Skewed distributions generated by truncated cauchy kernel

The aim of the present study is to explore skewed distributions extended from the skew symmetric distributions generated by Cauchy kernel. In the last two decades, there has been a growing interest in the construction of skew symmetric distributions. Different forms of the skewed distributions have been appeared in literature for data analysis and modelling. In particular, different forms of skew Cauchy symmetric distributions have been introduced and applied in different areas from physics to economics where researchers mostly have to deal with asymmetric data with heavy tails. However, the main weakness of skew Cauchy symmetric distributions is that they do not have finite moments and suffer from limited applicability. In the present study, we will introduce and explore the skew truncated Cauchy symmetric distributions to solve the problems related to infinite moments. A random variable X has skew symmetric distribution with probability density function of the form 2f(x)G(λx) where f is a density function which is symmetric around 0 and Gis distribution function of symmetric density function around 0 and λ is the skew ness parameter. In this study we will introduce skew truncated Cauchy normal, skew trunktaed Cauchy uniform, skew truncated Cauchy logistic, skew truncated Cauchy Laplace and skew truncated Cauchy student’s t model. For all of these new models, we will provide finite moments of all orders and solve the problems related to infinite moments. We will investigate some other mathematical properties such as distribution functions and characteristic functions. We will apply them to real applications. In particular, we will consider exchange rate data of Japanese Yen to the American Dollar from 1862 to 2003. On the other hand, the main feature of skew symmetric distribution is the new parameter which controls skewness and kurtosis and provides more flexible models. In this study, we will provide the ranges of possible values of skewness and kurtosis for all these models and compare them with skewness and kurtosis of truncated Cauchy distribution. According to the results, skew models with truncated Cauchy kernel will be more flexible than truncated Cauchy distribution. The simulation studies for these new models and graphical illustrations also will be provided.

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