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- Evaluation of kurtosis into the product of two normally distributed variablesPublication . Oliveira, Amilcar; Oliveira, Teresa; Seijas-Macias, J. AntonioKurtosis (k) is any measure of the "peakedness" of a distribution of a real-valued random variable. We study the evolution of the Kurtosis for the product of two normally distributed variables. Product of two normal variables is a very common problem for some areas of study, like, physics, economics, psychology, ... Normal variables have a constant value for kurtosis (k = 3), independently of the value of the two parameters: mean and variance. In fact, the excess kurtosis is defined as k - 3 and the Normal Distribution Kurtosis is zero. The product of two normally distributed variables is a function of the parameters of the two variables and the correlation between then, and the range for kurtosis is in [0;6] for independent variables and in [0;12] when correlation between then is allowed.
- The presence of distortions in the extended skew : normal distributionPublication . Seijas-Macias, J. Antonio; Oliveira, Amilcar; Oliveira, TeresaIn the last years, a very interesting topic has arisen and became the research focus not only for many mathematicians and statisticians, as well as for all those interested in modeling issues: The Skew normal distributions’ family that represents a generalization of normal distribution. The first generalization was developed by Azzalini in 1985, which produces the skew-normal distribution, and introduces the existence of skewness into the normal distribution. Later on, the extended skew-normal distribution is defined as a generalization of skew-normal distribution. These distributions are potentially useful for the data that presenting high values of skewness and kurtosis. Applications of this type of distributions are very common in model of economic data, especially when asymmetric models are underlying the data. Definition of this type of distribution is based in four parameters: location, scale, shape and truncation. In this paper, we analyze the evolution of skewness and kurtosis of extended skew-normal distribution as a function of two parameters (shape and truncation). We focus in the value of kurtosis and skewness and the existence of arange of values where tiny modification of the parameters produces large oscillations in the values. The analysis shows that skewness and kurtosis present an instability development for greater values of truncation. Moreover, some values of kurtosis could be erroneous. Packages implemented in software R confirm the existence of a range where value of kurtosis presents a random evolution.
- Distribution function for the ratio of two normal random variablesPublication . Oliveira, Amilcar; Oliveira, Teresa; Seijas-Macias, J. AntonioThe distribution of the ratio of two normal random variables X and Y was studied from [1] (the density function) and [2] (the distribution function). The shape of its density function can be unimodal, bimodal, symmetric, asymmetric, following several type of distributions, like Dirac Distribution, Normal Distribution, Cauchy Distribution or Recinormal Distribution. In this paper we study a different approximation for this distribution Z = X /Y , as a function of four parameters: ratio of the means of the two normal variables, ratio of the standard deviations of the two normal variables, the variation coefficient of the normal variable Y , and the correlation between the two variables. A formula for the Distribution function and the density function of Z is given. In addition, using graphical procedures we established singularity points for the parameters where the approximation given for Z has a non normal shape.
- Skewness into the product of two normally distributed variables and the risk consequencesPublication . Oliveira, Amilcar; Oliveira, Teresa A.; Seijas-Macias, J. AntonioThe analysis of skewness is an essential tool for decision-making since it can be used as an indicator on risk assessment. It is well known that negative skewed distributions lead to negative outcomes, while a positive skewness usually leads to good scenarios and consequently minimizes risks. In this work the impact of skewness on risk analysis will be explored, considering data obtained from the product of two normally distributed variables. In fact, modelling this product using a normal distribution is not a correct approach once skewness in many cases is different from zero. By ignoring this, the researcher will obtain a model understating the risk of highly skewed variables and moreover, for too skewed variables most of common tests in parametric inference cannot be used. In practice, the behaviour of the skewness considering the product of two normal variables is explored as a function of the distributions parameters: mean, variance and inverse of the coefficient variation. Using a measurement error model, the consequences of skewness presence on risk analysis are evaluated by considering several simulations and visualization tools using R software.
- An approach to distribution of the product of two normal variablesPublication . Oliveira, Amilcar; Seijas-Macias, J. AntonioThe distribution of product of two normally distributed variables come from the first part of the XX Century. First works about this issue were [1] and [2] showed that under certain conditions the product could be considered as a normally distributed. A more recent approach is [3] that studied approximation to density function of the product using three methods: numerical integration, Monte Carlo simulation and analytical approximation to the result using the normal distribution. They showed as the inverse variation coefficient µ/σ increases, the distribution of the product of two independent normal variables tends towards a normal distribution. Our study is focused in Ware and Lad approaches. The objective was studying which factors have more influence in the presence of normality for the product of two independent normal variables. We have considered two factors: the inverse of the variation coefficient value µ/σ and the combined ratio (product of the two means divided by standard deviation): (µ1µ2(/σ for two normal variables with the same variance. Our results showed that for low values of the inverse of the variation coefficient (less than 1) normal distribution is not a good approximation for the product. Another one, influence of the combined ratio value is less than influence of the inverse of coefficients of variation value.
- The uniform distribution product: an approach to the (Q,r) inventory model using RPublication . Oliveira, Amilcar; Oliveira, Teresa A.; Seijas-Macias, J. AntonioIn this work the probability density function (PDF) for the product of two uniformly distributed random variables is explored under the implementation of a new procedure in R language. Based on the Rohatgi theorem for the theoretical form of the product, different possibilities for the range of values of the limits of both distributions are considered. As an application, the management of a (Q,r) inventory model with the presence of lead-time and uniform demand forecasts is considered. Solution to this model looks up to minimize the total costs through the variables Q (reorder quantity) and r (the reorder point), and not always exists an analytical solution of the problem. We show a graphical procedure for the simulation results and a more exactly analytical solution. Implementation in R is straightforward.