An alternative to the Cauchy distribution

A few generalizations of the Cauchy distribution appear in the literature. In this paper, a new generalization of the Cauchy distribution is proposed, namely, the exponentiated-exponential Cauchy distribution (EECD). Unlike the Cauchy distribution, EECD can have moments for some restricted parameters space. The distribution has wide range of skewness and kurtosis values and has a closed form cumulative distribution function. It can be left skewed, right skewed and symmetric. Two different estimation methods for the EECD parameters are studied. • A new generalization of the Cauchy distribution is proposed, namely, exponentiated-exponential Cauchy distribution (EECD).• EECD has flexible shape characteristics. Moreover, EECD moments are defined under some restrictions on the parameter space.


Method details
The Cauchy distribution was first appeared in works of Pierre de Fermat and then studied by many researchers such as Isaac Newton, Gottfried Leibniz and others (see Ref. [1]). The Cauchy density was also used by Poisson [2] as counterexamples for some general results in probability. Based on Johnson et al. [1], the Cauchy distribution becomes associated with Cauchy [3] when Cauchy responded to an article by Bienayme' [4] criticizing a method of interpolation proposed by Cauchy. The fact that the Cauchy distribution has no moments, and therefore the law of large numbers does not apply, motivates researchers to generalize the Cauchy distribution. Few generalizations of the Cauchy distribution have appeared in the literature; Rider [5] proposed and study a generalization of the Cauchy distribution, Batschelet [6] proposed the wrapped-up Cauchy distribution, the skew-Cauchy distribution was proposed by Arnold and Beaver [7], another class of skew-Cauchy distribution was studied by Behboodian et al. [8], Huang and Chen [9] proposed a generalization of the skew-Cauchy distribution and recently Alshawarbeh et al. [10] used the beta family introduced by Eugene et al. [11] to generate the so called beta-Cauchy distribution.
In this paper, a new generalization of the Cauchy distribution is proposed. The proposed distribution is very flexible in terms of shapes, it can be left skewed, right skewed or symmetric. The moments are defined for some restricted values of the parameters. Also, the proposed distribution has a closed form cumulative distribution function (CDF) which adds more advantage to this distribution. The simplicity of the proposed distribution (closed from CDF) and the great flexibility in modeling real life data (see Application) will attract researchers to use this distribution as an alternative of the Cauchy distribution in modeling different scenarios.
Let rðtÞ be the probability density function (PDF) of a random variable T 2 ½a; b, for À1 a < b 1. Let Wð:Þ : ½0; 1 ! < be a link function satisfies the following conditions: Wð:Þ is absolutely continuous and monotonically non À decreasing function Wð0Þ ! a and Wð1Þ ! b: ' ð1:1Þ The CDF of the T-X family of distributions defined by Alzaatreh et al. [12] is given by where hðxÞ ¼ f ðxÞ=ð1 À FðxÞÞ and HðxÞ ¼ Àlogð1 À FðxÞÞ are, respectively, the hazard and cumulative hazard functions associated with f ðxÞ. Some general properties of the T-X in (1.4) have been recently studied, for more details see Alzaatreh et al. [12,13,31] and Lee et al. [14]. Also the discrete analogue of the T-X family is studied by Alzaatreh et al. [15].
If a random variable T follows the exponentiated exponential distribution (EED) with parameters a and l,rðtÞ ¼ la e Àlt ð1 À e Àlt Þ aÀ1 ; t ! 0, the definition in (1.4) leads to the exponentiated exponential-X family (EE-X) with the PDF g F ðxÞ ¼ alf ðxÞ 1 À 1 À FðxÞ ð Þ l aÀ1 1 À FðxÞ ð Þ lÀ1 : ð1:5Þ When a ¼ 1 and l ¼ n where n is a positive integer, the EE-X family in (1.5) reduces to the distribution of the first order statistics, X ð1Þ , from a sample of size n from f ðxÞ. When a ¼ n and l ¼ 1, the EE-X family reduces to the distribution of the n th order statistics, X ðnÞ , from a sample of size n from f ðxÞ. When a ¼ 1, the EE-X reduces to the exponentiated 1 À FðxÞ distribution with parameter l. The parameters a and l controls the skewness and kurtosis of the family. Also, as x ! À1, g F $ al a f F aÀ1 and as x ! 1, g F $ alf ð1 À FÞ lÀ1 .
The paper is outlined as follows. First we define using (1.5) a new generalization of the Cauchy distribution, namely, the exponentiated-exponential Cauchy (EEC) distribution. Then we study some properties of EEC distribution including quantile skeweness and kurtosis, Shannon entropy and moments. Also, different characterizations of the EE-X family based on truncated moments are discussed. Parameter Estimation deals with estimation methods of the EEC distribution. Applications of the EEC distribution to real data sets are provided.

The exponentiated-exponential Cauchy distribution
If X is a Cauchy random variable with parameter u > 0, then f ðxÞ ¼ ðpuð1 þ ðx=uÞ 2 ÞÞ À1 ; À1 < x < 1; and FðxÞ ¼ Therefore, the density in (2.1) is a generalization of the Cauchy density. From (2.1), we obtain the CDF of the EEC distribution as The hazard function associated with the EEC distribution is hðxÞ ¼ gðxÞ 1 À GðxÞ ¼ al 1 À 0:5 À p À1 tan À1 ðx=uÞ À Á l aÀ1 0:5 À p À1 tan À1 ðx=uÞ À Á lÀ1 pu 1 þ ðx=uÞ 2 1 À 1 À 0:5 À p À1 tan À1 ðx=uÞ À Á l a ; x 2 <: ð2:3Þ A physical interpretation of the EECD in (2.2) is possible for integer values of a and l. For example, let us start with a system F which consists of a independent components say X i ; i ¼ 1; Á Á Á ; a. Suppose that each component consists of l subcomponents. Assume that the system F fails if all of the a components fail (i.e. parallel system with respect to a components). Assume further that each of the a components fails if at least one of the l subcomponents fails. Now, let X denote the lifetime of the system F and X i1 ; Á Á Á ; X il ; i ¼ 1; Á Á Á ; a be independent random variables follow Cauchy (u). Then the CDF of X can be computed as follows In the next section, some general properties of the EEC distribution will be addressed including transformations, limiting behavior, quantile function and the Shannon entropy. Remark 1. The connection of EEC with some known distributions can be seen as follows; If a random variable Y follows Kumaraswamy's distribution with parameters a and l, then the random variable X ¼ ucotðpYÞ follows EEC a; l; u ð Þ. Also, if a random variable Y follows beta distribution with shape parameters a ¼ 1 and b ¼ a, then the random variable X ¼ ucotðpY 1=l Þ follows EEC a; l; u ð Þ. Finally, if a random variable Y follows exponentiated-exponential distribution with parameters a and l, then the random variable X ¼ ucotðpe ÀY Þ follows EEC a; l; u ð Þ.
The mode of the EEC distribution is the solution of kðwÞ ¼ 0 where kðwÞ ¼ ð1 À w l Þð1 À la À 2pwcotðpwÞÞ þ lða À 1Þ; w 2 ð0; 1Þ ð2:4Þ and w ¼ 1 2 À 1 p tan À1 ðx=uÞ. To find the mode of the EECD; first find w 0 such that kðw 0 Þ ¼ 0 and then obtain the mode at x 0 ¼ ucotðpw 0 Þ. If a ¼ l ¼ 1, then (2.4) implies that x 0 ¼ 0 is the only mode. This result agrees with the mode of the standard Cauchy distribution.

displays the Galton's skewness and
Moors' kurtosis for the EECD when u ¼ 1. From Fig. 4, the EECD takes wide range of skewness and  kurtosis values. This indicates that the EECD can be very effective in modeling real data sets with various skewness and kurtosis values.
Galtons' skewness is also used to determine the regions in which the EEC distribution is left skewed or right skewed. A numerical method is used to determine the points where the Galtons' skewness equals to zero. Fig. 5 shows the regions in which the EEC distribution is left skewed or right skewed. The quadratic function in Fig. 5 connects the points where EEC distribution is symmetric.

Some properties of EEC distribution
The entropy of a random variable X is a measure of variation of uncertainty [19]. Shannon entropy [20] for a random variable X with PDF g(x) is defined as E Àlog gðXÞ ð Þ f g. Since 1948, Shannon entropy has been used in many fields such as communication theory, engineering, physics and biology. Alzaatreh et al. [12] derived the Shannon entropy for the T-X family of distributions. Also, Ghosh and Alzaatreh [21] derived the Shannon entropy for the exponentiated exponential-X in (1.5) as À logðlaÞ þ ð1 À 1=lÞðcðaÞ À cð1ÞÞ À ðalÞ À1 þ 1;

ð3:1Þ
where c is the digamma function and T is the exponentiated-exponential random variable with parameters a and l. In the following theorem, we derive the Shannon entropy for the EEC distribution.  Theorem 2. The Shannon entropy for the EEC distribution is given by where HðaÞ is the harmonic number of a.
Proof. We first need to find ÀE logf F À1 1 À e ÀT À Á n o , where f ðxÞ and FðxÞ are, respectively, the PDF and the CDF of the Cauchy distribution. It is easy to show that logf ðF À1 ð1 À e ÀT ÞÞ ¼ ÀlogðpuÞ þ 2logðsinðpe ÀT ÞÞ and hence, Now, using Gradshteyn and Ryzhik ( [22], p. 55), logðsinpxÞ can be written as where B n is the Bernoulli number. By letting u ¼ e Àt and using the series representation of logðsinpxÞ in (3.4), (3.3) reduces to where Bða; bÞ ¼ GðaÞGðbÞ=Gða þ bÞ, the beta function. Now, HðaÞ. The results in (3.1) followed by using the above result and substituting (3.5) in (3.2) and using the fact that HðaÞ ¼ cða þ 1Þ À cð1Þ. &

Moments
On using Remark 1, the rth moments for the EEC distribution can be written as where v 0 ¼ p Àr , v m ¼ pm À1 P m k¼1 ðkr À m þ kÞw k v mÀk ; m ! 1 Hence, from (3.6) the rth moments for the EEC distribution can be written as EðX r Þ ¼ au r X 1 k¼0 v k B½a; l À1 ð2k À rÞ þ 1:

ð3:9Þ
The rth moments of the EEC do not always exist. The following theorem gives a necessary and sufficient condition for the existence of the rth moments of the EEC distribution. Proof. Consider the following integrals where gðxÞ is defined in (2.1).
Without loss of generality assume u ¼ 1. Since the middle integral above exists, it suffices to investigate the existence of the first and third integrands. Let dðxÞ ¼ alp Àl x ÀðlÀkþ1Þ for x ! 1. Then R 1 1 dðxÞdx exists iff l > k: One can easily show that x k gðxÞ $ dðxÞ as x ! 1 and hence Fig. 6 provides the mean and variance of the EEC distribution when the scale parameter u ¼ 1 and for various combinations of a and l. From Fig. 6 it appears that for fixed l, the mean is an increasing functions of a while the variance is a decreasing function of a. Also, for fixed a, the mean is a decreasing function of l.
The rth incomplete moments for the EEC distribution is defined as x r gðxÞdx: ð3:10Þ On using the substitution u ¼ 0:5 À p À1 tan À1 ðx=uÞ, (3.10) can be written as where u m ¼ 0:5 À p À1 tan À1 ðm=uÞ. where B x ða; bÞ ¼ R x 0 u aÀ1 ð1 À uÞ bÀ1 du, is the incomplete beta function. The first incomplete moment is used to find the deviations from the mean and median. The deviation from the mean and the deviation from the median are used to measure the dispersion and Proof. By definitions of DðmÞ and DðMÞ, it is easy to see that The rest of the proof follows from (3.12). &

Some characterizations of the EE-X family based on truncated moments
Glänzel [25] provides characterizations based on the truncated moments for some important distributions including the standard Cauchy distribution. For more information, one is referred to Hamedani [26]. Below, we provide some results from Glänzel [25] which will be used to show Theorem 5.
The following theorem provides a characterization for the EE-X family of distributions in (1.5).
Theorem 5. Let Y : V ! H be a continuous random variable. Then Y follows the EE-X family in (1.5) if and only if the functions in Theorem 4 can be chosen as: h ¼ 0:5ð1 À FÞ l , g 1 ¼ g 2 ð1 À FÞ l and g 2 ¼ ð1 À ð1 À FÞ l Þ 1Àa , x 2 H; a; l > 0, where F is the CDF of the random variable X defined in (1.5).

Maximum likelihood estimation method (MLE)
Let a random sample of size n be taken from the EEC distribution. The log-likelihood function for the EEC distribution in (2.1) is given by logLða; l; uÞ ¼ X n i¼1 loggðx i ; a; l; uÞ where z i ¼ 0:5 À p À1 tan À1 ðx i =uÞ.
The MLEâ,l andû can be obtained by maximizing the log likelihood function in (4.1) numerically.
The initial value for u is taken to be the MLE of u by assuming the data, x 1 ; x 2 ; Á Á Á ; x n , follows the Cauchy distribution. The initial values for the parameters a and l are taken as follows: From Remark 1, the initial values of a and l are taken to be the MLEs of a and l by assuming the data y i ¼ u 0 cotðpe Àx i Þ; i ¼ 1; Á Á Á ; n follows EEða; bÞ. PROC NLMIXED in SAS is used to maximize the loglikelihood function in (4.1). In addition to the goodness of fit statistics, PROC NLMIXED gives the parameter estimates with their standard errors, which are the square roots of the diagonal entries in the estimated covariance matrix.

Alternative method of moment estimation (AMM)
Since the moments of the EECD do not always exist, we consider in this section an alternative method of moment estimation which was first proposed by Zografos and Balakrishnan [28].
Theorem 7. If X follows EECD with parameters a, l and u, then for any r 2 N,

ð4:2Þ
The alternative method of moments estimatesã; l andũ are obtained by solving the equations in (4.2) iteratively.

Simulation study
To evaluate the performance of the MLE and the AMM methods, a simulation study for both methods is conducted for a total of five parameter combinations and the process is repeated 1000 times. Three different sample sizes n = 50, 70 and 100 are considered. The bias (estimate-actual), and the mean square errors (MSE) of the parameter estimates for the MLE and the AMM are presented in Tables 1 and 2 respectively. From the results in Table 2, it appears that the mean square errors for some parameters using the AMM method are unacceptably high. This can be seen more clearly for the parameter u. The results in Table 1 show that the ML estimates, in most cases, have smaller mean square errors than the AMM estimates. Also, the bias using MLE method is acceptable. These results suggest using the MLE methods for data fitting. Also, a close look at the results from the small simulation study in Table 1, it is noticed that when a 1(a > 1), the MLE of a is overestimated (underestimated). Also, when a ! 1(a < 1), the MLE of l is overestimated (underestimated). Furthermore, Table 1 indicates that the MLE of u is always overestimated.

Application
To illustrate the applications of the EEC distribution, the EEC distribution is fitted to two data sets. The first data set in Table 3 (http://www.ibge.gov.br/seriesestatisticas/exibedados.php?idnivel=-BR&idserie=PRECO101), is the INPC data which represents the national index of consumer prices of Brazil since 1979. The INPC index measures the cost of living of households with heads employees. The second data set in Table 4 from Weisberg [29], represents the sum of skin folds in102 male and 100 female athletes collected at the Australian Institute of Sports. The data sets are fitted to the EEC distribution and compared with the two-parameter Cauchy, the three parameter skew-Cauchy [8] and the beta-Cauchy [10] distributions. The following are the PDF of the Cauchy, skew-Cauchy and beta-Cauchy distributions respectively;  The maximum likelihood estimates, the log-likelihood value, the AIC (Akaike Information Criterion), the Kolmogorov-Smirnov (K-S) test statistic, and the p-value for the K-S statistic for the fitted distributions to the data sets are reported in Tables 5 and 6.
The results in Tables 5 and 6 show that the Cauchy distribution does not provide adequate fit to both data sets. The Skew-Cauchy distribution does not provide adequate fit to data sets in Table 4 and provides adequate fit to the data in Table 3. The EEC and beta-Cauchy distributions provide the best fit to the two data sets. The fact that EECD has less number of parameters compared with beta-Cauchy distribution makes EECD a better choice for fitting both data sets.
From Table 5, the estimated value for the parameter u for the EEC distribution is approximately 1.
Therefore, the two-parameter EEC distribution can be a natural choice for this data set. The likelihood ratio test for the hypothesis H 0 : u ¼ 1 against H a : u 6 ¼ 1 confirms that the two-parameter EEC The likelihood ratio statistic in this case is based on l ¼ L 0 ðã;lÞ=L a ðâ;l;ũÞ, where L 0 and L a are the likelihood values for the standard EEC and the three-parameter EEC distributions respectively. The quantity À2logl asymptotically follows a chi-square distribution with 1 degree of freedom. In this case, we have À2logl ¼ 0:0004 and the p-value is 1.0000. Fig. 7 displays the empirical and the fitted cumulative distribution functions for the data sets in Tables 3 and 4. In this Figure, the standard EEC  distribution is used for the data set in Table 3 and the three-parameter EEC distribution is used for the data set in Table 4. The figure supports the results from Tables 5 and 6.

Summary and conclusion
In this article, a generalization of the Cauchy distribution, the EECD, is defined and studied. Several properties of the proposed distribution are studied in detail including mode, moments, skewness, kurtosis and Shannon entropy. Two real data sets are fitted to the EECD and compared with Cauchy, skew Cauchy and beta-Cauchy distributions. The results show the great flexibility of the proposed model. Based on Figs. 4 and 5, the EEC indeed can fit different data sets with wide range of skewness values including left and right skewness. Furthermore, a simulation study is conducted for various parameter and sample size values to generate highly left and right skewed data sets from EECD. The results, based on K-S statistic, showed that the EECD produces good fit to various highly skewed (left and right) data sets. To conserve space, the results were not included in the article.
For future research, one can propose methods of discrimination between two or more members of the EE-X family based on the ratio of the Shannon entropies. For more information about this problem, one is referred to Zografos and Balakrishnan [28]. Furthermore, one can use the kullback-Leibler divergence [30] to discriminate between member of EE-X family and other family such as the beta family [11]. Also, it is noteworthy to mention that the method of discrimination between members of EE-X family using the idea proposed by Zografos and Balakrishnan [28] can be extended to cover the gamma-X family [13] or even the T-X family [12].