Abstract
In this study we introduce a new method of adding two shape parameters to any baseline bivariate distribution function (df) to get a more flexible family of bivariate df's. Through the additional parameters we can fully control the type of the resulting family. This method is applied to yield a new two-parameter extension of the bivariate standard normal distribution, denoted by BSSN. The statistical properties of the BSSN family are studied. Moreover, via a mixture of the BSSN family and the standard bivariate logistic df, we get a more capable family, denoted by FBSSN. Theoretically, each of the marginals of the FBSSN contains all the possible types of df's with respect to the signs of skewness and excess kurtosis. In addition, each possesses very wide range of the indices of skewness and kurtosis. Finally, we compare the families BSSN and FBSSN with some important competitors (i.e., some generalized families of bivariate df's) via real data examples. AMS 2010 Subject Classification: 62-07; 62E10; 62F99.
Highlights
Multivariate data is obtained when we observe more than one statistical outcome variable at a time
The popularity of normal distribution is due to the ease of simulation and the possibility of deriving closed-form theoretical results
These tables show that the FBSSN family has the smallest values of the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) criteria in the three tables, followed by the BSSN family
Summary
Multivariate data is obtained when we observe more than one statistical outcome variable at a time. The g-and-h univariate distribution was suggested by Tukey (1977) and discussed by Hoaglin and Peters (1979) and Hoaglin (1983) This distribution is defined by transforming the standard normal variable Z to: X. where, g is a real constant controlling the skewness and h is a nonnegative real constant controlling the kurtosis, or elongation (for the definition of g-and-h distribution when h ∈ R, see Martinez and Iglewicz, 1984). In Azzalini and Dalla Valle (1996) pointed out the disadvantage of the model defined in (2.4) and suggested the following version for bivariate SN distribution, denoted by BSN2. Barakat (2015) suggested a new univariate generalized family, denoted by SSN, by taking the standard normal distribution as the baseline distribution of the stable symmetric family.
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