International Mathematical Forum, Vol. 11, 2016, no. 20, 975 - 988
HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2016.6791
A Skewed Truncated Cauchy Logistic
Distribution and its Moments
Zahra Nazemi Ashani
Department of mathematics, Universiti Putra Malaysia, Malaysia
Mohd Rizam Abu Bakar
Institute for mathematical research, Universiti Putra Malaysia, Malaysia
Copyright © 2016 Zahra Nazemi Ashani and Mohd Rizam Abu Bakar. This article is
distributed under the Creative Commons Attribution License, which permits unrestricted use,
distribution, and reproduction in any medium provided the original work is properly cited.
Abstract
Recently, a high number of researches have been done in the field of skew
distributions. Using the symmetric distributions for modeling the asymmetric data
is not appropriate and may be caused loss of information. Base on this fact,
studies for asymmetric distributions have been as important as symmetric
distributions since the past two decades. Between skew distributions, we focus on
skew distributions with Cauchy kernel because of lack of finite moments. We
follow the paper of Nadarajah and Kotz (2007) and introduce skew truncated
Cauchy logistic distribution with the pdf of the form , where is
density function of truncated Cauchy distribution and is considered to be
logistic distribution function. This distribution could be a better model than skew
Cauchy logistic distribution because it has finite moments of all orders.
Cumulative distribution function and the finite moments of all orders are
computed. The simulation study also is provided. Finally, with considering the
data of exchange rate from Japanese Yen to American Dollar from 1862 to 2003,
we illustrate the application of this model in the economics. In addition, the
flexibility of the model is illustrated by the range of skewness and kurtosis for
. The figures of the model for different values of also are provided.
Keywords: kurtosis, logistic distribution, skewness, skew symmetric distribution,
truncated Cauchy distribution
976 Zahra Nazemi Ashani and Mohd Rizam Abu Bakar
1 Introduction
Many statistical researchers have desired to introduce flexible models since
coming to existence of statistics science. They always have been trying to
eliminate unnecessary assumptions in the process of data analyzing. In the case of
continuous observations, sometimes data are not symmetric and have a slight
skewness. In such situations, it is better to use mixed distributions instead of
symmetric distributions in a way that symmetric distributions are a special case of
mixed distributions. The family of distributions which includes the skew and
symmetric distributions together is called skew symmetric distributions. The
construction of skew symmetric distributions is based on the lemma which
specified as follows:
Lemma1: If be pdf of a symmetric distribution around and is a cdf of a
continuous symmetric distribution around , then
is a probability density function. As a matter of fact, the story of skew symmetric
distributions commenced with the paper of Azzalini (1985). Azzalini (1985) used
the lemma and obtained skew normal distribution as follow:
where and are the pdf and cdf of standard normal distribution. In addition
some mathematical properties such as characteristic function and moments of all
orders were provided. Gupta et al. (2002) obtained some skew symmetric
distributions so that and belonged to the same family. So they introduced
skew students’t, Cauchy, Laplace, logistic and uniform probability density
function. Mukhopadhyay & Vidakovic (1995) proposed the method that and
could be come from different families of probability density functions. Therefore,
researchers could introduce many different univariate and multivariate skew
symmetric distributions. For example, Nadarajah & Kotz (2003-2009) obtained
skew symmetric distributions with the normal, students’t, logistic, Cauchy,
Laplace and uniform kernel. They also provided some mathematical properties
such as characteristic function and moments of all orders except for skew
distributions with Cauchy kernel. Nadarajah & Kotz (2005) introduced skew
symmetric distributions with the Cauchy kernel so that, according to the lemma,
was replaced with Cauchy density function and considered to be normal,
A skewed truncated Cauchy logistic distribution and its moments 977
students’t, Cauchy, Laplace, logistic and uniform distribution functions. They
provided characteristic function for these models however, they were not able to
obtain finite moments of all orders. Actually, skew Cauchy symmetric
distributions have the same problem as Cauchy distribution. Therefore, they tried
to solve the problem in later stages of their research. At the first step, Nadarajah &
Kotz (2006) partially solved this problem with obtaining the finite moments of all
orders for the truncated Cauchy distribution with the following structure
where and . Finally, Nadarajah & Kotz
(2007) solved the problem for skew Cauchy distribution by introducing skew
truncated Cauchy distribution and providing finite moments of all orders.
The applications of skew Cauchy symmetric distributions remain fairly limited
because of the lack of finite moments. In fact, skew Cauchy symmetric
distributions are important distributions for illustrating different phenomena in a
wide range of fields from physics to economy where researchers deal with
asymmetric data with heavy tails. In this paper, we follow the paper of Nadarajah
& Kotz (2005) and consider skew Cauchy logistic distribution. A random variable
is said to have a skew Cauchy logistic distribution if its pdf is ,
where denote to the pdf of Cauchy distribution and is the cdf of logistic
distribution. This distribution, despite of its simplicity, seems not to have been
discussed in detail because of lack of finite moments. In this paper, we try to solve
the problem with introducing skew truncated Cauchy logistic distribution specified
by
where 0 . Using Taylor series expansion for
, then
978 Zahra Nazemi Ashani and Mohd Rizam Abu Bakar
Actually, we replaced f with pdf of truncated Cauchy distribution and G with
distribution function of logistic distribution. When , it reduces to the
truncated Cauchy distribution. We consider without loss of generality that .
Using the fact we have the same result for The
objective of this paper is to study skew truncated Cauchy logistic distribution and
provide finite moments of all orders. Therefore, the rest of paper organized as
follows. In section 2 we study the basic of skew truncated Cauchy logistic
distribution more accurate and provide cumulative distribution function. In section
3 we identify the finite moments of all orders for skew truncated Cauchy logistic
distribution. In section 4, we do the simulation study and plot p-p plot for
simulated data. At the end, in section 5, we present the application of this new
model in economy using the maximum likelihood method and exchange rate data
from Japanese Yen to United State Dollars from 1862 to 2003. Also, for
illustrating the flexibility of skew truncated Cauchy logistic distribution, we
obtain ranges of skewness and kurtosis for . Furthermore, the figures
of the pdf for different values of are provided.
For calculation, we need to use the following equation (Equation (3.194.5)
Gradshteyn & Ryzhik, 2000):
For ,
where
and also we use
A skewed truncated Cauchy logistic distribution and its moments 979
2 The basic of skew truncated Cauchy logistic distribution
The density function of Cauchy distribution is as follows:
This density function is symmetric around and has a fatter tails compared to the
normal distribution. Cauchy distribution is becoming as popular as a normal
distribution. It is used in different areas such as biological analysis, stochastic
modeling of decreasing failure rate life components, reliability and extreme risk
analysis. This is because of function’s tails. In fact, this distribution is more
realistic in the real world applications. However, the main weakness of Cauchy
distribution is the fact that it does not have finite moments. For example, for
computing the expectation value of Cauchy distribution when and ,
we have
It is clear that this integral is not completely convergence. Johnson & Kotz (1970)
overcame this weakness by introducing the truncated Cauchy distribution.
Nadarajah & Kotz (2006) found finite moments of all orders for truncated Cauchy
distribution.
Truncated Cauchy distribution is useful and used in many industrial setting. For
example, inspection of final products before sending to the customers or
inspection of products at every stage of the multistage production process are
samples of using truncated distribution. In addition, truncated Cauchy distribution
is a common prior for Bayesian models mainly with respect to economic data.
Nadarajah & Kotz (2005) defined the skew distribution with the Cauchy kernel.
These models are used in many different areas. For example in economics where
many finance returns on risky financial assets do not follow the normal
distribution. The real data is distributed skewed and fat-tailed. However, these
models suffer from limited applicability because they don’t have finite moments.
In fact, there is no reasonable reason to identify that empirical moment of all
orders should be infinite. Therefore, Nadarajah & Kotz (2007) introduced skew
truncated Cauchy distribution and overcame the limitation for skew Cauchy distri-
980 Zahra Nazemi Ashani and Mohd Rizam Abu Bakar
bution. In this paper, we focus on skew Cauchy logistic distribution and introduce
skew truncated Cauchy logistic distribution. The cumulative distribution function
can be easily computed as follows:
Proof:
When
Using the equation (1.3.2.31) in volume 1 Prudnikov et al. (1986),
, the integral can be calculated as
When
A skewed truncated Cauchy logistic distribution and its moments 981
Using the equation (1.3.2.31) in volume 1 Prudnikov et al. (1986), the integral can
be calculated as follows:
3 Moments
Theorem 1 and 2 provide the moments of all orders for skew truncated Cauchy
logistic distribution when is odd and is even.
Theorem1: If has the pdf of (3) then
for odd.
982 Zahra Nazemi Ashani and Mohd Rizam Abu Bakar
Proof:
Use the Taylor series expansion for exponential function ,
therefore
By using the equation (3.194.5) in Gradshteyn & Ryzhik (2000), the nth moments
of when is odd, can be calculated.
Theorem2:
If has the pdf (3)
for even.
Proof:
Using Maclurin series expansion , then
A skewed truncated Cauchy logistic distribution and its moments 983
By applying the equation (3.194.5) in Gradshteyn & Ryzhik (2000), the nth
moments of when is even, can be calculated.
Figure 1 shows the graphic for skewness coefficient for and .
The skewness range is given by for from 10 to
Figure 1: Graph of skewness for skew truncated Cauchy logistic distribution
The graphic for kurtosis coefficient for skew truncated Cauchy logistic model is
also provided in figure 2. The range for the kurtosis coefficient is provided
similarly for from to .
984 Zahra Nazemi Ashani and Mohd Rizam Abu Bakar
Figure 2: Graph of kurtosis for skew truncated Cauchy logistic distribution
4 Simulation study
In this section, we illustrate the flexibility of skew truncated Cauchy logistic
distribution over truncated Cauchy distribution by performing simulation study.
The technique for simulation is based on the formula in the paper of Azzalini
(1986). This method of generating random variables is twice more efficient than
the acceptance-rejection method. We simulated two independent samples with
. For one of the samples, the parameters are selected as
and . For the second sample, the parameters are
selected as and . We fitted both these samples to
truncated Cauchy distribution and skew truncated Cauchy logistic distribution.
Figure 4 presents the p-p plots according to these fits.
p-p plots for and
A skewed truncated Cauchy logistic distribution and its moments 985
p-p plots for and
It can be easily seen that the skew truncated Cauchy logistic distribution is a more
proper model than truncated Cauchy distribution for positively and negatively
skewed type of data.
5 Discussions
Cauchy distribution is one of the important distributions which has been applied in
different fields. For example, it is considered as a model in the economy,
biological and survival analysis. There is no reason for data in these situations that
the empirical moments of any orders should be infinite. Therefore, selection of the
Cauchy distribution or skew Cauchy symmetric distributions is unrealistic. In this
paper, the model of skew truncated Cauchy logistic distribution was introduced
which can be a more appropriate model than skew Cauchy distribution. For
example, we consider exchange rate data for Japanese Yen to the United States
Dollars between 1862 and 2003. Data come from the Global Financial Data
organization and are available through the website
http://www.globalfinancialdata.com/. Global Financial Data organization provides
financial and economic data from the 1200s to present. We transform data using
logarithms and relative change from one year to the next to get logical fit. The
benefits of applying relative change are that it makes data pure numbers and
independent from units of measurement. We fit both skew Cauchy distribution and
skew truncated Cauchy logistic distribution to the data using the method of
maximum likelihood. The maximum likelihood method estimates the parameters
of the model and test hypothesis about parameters. It also uses to compare two
models of the same data. There are two other kinds of tests for comparing two
models of data which called Wald test and Score test. When sample size is very
large, three of them are convergence. However, when the sample size is small,
maximum likelihood test is a more proper test and most of the statisticians prefer
to use maximum likelihood ratio test. For solving the likelihood equation, we
utilize quasi-Newton algorithm nlm in R software. In this example, we consider
h=2 and . The results are as follows:
986 Zahra Nazemi Ashani and Mohd Rizam Abu Bakar
For skew Cauchy distribution
and for skew truncated Cauchy logistic distribution
According to the likelihood ratio test, skew truncated Cauchy logistic is a better
model than skew Cauchy distribution for these data.
On the other hand, the main feature of skew symmetric distributions is the new
parameter which controls skewness and kurtosis and provides more flexible
models. According to the paper of Azzanili (1986), we calculate skewness and
kurtosis for a new model and truncated Cauchy distribution with and
on The skewness and kurtosis for truncated Cauchy distribution
are and , respectively. However, the skewness and kurtosis for skew
truncated Cauchy logistic distribution are and
respectively for from to . It is obvious that the new model present the
negative skewness and higher degree of peakness.
The figure shows the shapes of skew truncated Cauchy logistic for various values
of .
Figure 3: Examples of skew truncated Cauchy logistic distribution for
0,2,5,10, 1, 1 and h=1.
A skewed truncated Cauchy logistic distribution and its moments 987
Acknowledgments: The Authors acknowledge support from the Malaysia
Ministry of Higher Learning Grant No. 01-01-15-1705FR.
References [1] A. Azzalini, A class of distributions which includes the normal ones,
Scandinavian Journal of Statistic, 12 (1985), 171-178.
[2] A. Azzalini, Further results on a class of distributions which includes the
normal ones, Statistica, 46 (1986), 199-208. http://dx.doi.org/10.6092/issn.1973-2201/711
[3] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Product,
6th ed., Academic Press, San Diego, 2000.
[4] A. K. Gupta, F. C. Chang and W. J. Huang, Some skew-symmetric models,
Random Operators and Stochastic Equations, 10 (2002), 133-140.
http://dx.doi.org/10.1515/rose.2002.10.2.133
[5] N. L. Johnson and S. Kotz, Continuous Univariate Distributions, John Wiley
and Sons, New York, 1970.
[6] S. Mukhopadhyay and B. Vidakovic, Efficiency of linear Bayes rules for a
normal mean: skewed priors class, The Statistician, 44 (1995), 389-396.
http://dx.doi.org/10.2307/2348710
[7] S. Nadarajah and S. Kotz, Skewed distributions generated by the normal
kernel, Statistics and Probability Letters, 65 (2003), 269-277.
http://dx.doi.org/10.1016/j.spl.2003.07.013
[8] S. Nadarajah, Skewed distributions generated by the uniform kernel, Random
Operators and Stochastic Equations, 11 (2003), 297-305.
http://dx.doi.org/10.1515/156939703771378635
[9] S. Nadarajah and S. Kotz, Skewed distributions generated by the Laplace
kernel, American Journal of Mathematical and Management Sciences, 24
(2004), 321-349. http://dx.doi.org/10.1080/01966324.2004.10737636
[10] S. Nadarajah and S. Kotz, Skewed distributions generated by the Cauchy
kernel, Brazilian Journal of Probability and Statistic, 19 (2005), 39-51.
[11] S. Nadarajah and S. Kotz, A truncated Cauchy distribution, International
Journal of Mathematical Education in Science and Technology, 37 (2006),
605-608. http://dx.doi.org/10.1080/00207390600595223
988 Zahra Nazemi Ashani and Mohd Rizam Abu Bakar
[12] S. Nadarajah and S. Kotz, A skewed truncated Cauchy distribution with
application in economics, Applied Economics Letters, 14 (2007), 957-961.
http://dx.doi.org/10.1080/13504850600705950
[13] S. Nadarajah, Skewed distributions generated by the student’s t kernel,
Monte Carlo Methods and Applications, 13 (2008), 389-404.
http://dx.doi.org/10.1515/mcma.2007.021
[14] S. Nadarajah, The skew logistic distribution, AStA Advances in Statistical
Analysis, 93 (2009), 187-203.
http://dx.doi.org/10.1007/s10182-009-0105-6
[15] A. P. Prudnikov, Y. A. Brychkov and O. I. Marichev, Integrals and Series,
volume 1, Gordon and Breach Science Publisher, Amsterdam, 1986.
Received: July 19, 2016; Published: October 11, 2016
