 Methodology
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The generalized Cauchy family of distributions with applications
Journal of Statistical Distributions and Applications volume 3, Article number: 12 (2016)
Abstract
A family of generalized Cauchy distributions, TCauchy{Y} family, is proposed using the TR{Y} framework. The family of distributions is generated using the quantile functions of uniform, exponential, loglogistic, logistic, extreme value, and Fréchet distributions. Several general properties of the TCauchy{Y} family are studied in detail including moments, mean deviations and Shannon’s entropy. Some members of the TCauchy{Y} family are developed and one member, gammaCauchy{exponential} distribution, is studied in detail. The distributions in the TCauchy{Y} family are very flexible due to their various shapes. The distributions can be symmetric, skewed to the right or skewed to the left.
1. Introduction
The Cauchy distribution, named after Augustin Cauchy, is a simple family of distributions for which the expected value does not exist. Also, the family is closed under the formation of sums of independent random variables, and hence is an infinitely divisible family of distributions. The Cauchy distribution was used by Stigler (1989) to obtain an explicit expression for P(Z _{1} ≤ 0, Z _{2} ≤ 0) where (Z _{1}, Z _{2})^{T} follows the standard bivariate normal distribution. The Cauchy distribution has been used in many applications such as mechanical and electrical theory, physical anthropology, measurement problems, risk and financial analysis. It was also used to model the points of impact of a fixed straight line of particles emitted from a point source (Johnson et al. 1994). In Physics, it is called a Lorenzian distribution, where it is the distribution of the energy of an unstable state in quantum mechanics.
Eugene et al. (2002) introduced the betagenerated family of distributions using the beta as the baseline distribution. Based on the betagenerated family, Alshawarbeh et al. (2013) proposed the betaCauchy distribution. The betagenerated family was extended by Alzaatreh et al. (2013) to the TR(W) family. The cumulative distribution function (CDF) of the TR(W) distribution is \( G(x)={\displaystyle {\int}_a^{W\left(F(x)\right)}r(t)dt,} \) where r(t) is the probability density function (PDF) of a random variable T with support (a, b) for − ∞ ≤ a < b ≤ ∞. The link function W : [0, 1] → ℝ is monotonic and absolutely continuous with W(0) → a and W(1) → b.
Aljarrah et al. (2014) considered the function W(.) to be the quantile function of a random variable Y and defined the TR{Y} family. In the TR{Y} framework, the random variable T is a ‘transformer’ that is used to ‘transform’ the random variable R into a new family of generalized distributions of R. Many families of generalized distributions have appeared in the literature. Alzaatreh et al. (2014, 2015) studied the Tgamma and the Tnormal families. Almheidat et al. (2015) studied the TWeibull family. In this paper, a family of generalized Cauchy distribution is proposed and studied.
This article focuses on the generalization of the Cauchy distribution and studies some new distributions and their applications. The article gives a brief review of the TR{Y} framework and defines several new generalized Cauchy subfamilies. It contains some general properties of the TCauchy{Y} distributions. A member of the TCauchy{Y} family, the gammaCauchy{exponential} distribution, is studied. The study includes moments, estimation and applications. Some concluding remarks were provided.
2. The TCauchy{Y} family of distributions
The TR{Y} framework defined in Aljarrah et al. (2014) (see also Alzaatreh et al. 2014) is given as follows. Let T, R and Y be random variables with CDF F _{ Z }(x) = P(Z ≤ x), and corresponding quantile function Q _{ Z }(p), where Z = T, R, Y and the quantile function is defined as Q _{ Z }(p) = inf{z : F _{ Z }(z) ≥ p}, 0 < p < 1. If densities exist, we denote them by f _{ Z }(x), for Z = T, R and Y. Now assume the random variables T, Y ∈ (a, b) for − ∞ ≤ a < b ≤ ∞. The random variable X in TR{Y} family of distributions is defined as
The corresponding PDF associated with (1) is
Alternatively, (2) can be written as
The hazard function of the random variable X can be written as
Alzaatreh et al. (2013, 2014, 2015) studied, respectively, the TR{exponential}, Tnormal{Y} and Tgamma{Y} families of distributions. Aljarrah et al. (2014) studied some general properties of the TR{Y} family. Next, we define the TCauchy{Y} family.
Let R be a random variable that follows the Cauchy distribution with PDF f _{ R }(x) = f _{ C }(x) = π ^{− 1} θ ^{− 1}(1 + (x/θ)^{2})^{− 1} and CDF F _{ R }(x) = F _{ C }(x) = 0.5 + π ^{− 1} tan^{− 1}(x/θ), x ∈ ℝ, θ > 0, then (3) reduces to
Hereafter, the family of distributions in (5) will be called the TCauchy{Y} family. It is clear that the PDF in (5) is a generalization of Cauchy distribution. From (1), if \( T\overset{d}{=}Y, \) then \( X\overset{d}{=}\mathrm{Cauchy}\left(\theta \right). \) Also, if \( Y\overset{d}{=}\mathrm{Cauchy}\left(\theta \right), \) then \( X\overset{d}{=}T. \) Furthermore, when T ~ beta(a, b) and Y ~ uniform(0, 1), the TCauchy{Y} reduces to the betaCauchy distribution (Alshawarbeh et al. 2013). When T ~ Power(a) and Y ~ uniform(0, 1), the TCauchy{Y} reduces to the exponentiatedCauchy distribution (Sarabia and Castillo 2005). Table 1 gives six quantile functions of known distributions (in standard form) which will be applied to generate TCauchy{Y} subfamilies in the following subsections. It is straightforward to use nonstandard quantile functions. By using nonstandard quantile functions, many resulting distributions in the TR{Y} family will have more than five parameters, which are not practically useful (Johnson et al. 1994, p. 12). Thus, we focus on the standard quantile functions in this paper.
2.1 TCauchy{uniform} family of distributions
By using the quantile function of the uniform distribution in Table 1, the corresponding CDF to (1) is
and the corresponding PDF to (6) is
2.2 TCauchy{exponential} family of distributions
By using the quantile function of the exponential distribution in Table 1, the corresponding CDF to (1) is
and the corresponding PDF to (8) is
Note that the CDF and the PDF in (8) and (9) can be written as F _{ X }(x) = F _{ T }(H _{ C }(x)) and f _{ X }(x) = h _{ C }(x)f _{ T }(H _{ C }(x)) where h _{ C }(x) and H _{ C }(x) are the hazard and cumulative hazard functions for the Cauchy distribution, respectively. Therefore, the TCauchy{exponential} family of distributions arises from the ‘hazard function of the Cauchy distribution’.
2.3 TCauchy{loglogistic} family of distributions
By using the quantile function of the loglogistic distribution in Table 1, the corresponding CDF to (1) is
and the corresponding PDF is
which is a family of generalized Cauchy distributions arising from the ‘odds’ of the Cauchy distribution.
2.4 TCauchy{logistic} family of distributions
By using the quantile function of the logistic distribution in Table 1, the corresponding CDF to (1) is
and the corresponding PDF is
Note that (13) can be written as \( {f}_X(x)=\frac{h_C(x)}{F_C(x)}\times {f}_T\left( \log \left(\frac{F_C(x)}{1{F}_C(x)}\right)\right) \), which is a family of generalized Cauchy distributions arising from the ‘logit function’ of the Cauchy distribution.
2.5 TCauchy{extreme value} family of distributions
By using the quantile function of the extreme value distribution in Table 1, the corresponding CDF to (1) is
and the corresponding PDF is
The CDF in (14) and the PDF in (15) can be written as F _{ x }(x) = F _{ T }(log H _{ C }(x)) and f _{ X }(x) = {h _{ C }(x)/H _{ C }(x)}f _{ T }(log H _{ C }(x)) respectively.
2.6 TCauchy{Fréchet} family of distributions
By using the quantile function of the Fréchet distribution in Table 1, the corresponding CDF to (1) is
and the corresponding PDF is
Figures 1 and 2 show some examples of two members of the TCauchy{Y} family. The first example is WeibullCauchy{exponential} distribution which can be obtained by replacing the random variable T in (9) with Weibull(c, γ) random variable. The second example is LomaxCauchy{loglogistic} distribution which can be obtained by replacing the random variable T in (11) with Lomax(α, λ) random variable. From the figures, it appears that the shapes of the distributions can be leftskewed, rightskewed or symmetric.
3. Some properties of the TCauchy{Y} family of distributions
In this section, we discuss some general properties of the TCauchy family of distributions. The proofs are omitted for straightforward results.
Lemma 1: Let T be a random variable with PDF f _{ T }(x), then

(i)
The random variable X = − θ cot(πF _{ Y }(T)) follows the TCauchy{Y} distribution.

(ii)
The quantile function for TCauchy{Y} family is Q _{ X }(p) = − θ cot(πF _{ Y }(Q _{ T }(p))).
The Shannon’s entropy (Shannon 1948) of a random variable X is a measure of variation of uncertainty and it is defined as η _{ X } = − E{log(f(X))}. The following proposition provides an expression for the Shannon’s entropy for the TCauchy{Y} family.
Proposition 1: The Shannon’s entropy for the TCauchy{Y} family is given by
Proof: By using the result in Aljarrah et al. (2014), the Shannon’s entropy for the TCauchy{Y} is
Now, one can show that
On using the following series expansion from Gradshteyn and Ryzhik (2007, p. 55)
where \( {V}_j=\frac{{\left(1\right)}^j{\left(2\pi \right)}^{2j}{B}_{2j}}{2j(2j)!} \) and B _{ j } is the Bernoulli number, we get the result in (18).□
Next proposition gives a general expression for the rth moment for the TCauchy{Y} family.
Proposition 2: The rth moment for the TCauchy{Y} family of distributions is given by
where c _{0} = π ^{− r}, \( {c}_m=\pi {m}^{1}{\displaystyle {\sum}_{k=1}^m\left(krm+k\right){w}_k{c}_{mk},\kern0.24em m\ge 1} \) and \( {w}_k=\frac{{\left(1\right)}^k{2}^{2k}{B}_{2k\;}{\pi}^{2k1}}{(2k)!}. \)
Proof: From Lemma 1(i), the rth moment for the TCauchy{Y} family can be written as E(X ^{r}) = (−1)^{r} θ ^{r} E(cot π(F _{ Y }(T)))^{r}. Now, using the following series expansion (see Abramowitz and Stegun 1964, p.75), \( \cot \kern0ex \left(\pi x\right)={\displaystyle \sum_{k=0}^{\infty }{w}_k{x}^{2k1},\leftx\right}<\pi, \) where \( {w}_k=\frac{{\left(1\right)}^k{2}^{2k}{B}_{2k\;}{\pi}^{2k1}}{(2k)!}. \) Therefore,
where c _{0} = π ^{− r}, \( {c}_m=\pi {m}^{1}{\displaystyle \sum_{k=1}^m\left(krm+k\right){w}_k{c}_{mk},}\kern0.5em m\ge 1 \) [see Gradshteyn and Ryzhik 2007, p. 17]. □
As an example of the applicability of the results in Lemma 1 and Propositions 1 and 2, we use these results and apply them on the TCauchy{exponential}. One can get similar results by choosing any of the TCauchy{Y} families.
Corollary 1: Based on Lemma 1, if T is a random variable with PDF f _{ T }(x), then

(i)
The random variable X = θ cot(πe ^{− T}) follows a distribution in the TCauchy{exponential} family.

(ii)
The quantile functions for TCauchy{exponential} family is \( {Q}_X(p)=\theta \cot \left(\pi {e}^{{Q}_T(p)}\right). \)
Corollary 2: The Shannon’s entropy for the TCauchy{exponential} family is given by
where M _{ T }(.) is the moment generating function of the random variable T.
Proof: The result follows from Proposition 1 and the fact that E(log f _{ Y }(T)) = μ _{ T }.□
Corollary 3: The rth moment for the TCauchy{exponential} family of distributions is given by
where c _{ k } is defined in Proposition 2.
Proof: The result follows from Proposition 2 and the fact that cot(πF _{ Y }(u)) = − cot(πe ^{− u}).□
Proposition 3: The mode(s) of the TCauchy{exponential} family are the solutions of the equation
Proof: For Cauchy distribution, one can show that \( {f}_C^{\prime }(x)=2\pi {\theta}^{1}x{f}_C^2(x) \) and \( {h}_C^{\prime }(x)=2\pi {\theta}^{1}x{h}_C(x)+{h}_C^2(x). \) On finding \( {f}_X^{\prime }(x) \) by using Eq. (9) and setting the derivative to zero, it is easy to get the result in (24). □
4. GammaCauchy{exponential} distribution
For the remaining sections, we investigate in details the properties, parameter estimation and applications of a new distribution of the TCauchy{Y} family, the gammaCauchy{exponential} distribution. This distribution is interesting as it consists of special cases of exponentiated Cauchy and distributions of record values from the Cauchy distribution. Let T be a random variable that follows the gamma distribution with parameters \( \alpha \) and β. From Eqs. (8) and (9), the PDF and CDF of gammaCauchy{exponential} distribution are, respectively, given by
where \( \gamma \left(\alpha, x\right)={\displaystyle {\int}_0^x{t}^{\alpha 1}}{e}^{t}dt \) is the incomplete gamma function. For simplicity, a random variable X with PDF f(x) in (25) is said to follow the gammaCauchy{exponential} distribution and is denoted by GC(α, β, θ).
Some special cases are worth mentioning:

(i)
GC(1, β, θ) is the exponentiated Cauchy distribution proposed by Sarabia and Castillo (2005). In particular GC(1,1,1) is the standard Cauchy distribution.

(ii)
GC(1, n ^{− 1}, θ), n ∈ ℕ is the distribution of the minimum of n independent Cauchy random variables.

(iii)
GC(n + 1, 1, θ), n ∈ ℕ is the distribution of the nth upper record in a sequence of independent Cauchy random variables.
Remarks: The following results follow from Corollary 1, Corollary 2 and Proposition 3.

(i)
If a random variable Y follows a gamma distribution with parameters \( \alpha \) and β, then X = θ cot(πe ^{− Y}) follows the GC(α, β, θ) distribution.

(ii)
The quantile function of GC(α, β, θ) is \( Q(p)=\theta \cot \left(\pi {e}^{\beta {\gamma}^{1}\left[\alpha, p\varGamma \left(\alpha \right)\right]}\right),\kern0.5em 0<p<1. \)

(iii)
The Shannon’s entropy for the GC(α, β, θ) distribution is given by
\( {\eta}_X=\alpha \left(1+\beta \right)+ \log \left({\pi}^{1}\theta \beta \varGamma \left(\alpha \right)\right)+\left(1\alpha \right)\psi \left(\alpha \right)2{\displaystyle \sum_{j=1}^{\infty }{V}_j{\left(1+2j\beta \right)}^{\alpha },} \) where ψ(.) is the digamma function and V _{ j } is defined in Eq. (21).
Proposition 4: The GC(α, β, θ) distribution is unimodal and the mode is at m = θx where x is the solution of the equation
Proof: It is not difficult to show that the mode of f(x) in (25) is the solution of k(x/θ) = 0, where k(x) is defined above. Therefore, the mode of f(x) is at m = θx where k(x) = 0. To show the unimodality of f(x), consider A(x) = log(π ^{− 1} cot^{− 1}(x)) and B(x) = 2x cot^{− 1}(x). Clearly A(x) is a strictly decreasing function (since it is equal to log(1 − F _{ C }(x))). Furthermore, A(x) < 0 for all x ∈ ℝ. Now, B′(x) = 2[−x/(1 + x ^{2}) + cot^{− 1}(x)]. Therefore, B′(x) > 0 for all x ≤ 0. If x > 0, we have B′(x) < B′(0) = π/2 since B″(x) < 0. Since \( \underset{x\to \infty }{ \lim }{B}^{\prime }(x)=0. \) we get B′(x) > 0 for all x > 0. Therefore, B(x) is strictly increasing for all x ∈ ℝ. Now, let us prove the claim that η(x) = A(x)B(x) is a decreasing function on ℝ.
Proof of the claim: Let 0 ≤ x ≤ y, then 0 ≤ − A(x) ≤ − A(y) and 0 ≤ B(x) ≤ B(y). This implies that η(x) ≥ η(y). Now let x < 0, then η′(x) = − 2x/(x ^{2} + 1) − 2(x ^{2} + 1)^{− 1} x log(π ^{− 1} cot^{− 1}(x)) + 2 cot^{− 1}(x)log(π ^{− 1} cot^{− 1}(x)). Since the middle term in η′(x) is negative, consider \( \psi (x)=\frac{x}{x^2+1}{ \cot}^{1}(x) \log \left({ \cot}^{1}(x)/\pi \right). \) On differentiation, \( {\psi}^{\prime }(x)=\frac{1}{x^2+1}\left\{\frac{2}{x^2+1}+ \log \left({ \cot}^{1}(x)/\pi \right)\right\}. \) It is easy to show that the term \( \zeta (x)=\frac{2}{x^2+1}+ \log \left({ \cot}^{1}(x)/\pi \right) \) is strictly increasing on x ≤ 0 with ζ(0) > 0 and ζ(−∞) → 0. Thus, ζ(x) > 0 for all x < 0. This implies that ψ(x) is strictly increasing on x < 0 with ψ(0) > 0 and ψ(−∞) → 0. That is, ψ(x) > 0 for all x < 0. Therefore η′(x) ≤ 0 for all x < 0. Hence, η(x) = A(x)B(x) is a decreasing function in ℝ. This completes the proof of the claim. The fact that η(−∞) → 2 and η(∞) → − ∞ implies that η(x) = 0 has a unique solution. Now, B(x) − 1 + 1/β is only a shift by − 1 + 1/β and therefore remains a strictly increasing function. One can show that the term A(x)[B(x) − 1 + 1/β] remains a decreasing function for all x ∈ ℝ and hence k(x) remains a decreasing function in ℝ with k(−∞) → α + 1 > 0 and k(∞) → − ∞. This ends the proof.□
In Fig. 3, various graphs of f(x) are provided for different parameter values of α and β where θ = 1. The plots indicate that the gammaCauchy{exponential} distribution can be symmetric, rightskewed or leftskewed. Also, it appears that gammaCauchy{exponential} is symmetric only for the trivial case when α = β = 1.
In the following subsection, we provide some results related to the moments of GC(α, β, θ) distribution.
4.1 Moments of gammaCauchy{exponential} distribution
From Corollary 3, the rth moment of the GC(α, β, θ) can be written as
where c _{ k } is defined in Eq. (22). Therefore, the mean of GC(α, β, θ) is
where c _{ k } is defined in (22) with r = 1. Note that \( {\mu}_1^{\prime}\left(\alpha, \beta, \theta \right) \) is defined here for α > 1 and β < 1.
The next proposition establishes the condition for the existence of rth moment of the GC(α, β, θ) distribution.
Proposition 5: The rth moment of the GC(α, β, θ) distribution exists if and only if α > r and β ^{− 1} > r.
Proof: Without loss of generality, we assume θ = 1 and apply a similar idea as in Alshawarbeh et al. (2012). We write
Since the middle integrand is bounded by 2, it suffices to investigate the existence of the first and third integrands of the right hand side of Eq. (28). Consider \( {I}_1={\displaystyle {\int}_1^{\infty }{x}^r}g(x)dx \) and \( {I}_2={\displaystyle {\int}_{\infty}^{1}{x}^r}g(x)dx. \) Consider the following inequality from Abramowitz and Stegun (1964), p. 68
On using the inequality in (29) and for α > 1, we have
Let us write \( \delta (x)=\frac{x^r}{1+{x}^2}{\left(1/2+{\pi}^{1}ta{n}^{1}(x)\right)}^{\alpha 1}{\left(1/2{\pi}^{1}ta{n}^{1}(x)\right)}^{1/\beta \alpha } \). Then one can show that as x → ∞, δ(x) ~ x ^{− (1/β − α − r + 2)}. Therefore, I _{1} exists if and only if 1/β − α > r − 1. Since α > 1, this implies 1/β > r. Now, if α < 1, the inequality in (29) implies that
Let \( \zeta (x)=\frac{x^r}{1+{x}^2}{\left(1/2+{\pi}^{1}ta{n}^{1}(x)\right)}^{\alpha 1}{\left(1/2{\pi}^{1}ta{n}^{1}(x)\right)}^{1/\beta 1}. \) As x → ∞, ζ(x) ~ x ^{− (1/β − r + 1)}. So I _{1} exists if and only if 1/β > r. Similarly one can show that I _{2} exists if and only if α > r.□
Next, we consider recursive relation for the rth moment of the GC(α, β, θ) distribution.
Proposition 6: Let X ~ GC(α, β, 1) and n ∈ ℕ. Then

(i)
\( {\mu}_{2n}^{\prime}\left(\alpha, \beta \right)=\frac{1}{\pi \beta {\left(1\beta \right)}^{\alpha 1}}{\displaystyle \sum_{j=1}^n\frac{{\left(1\right)}^{j1}}{2n2j+1}}\left\{{\mu}_{2n2j+1}^{\prime}\left(\alpha, \frac{\beta }{1\beta}\right){\mu}_{2n2j+1}^{\prime}\left(\alpha 1,\frac{\beta }{1\beta}\right)\right\}+{\left(1\right)}^n. \)

(ii)
\( {\mu}_{2n+1}^{\prime}\left(\alpha, \beta \right)=\frac{1}{\pi \beta {\left(1\beta \right)}^{\alpha 1}}{\displaystyle \sum_{j=1}^n{\displaystyle \sum_{i=0}^j\frac{{\left(1\right)}^{nj}}{2j}}}\left(\begin{array}{c}\hfill n\hfill \\ {}\hfill j\hfill \end{array}\right)\left(\begin{array}{c}\hfill j\hfill \\ {}\hfill i\hfill \end{array}\right)\left\{{\mu}_{2i}^{\prime}\left(\alpha, \frac{\beta }{1\beta}\right){\mu}_{2i}^{\prime}\left(\alpha 1,\kern0.5em \frac{\beta }{1\beta}\right)\right\}+{\left(1\right)}^n{\mu}^{\prime}\left(\alpha, \beta \right). \)
Proof: From (25) and using the substitution u = tan ^{− 1}(x), we have
where
and
It is easy to see that I _{0} = (−1)^{n} π β ^{α} Γ(α). Also, it is not difficult to show that
Substituting (31) in (30), we get the result in (i). For the proof of (ii), one can easily see that
where
The rest of the proof is not difficult to show.□
Table 2 provides the mean, variance, skewness and kurtosis of the GC(α, β, 1) for various values of \( \alpha \) and β. For fixed \( \alpha \), the mean and skewness are increasing functions of β. Also, for fixed β, the mean is an increasing function of \( \alpha \). Furthermore, the values of the skewness from Table 2 show that the distribution is very flexible in terms of shapes and the distribution can be left or right skewed.
Skewness and kurtosis of a distribution can be measured by β _{1} = μ _{3}/σ ^{3} and β _{2} = μ _{4}/σ ^{4}, respectively. However, the third and fourth moments of GC(α, β, θ) do not always exist (see Proposition 5). Alternatively, we can define the measure of asymmetry and tail weight based on quantile function. The Galton’s skewness S defined by Galton (1883) and the Moors’ kurtosis K defined by Moors (1988) are given by
When the distribution is symmetric, S = 0 and when the distribution is right (or left) skewed S > 0 (or S < 0). As K increases the tail of the distribution becomes heavier. To investigate the effect of the two shape parameters \( \alpha \) and β on the GC(α, β, θ) distribution, Eqs. (32) and (33) are used to obtain the Galtons’ skewness and Moors’ kurtosis where the quantile function is defined in Remark (ii). Figure 4 displays the Galton’s skewness and Moors’ kurtosis for the GC(α, β, 1). From this figure, it appears that the GC(α, β, θ) distribution has a wide range of skewness and kurtosis. It can be left skewed, right skewed or symmetric.
4.2 Estimation of parameters by maximum likelihood method
Let X _{1}, X _{2}, …, X _{ n } be a random sample of size n drawn from the GC(α, β, θ). The loglikelihood function is given by
where z _{ i } = 0.5 − π ^{− 1} tan ^{− 1}(x _{ i }/θ).
The derivatives of (34) with respect to \( \alpha \), β and θ are given by
where ψ(α) = ∂ log Γ(α)/∂α, is the digamma function.
Therefore, the MLE \( \widehat{\alpha}, \) \( \widehat{\beta} \) and θ are obtained by setting the Eqs. (35), (36) and (37) to zero and solving them simultaneously. Note that the number of equations can be reduced to two by using Eq. (36) to get \( \beta ={\displaystyle \sum_{i=1}^n\frac{ \log \left({z}_i\right)}{n\alpha }}. \) The initial value for the parameter θ can be taken as θ _{0} = 1. From Remark (i) in GammaCauchy{exponential} distribution, the random variable Y _{ i } = − log[0.5 − π ^{− 1} tan^{− 1}(X _{ i }/θ _{0})], i = 1, 2, …, n follows a gamma distribution with parameters α and β. Therefore, by equating the sample mean and sample variance of Y _{ i } with the corresponding population mean and variance, the initial estimates for α and β are, respectively, \( {\alpha}_0={\overline{y}}^2/{s}_y^2 \) and \( {\beta}_0={s}_y^2/\overline{y} \) where \( \overline{y} \) and \( {s}_y^2 \) are the sample mean and sample variance for y _{ i }, i = 1, …, n.
4.3 Simulation study
A simulation study is conducted to evaluate the MLE in terms of estimates and standard deviations for various parameter combinations and different sample sizes. We consider the values 0.5, 0.9, 1, 2, 5 for the parameter \( \alpha \), 0.5, 1, 3 for the parameter β, and 1, 2 for the parameter θ. The simulation study for the MLE is conducted for a total of six parameter combinations and the process is repeated 200 times. Three different sample sizes n = 50, 100, 300 are considered. The ML estimates and the standard deviations are presented in Table 3. From this table, it appears that the ML estimates of \( \alpha \) and θ tend to be overestimated. As expected, as the sample size increases, the bias and standard deviation values for all the estimates decrease.
4.4 Applications
In this section, the GC(α, β, θ) distribution is fitted to two data sets. The first data set from Bjerkedal (1960), represents the survival time in days of 72 guinea pigs infected with virulent tubercle bacilli. The first data set is
10, 33, 44, 56, 59, 72, 74, 77, 92, 93, 96, 100, 100, 102, 105, 107, 107, 108, 108, 108, 109, 112, 121, 122, 122,124,130, 134, 136, 139, 144,146, 153, 159, 160, 163, 163,168, 171, 172, 176,113, 115, 116, 120, 183,195, 196, 197, 202, 213, 215, 216, 222, 230,231, 240, 245, 251, 253, 254, 255, 278, 293, 327, 342, 347, 361,402, 432, 458, 555 
The data is skewedtothe right with skewness = 1.3134 and kurtosis = 3.8509.
The second data set from Durbin and Koopman (2001), represents the measurements of the annual flow of the Nile River at Ashwan from 1871–1970. The second data set is
1120, 1160, 963, 1210, 1160, 1160, 813, 1230, 1370, 1140, 995, 935, 1110, 994, 1020, 960, 1180, 799, 958, 1140, 1100, 1210, 1150, 1250, 1260, 1220, 1030, 1100, 774, 840, 874, 694, 940, 833, 701, 916, 692, 1020, 1050, 969, 831, 726, 456, 824, 702, 1120, 1100, 832, 764, 821, 768, 845, 864, 862, 698, 845, 744, 796, 1040, 759, 781, 865, 845, 944, 984, 897, 822, 1010, 771, 676, 649, 846, 812, 742, 801, 1040, 860, 874, 848, 890, 744, 749, 838, 1050, 918, 986, 797, 923, 975, 815, 1020, 906, 901, 1170, 912, 746, 919, 718, 714, 740 
The data is approximately symmetric with skewness = 0.3175 and kurtosis = 2.6415.
We fitted the two data sets to the GC(α, β, θ) distribution and compared the results with Cauchy, gammaPareto proposed by Alzaatreh et al. (2012) and betaCauchy distributions proposed by Alshawarbeh et al. (2013). The maximum likelihood estimates, the loglikelihood value, the AIC (Akaike Information Criterion), the Kolmogorov Smirnov (KS) test statistic, and the pvalue for the KS statistic for the fitted distributions to the data sets 1 and 2 are reported in Tables 4 and 5 respectively.
The results in Tables 4 and 5 show the GC(α, β, θ) and betaCauchy provide an adequate fit to the survival time data while the GC(α, β, θ) distribution provides the best fit (based on KS pvalue) to the annual flow of the Nile River data. The fact that GC(α, β, θ) distribution has only three parameters compared with the betaCauchy distribution makes GC(α, β, θ) a natural choice for fitting these two data sets. A closer look at the parameter estimates for the betaCauchy distribution indicates that the estimates of α, β and θ in the betaCauchy distribution are not statistically significant for the two examples. This is an indication that betaCauchy is overparameterized for fitting these two data sets. This supports the point that the threeparameter GC(α, β, θ) distribution should be used to fit the two data sets. Figure 5 displays the histogram and the fitted density functions for the two data sets, which support the results in Tables 4 and 5.
5. Concluding remarks
A family of generalized Cauchy distributions, TCauchy{Y} family, is proposed using the TR{Y} framework. Several properties of the TCauchy{Y} family are studied including moments and Shannon’s entropy. Some members of the TCauchy{Y} family are presented. A member of the TCauchy{Y} family, the gammaCauchy{exponential} distribution, is studied in detail. This distribution is interesting as it consists of exponentiated Cauchy distribution and distributions of record values of Cauchy distribution as special cases. Various properties of the gammaCauchy{exponential} distribution are studied, including mode, moments and Shannon’s entropy. Unlike the Cauchy distribution, the gammaCauchy{exponential} distribution can be rightskewed or leftskewed. Also, the moments of the gammaCauchy{exponential} distribution exist under certain restrictions on the parameters. In particular, the rth moment for the gamma Cauchy{exponential} distribution exists if and only if α, β ^{− 1} > r and this is not the case for the Cauchy distribution. The flexibility of the gammaCauchy{exponential} distribution and the existence of the moments in some cases make this distribution as an alternate to the Cauchy distribution in situations where the Cauchy distribution may not provide an adequate fit.
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Acknowledgement
The first author gratefully acknowledges the support received from the Social Fund Policy Grant at Nazarbayev University.
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The authors, AA, CL, FF and IG with the consultation of each other carried out this work and drafted the manuscript together. All authors read and approved the final manuscript.
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The authors declare that they have no competing interests.
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Alzaatreh, A., Lee, C., Famoye, F. et al. The generalized Cauchy family of distributions with applications. J Stat Distrib App 3, 12 (2016). https://doi.org/10.1186/s4048801600503
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DOI: https://doi.org/10.1186/s4048801600503