through a simulation study. Additionally, the asymptotic distribution is used to construct an asymptotic confidence interval for Ru. Finally, we use two real datasets to demonstrate the practical applicability of our proposed method. [ABSTRACT FROM AUTHOR]

Spatial datasets may contain extreme values and exhibit heavy tails. So, the Gaussianity assumption for the corresponding random field is not reasonable. A sub-Gaussian α-stable (SGαS) random field may be more suitable as a model for heavy-tailed spatial data. This paper focuses on geostatistical data and presents an algorithm for simulating SGαS random fields. [ABSTRACT FROM AUTHOR]

Pitman measure of closeness (PMC) is a criterion to show how much an estimator is close to its parameter with respect to another estimator. We develop a new similarity measure which uses PMC on the basis of the similarity measure for hierarchical clustering algorithm that has better performance for heavy-tailed data. [ABSTRACT FROM AUTHOR]

Classical regression approaches are not robust when errors are heavy-tailed or asymmetric. That may be due to the non-existence of the mean or variance of the error distribution. Estimation based on trimmed data, which ignored outlier or leverage points, has an old history and frequently used. This procedure chooses fixed cut-off points. In this work, we use this idea recently applied for initial estimates of regression coefficients with heavy-tailed stable errors. We propose an effective procedure to calculate the cut-off points based on the tail index and skewness parameters of errors. We use the property of the existence of some moments of stable distribution order statistics. Data are trimmed based on ordered residuals of a least square regression. However, the trimmed data's optimal number is determined based on the number of error order statistics whose variance exists. Then, we use the rest of the ordered data to estimate the regression coefficients. Based on these order statistics' joint distribution, we analytically compute the bias and variance of the introduced estimator of regression parameters that was impossible for regression with stable errors. [ABSTRACT FROM AUTHOR]

Recently, some statistical studies have been done using the shape data. One of these studies is clustering shape data, which is the main topic of this paper. We are going to study some clustering algorithms on shape data and then introduce the best algorithm based on accuracy, speed, and scalability criteria. In addition, we propose a method for representing the shape data that facilitates and speeds up the shape clustering algorithms. Although the mentioned method is not very accurate, it is fast; therefore, it is useful for datasets with a high number of landmarks or observations, which take a long time to be clustered by means of other algorithms. It should be noted that this method is not new, but in this article we apply it in shape data analysis. [ABSTRACT FROM AUTHOR]