Carolina Marchant

Capability indices for Birnbaum–Saunders processes applied to electronic and food industries

Process capability indices (PCIs) are tools widely used by the industries to determine the quality of their products and the performance of their manufacturing processes. Classic versions of these indices were constructed for processes whose quality characteristics have a normal distribution. In practice, many of these characteristics do not follow this distribution. In such a case, the classic PCIs must be modified to take into account the non-normality. Ignoring the effect of this non-normality can lead to misinterpretation of the process capability and ill-advised business decisions. An asymmetric non-normal model that is receiving considerable attention due to its good properties is the Birnbaum–Saunders (BS) distribution. We propose, develop, implement and apply a methodology based on PCIs for BS processes considering estimation, parametric inference, bootstrap and optimization tools. This methodology is implemented in the statistical software {\tt R}. A simulation study is conducted to evaluate its performance. Real-world case studies with applications for three data sets are carried out to illustrate its potentiality. One of these data sets was already published and is associated with the electronic industry, whereas the other two are unpublished and associated with the food industry.

A nonparametric method for estimating asymmetric densities based on skewed Birnbaum–Saunders distributions applied to environmental data

In this paper, we introduce a new nonparametric kernel method for estimating asymmetric densities based on generalized skew-Birnbaum–Saunders distributions. Kernels based on these distributions have the advantage of providing flexibility in the asymmetry and kurtosis levels. In addition, the generalized skew-Birnbaum–Saunders kernel density estimators are boundary bias free and achieve the optimal rate of convergence for the mean integrated squared error of the nonnegative asymmetric kernel estimators. We carry out a data analysis consisting of two parts. First, we conduct a Monte Carlo simulation study for evaluating the performance of the proposed method. Second, we use this method for estimating the density of three real air pollutant concentration data sets. These numerical results favor the proposed nonparametric estimators.

Modeling wind energy flux by a Birnbaum–Saunders distribution with an unknown shift parameter

In this paper, we discuss a Birnbaum–Saunders distribution with an unknown shift parameter and apply it to wind energy modeling. We describe structural aspects of this distribution including properties, moments, mode and hazard and shape analyses. We also discuss estimation, goodness of fit and diagnostic methods for this distribution. A computational implementation in R language of the obtained results is provided. Finally, we apply such results to two unpublished real wind speed data from Chile, which allows us to show the characteristics of this statistical distribution and to model wind energy flux.

Generalized Birnbaum-Saunders kernel density estimators and an analysis of financial data

The kernel method is a nonparametric procedure used to estimate densities with support in R. When nonnegative data are modeled, the classical kernel density estimator presents a bias problem in the neighborhood of zero. Several methods have been developed to reduce this bias, which include the boundary kernel, data transformation and reflection methods. An alternative proposal is to use kernel estimators based on distributions with nonnegative support, as is the case of the Birnbaum-Saunders (BS), gamma, inverse Gaussian and lognormal models. Generalized BS (GBS) distributions have received considerable attention, due to their properties and their flexibility in modeling different types of data. In this paper, we propose, characterize and implement the kernel method based on GBS distributions to estimate densities with nonnegative support. In addition, we provide a simple method to choose the corresponding bandwidth. In order to evaluate the performance of these new estimators, we conduct a Monte Carlo simulation study. The obtained results are illustrated by analyzing financial real data.

A family of autoregressive conditional duration models applied to financial data

The Birnbaum-Saunders distribution is receiving considerable attention due to its good properties. One of its extensions is the class of scale-mixture Birnbaum-Saunders (SBS) distributions, which shares its good properties, but it also has further properties. The autoregressive conditional duration models are the primary family used for analyzing high-frequency financial data. We propose a methodology based on SBS autoregressive conditional duration models, which includes in-sample inference, goodness-of-fit and out-of-sample forecast techniques. We carry out a Monte Carlo study to evaluate its performance and assess its practical usefulness with real-world data of financial transactions from the New York stock exchange.

Air Contaminant Statistical Distributions with Application to PM10 in Santiago, Chile

The use of statistical distributions to predict air quality is valuable for determining the impact of air chemical contaminants on human health. Concentrations of air pollutants are treated as random variables that can be modeled by a statistical distribution that is positively skewed and starts from zero. The type of distribution selected for analyzing air pollution data and its associated parameters depend on factors such as emission source and local meteorology and topography. International environmental guideline use appropriate distributions to compute exceedance probabilities and percentiles for setting administrative targets and issuing environmental alerts. The distribution bears a relationship to the normal distribution, and there are theoretical - and physical-based mechanistic arguments that support its use when analyzing air-pollutant data. Others distribution have also been used to model air population data, such as the beta, exponential, gamma, Johnson, log-logistic, Pearson, and Weibull distribution. One model also developed from physical-mechanistic considerations that has received considerable interest in recent year is the Birnbaum-Saunders distribution. This distribution has theoretical arguments and properties similar to those of the log-normal distribution, which renders it useful for modeling air contamination data. In this review, we have addressed the range of common atmospheric contaminants and the health effects they cause. We have also reviewed the statistical distributions that have been use to model air quality, after which we have detailed the problem of air contamination in Santiago, Chile. We have illustrated a methodology that is based on the Birnbaum-Saunders distributions to analyze air contamination data from Santiago, Chile. Finally, in the conclusions, we have provided a list of synoptic statements designed to help readers understand the significance of air pollution in Chile, and in Santiago, in particular, but that can be useful to other cites and countries.

Diagnostics in multivariate generalized Birnbaum-Saunders regression models

ABSTRACT Birnbaum–Saunders (BS) models are receiving considerable attention in the literature. Multivariate regression models are a useful tool of the multivariate analysis, which takes into account the correlation between variables. Diagnostic analysis is an important aspect to be considered in the statistical modeling. In this paper, we formulate multivariate generalized BS regression models and carry out a diagnostic analysis for these models. We consider the Mahalanobis distance as a global influence measure to detect multivariate outliers and use it for evaluating the adequacy of the distributional assumption. We also consider the local influence approach and study how a perturbation may impact on the estimation of model parameters. We implement the obtained results in the R software, which are illustrated with real-world multivariate data to show their potential applications.

A Multivariate Log-Linear Model for Birnbaum-Saunders Distributions

Univariate Birnbaum-Saunders models have been widely applied to fatigue studies. Calculation of fatigue life is of great importance in determining the reliability of materials. We propose and derive new multivariate generalized Birnbaum-Saunders regression models. We use the maximum likelihood method and the EM algorithm to estimate their parameters. We carry out a simulation study to evaluate the performance of the corresponding maximum likelihood estimators. We illustrate the new models with real-world multivariate fatigue data.

Robust multivariate control charts based on Birnbaum–Saunders distributions

ABSTRACT Multivariate control charts are powerful and simple visual tools for monitoring the quality of a process. This multivariate monitoring is carried out by considering simultaneously several correlated quality characteristics and by determining whether these characteristics are in control or out of control. In this paper, we propose a robust methodology using multivariate quality control charts for subgroups based on generalized Birnbaum–Saunders distributions and an adapted Hotelling statistic. This methodology is constructed for Phases I and II of control charts. We estimate the corresponding parameters with the maximum likelihood method and use parametric bootstrapping to obtain the distribution of the adapted Hotelling statistic. In addition, we consider the Mahalanobis distance to detect multivariate outliers and use it to assess the adequacy of the distributional assumption. A Monte Carlo simulation study is conducted to evaluate the proposed methodology and to compare it with a standard methodology. This study reports the good performance of our methodology. An illustration with real-world air quality data of Santiago, Chile, is provided. This illustration shows that the methodology is useful for alerting early episodes of extreme air pollution, thus preventing adverse effects on human health.

Multivariate Generalized Birnbaum-Saunders Models Applied to Case Studies in Bio-Engineering and Industry

Birnbaum-Saunders models are receiving considerable attention in the literature. Multivariate regression models are a useful tool in the multivariate analysis, which takes into account the correlation between variables. Diagnostic analysis is an important aspect to be considered in the statistical modeling. In this work, we formulate a statistical methodology based on multivariate generalized Birnbaum-Saunders regression models and their diagnostics. We implement the obtained results in the R software, which are illustrated with two real-world multivariate data sets related to case studies in bio-engineering and industry to show their potential applications.