HALF CAUCHY EXPONENTIAL DISTRIBUTION: ESTIMATION AND APPLICATIONS Текст научной статьи по специальности «Науки о Земле и смежные экологические науки»

HALF CAUCHY - EXPONENTIAL DISTRIBUTION: ESTIMATION AND APPLICATIONS

K.Jayakumar

Department of Statistics, University of Calicut, Kerala -673 635, India [email protected]

Fasna.K

Department of Statistics, University of Calicut, Kerala -673 635, India [email protected]

Abstract

In this paper, we introduce a new two-parameter distribution called the new Half Cauchy - exponential distribution (HCE) for modeling lifetime data. The structural properties of the new distribution are discussed. Expressions for the quantiles, mode, mean deviation, and distribution of order statistics are derived. The model parameters of HCE distribution are estimated by the method of maximum likelihood, method of least square, method of Cramer-von-Mises, and Anderson-Darling methods. The existence and uniqueness of maximum likelihood estimates are proved. The importance of the new distribution is proved empirically by real-life data set.

Keywords: Half-Cauchy distribution, Method of least-squares, Method of Cramer-von-Mises, Maximum likelihood estimation, T-X family

1. Introduction

The Cauchy distribution named after Augustin Cauchy is a continuous probability distribution and is also known as the Lorentz distribution or Breit-Wigner distribution. It is also the distribution of the ratio of two independent normally distributed random variables with zero mean. Cauchy distribution is unimodal, symmetric, and bell-shaped with much heavier tails than normal distribution. It is also used for the analysis when outliers are presented in the data. Cauchy distribution has received applications in many areas including physics, mathematics, econometrics, engineering, spectroscopy, biological analysis, clinical trials, stochastic modeling of decreasing failure rate life components, queueing theory, and reliability. For more details and discussion, the reader is referred to [20], [3], and [8].

For the Cauchy distribution, the finite moments of order greater than or equal to one do not exist and hence the central limit theorem does not hold. Further, the maximum likelihood estimation of its parameters is not ideal because of no closed-form solution of the likelihood equations. The use of Edgeworth expansion to construct an accurate approximation to the sampling distribution of the maximum likelihood estimator of parameters of Cauchy distribution suggested by [1]. The method of moments is also not possible for this distribution.

Because of these facts, the Cauchy distribution serves as a counterexample for some well-accepted results and concepts in Statistics. This also makes the choice of the Cauchy distribution as an unrealistic model. That is why, modification of Cauchy distribution have been suggested in the

HALF CAUCHY - EXPONENTIAL DISTRIBUTION

K.Jayakumar, Fasna K.

literature to overcome the problem of the moments and other useful properties.

The Half-Cauchy (HC) distribution is the folded standard Cauchy distribution around the origin

so that positive values are observed.

A random variable X has the HC distribution with scale parameter a > 0, if its cumulative distribution function (cdf) is given by

Although some applications of the half Cauchy distribution exist in the literature, the fact that the finite moments of order greater than or equal to one do not exist, the central limit theorem does not hold. This fact reduces the applicability of this distribution in modeling real-life scenarios. As a heavy-tailed distribution, the HC distribution has been used as an alternative to exponential distribution to model dispersal distances by [4], as the former predicts more frequent long-distance dispersed events than the later. The HC distribution to model ringing data on two species of tits (Parus caeruleus and Parus major) in Britain and Ireland used by [6].

In the real situation, we come across non-normal data sets frequently. One usual way of dealing with non-normal data is to find a suitable transformation that makes the data more normal-like and to apply standard normal-based methods to the transformed data. Finding a suitable transformation can be difficult with data and it is often preferable to work with data without changing the original scale as the easy way of interpretation. These difficulties motivated for more-flexible parametric families of distributions to model non-normal data.

Our focus in this article is on continuous non-normal data. Because real data often deviate from normality in the tails or exhibit asymmetry in the distribution, there has been a growing interest in distributions with additional parameters regulating asymmetry and tails directly. Traditionally, log-normal or gamma distributions are used to model positively skewed data. As a viable and flexible alternative, in this study, we propose Half-Cauchy Exponential (HCE) distribution. In a number of domains such as medical applications, atmospheric sciences, microbiology, environmental science, and reliability theory among others, data are positive and right skewed. The suitable models used by researchers and practitioners to deal with this kind of data are usually parametric distributions such as log-normal, gamma, and Weibull. However, these distributions are not always enough to reach a good fit of the data. This has motivated the interest in the development of more flexible and better-adapted distributions, which have been generated using different strategies as the combination of known distributions.

In the last two decades, there has been an increased interest in defining new generators for univariate continuous distributions to model data in several areas such as engineering, actuarial, medical sciences, biological studies, demography, economics, finance, and insurance. However, in many applied areas like lifetime analysis, finance, and insurance, there is a clear need for extended forms of these distributions, that is, new distributions which are more flexible to model real data. The addition of parameters has been proved useful in exploring skewness and tail properties, and for improving the goodness-of-fit. Thus motivated in to introduce an extended form of HC distribution.

In this paper, we propose a new lifetime model using the technique introduced by [19]. A family of distributions generated by gamma random variables have introduced by [12]. This family of distributions has its cumulative distribution function (cdf) as

R(x) = n arctan ^X) > x > 0 The probability density function (pdf) corresponding to 1 is

(1)

(2)

K.Jayakumar, Fasna K. RT&A, No.3 (74) HALF CAUCHY - EXPONENTIAL DISTRIBUTION_Volume 18, September 2023

Using similiar approach [19] introduced a new family of distributions with cdf given by

r- ln[F(x)]

G(x) = 1 - r(t)dt. (4)

J a

In this paper the T-X family defined by [19] is used to create the half-Cauchy X family of distribution. Let T be a random variable having HC distribution with pdf, r(t) = na [1 + (a)2] , t > 0. Then, the pdf of the Half Cauchy - X family of distributions from equation (4) is

g(x)-

1 +| - ln(f(x))\2

naF(x)

The cdf corresponding to (6) is given by

■ - ln(F(x))

2

G(x) = 1--arctan

n

a

(5)

(6)

One of the main benefits of the Half Cauchy - X family is its ability of fitting skewed data that cannot be properly fitted by existing distributions.

The paper is organized as follows. In Section 2, we proposed Half Cauchy-Exponential (HCE) model, and discuss the shape of the density function and distribution function of the model. We derive the quantiles, mode, and Mean deviation. Analytical shapes of the reliability functions of the model under study and pdf of order statistics and their moments are derived in Section 3. In Section 4, the method of maximum likelihood estimation(MLE), method of least square (LSE), method of Cramer-von-Mises (CVME), and Anderson-Darling methods (ADE) are discussed. We explore the usefulness of the proposed distribution by means of real data set and estimation techniques are applied to calculate the model parameters in Section 5. In Section 6, concluding remarks are presented.

1

2. Half Cauchy-Exponential Distribution

In this section, we consider the case where f follows exponential distribution with parameter 9 > 0 and the cdf and pdf are respectively F(x) = 1 - e-9x and f (x) = 9e-9x; x > 0,9 > 0. The cdf and pdf of this new distribution are respectively, given by

G(x) = 1 - n arctan T - ln(1 ^^ , x > 0, 9 > 0, a > 0 (7)

and

29 e-9x

g(x) = na

1 - ln(1 - e-9x)

a

(8)

We call this new distribution Half Cauchy Exponential (HCE) distribution with parameters 9 and a. Evidently, the density function (8) does not involve any complicated function. Also, there is no functional relationship between the parameters. We denote the random variable X having pdf (8) as HCE(9,a).

The pdf plots of HCE(9, a) for various values of the parameters are presented in Figure 1. From the figure, it can be seen that the HCE distribution is well-suited for modelling right-skewed data.

The cdf plots of HCE(9, a) for various choices of the values of parameters are presented in Figure 2.

3. Properties of the Half Cauchy - Exponential Distribution

Lemma 1. The qth quantile xq of the HCE random variable is given by

xq = -1 In (1 - exp-^^) . (9)

1

-6 = 0.5, o = 0.1

\ -e = 0.9, a = 0.2

\ -9 = 0.1 ,o = 3

\ \ -6 = 1.5, a = 2.5

\ \ -9 = 7, o = 0.9

Figure 1: Plots of the pdf of HCE (6, a) distribution.

« = 0.9, o = 0.1

Figure 2: Plots of the cdf of HCE(6, a) distribution.

Proof. The qth quantile xq of the HCE random variable is defined as

q = P(X < xq) = G(xq), xq > 0 Using the distribution function of the HCE distribution, we have

q = G(xq) = 1 - — arctan ^

- ln(1 - e-6x)

u

That is,

Hence

arctan

- ln(1 - e-6x)\ _ n(1 - q)

u

xq = -1 In ( 1 - exp

-a tan

n(1-q)"

This completes the proof. ■

Using the usual inverse transformation method, random numbers can be sampled from the proposed model. Let U be a random number drawn from a uniform distribution on (0,1). Then a random number X following HCE(d, a) distribution is obtained by the equation (9) . In particular, the median is given by,

1

xo.5 = - Q ln(1 - exp a). Theorem 1. The mode of the HCE(6, a) is the solution of the equation k(x) = 0, where

- ln(1 - e-6x)'

k(x) = 2e-6x ln(1 - e-6x) - a

1 +

<j

Proof. The critical point of the HCE density function are the roots of the equation:

d log(g(x))

That is

d log(g(x)) dx

-9-

9e

-9 x

1 - e-9x

dx

+

0

29e ln(1 - e-9x)

t(1 - e-9x)

1 +( ^l-^) 2

The critical values of (11) are the solution of k(x) = 0. Hence the proof.

(11)

■

3.1. Mean Deviation

The mean deviation about the median can be used as a measure of the degree of scattering in a population. Let M be the median of the HCE distribution given by (10). The mean deviation about the median can be calculated as

/TO

|x - Mlg(x)dx,

-TO

Hence we obtain the following equation 5 = ^ - 2J(M) where J(q) is

J(q)=IL

i-q e-9x -to x 1- e-9x

1 +

- ln(1 - e-9x)

cr

dx.

(12)

One can easily compute this integral numerically in software such as MATLAB, Mathcad, R, and others and hence obtain the mean deviation about the median as desired.

1

2

3.2. Stochastic Ordering

Stochastic orders have been used during the last forty years, at an accelerated rate, in many diverse areas of probability and statistics. Such areas include reliability theory, survival analysis, queueing theory, biology, economics, insurance, and actuarial science (see, [5]). Let X and Y be two random variables having cdf's F and G respectively, and denote by F = 1 - F and G = 1 - G their respective survival functions, with corresponding pdf's f,g. The random variable X is said to be smaller than Y in the:

1. stochastic order (denoted as X <st Y) if F(x) < G (x) for all x;

2. likelihood ratio order (denoted as X <ir Y) if gxj is decreasing in x > 0;

3. hazard rate order (denoted as X <hr Y)if G^is decreasing in x > 0;

4. reversed hazard rate order (denoted as X <rhr Y) if G^ is decreasing in x > 0.

The four stochastic orders defined above are related to each other, have the following implications (see, [5]):

X <rhr Y ^ X <lr Y ^ X <hrY ^ X <st Y. (13)

The HCE is ordered with respect to the strongest likelihood ratio ordering as shown in the following theorem. It shows the flexibility of the two-parameter HCE distribution.

Theorem 2. Let X - HCE(91,a1) and Y - HCE(91,a1). If 91 = 92 = 9 and < (T?, then X <h Y hence X <rhr Y,X <hr Y and X <st Y.

Figure 3: Plots of reliability function of the HCE (6, a) distribution. Proof. The likelihood ratio is

gX(x) 02 1 + ( - ln(1-e-ex) \2 °2 J

gY (x) 01 1 + ( - ln(1-e-ex) j2

Thus,

d_ dx

gX (x) .gY(x).

29 e

-dx

1 - e

-ex

o22 + - ln(1 - e-ex) oi2 + - ln(1 - e-ex)_

Now, if e1 = e2 = e and oi < o2, then d X <rhr Y,X <hr Y and X <st Y.

gX (y) gY (y)

< 0, which implies that X <lr Y hence

■

Lemma 2. If a random variable Y follows the standard exponential distribution, then X

-a tan

-1 ln (1 - exp V 2 J J - HCE(6, a).

3.3. Reliability Analysis

The reliability function is the characterization of an explanatory that maps a set of events, usually associated with the failure of some system onto time. It is the probability that the system will survive beyond a specified time, which is defined by R(t) = 1 - G(t). The Reliability function of HCE(6, a) is given by,

2 ( - ln(1 - e-6t)'

2

R(t) = — arctan -

n V

a

(14)

The reliability behaviour of HCE(6, a) for various choices of the values of the parameters is presented in Figure 3. The other characteristic of interest of a random variable is the hazard rate function defined by

h(t) = g(t) h( ) 1 - G(t)

The hazard rate function of HCE(6, a) is given by,

2e e-

h(t)

no 1-e-e

1 +

ln(1-e-et)

2 (- ln(1-e-et)

n arctan '-^-

(15)

The behaviour of the hazard rate function of HCE(e, o) for various choices of the values of the parameters is presented in Figure 4. The cumulative hazard rate function of HCE distribution,

1

1

1

2

H(t) is given by,

Figure 4: Plots of hazard rate function of the HCE (9, t) distribution.

H(t) = - ln R(t)

, 2 ( - ln(1 - e-et)

— ln — arctan '

n

a

(16)

It is important to know that the units for H(t) are the cumulative probability of failure per unit of time, distance, or cycles.

3.4. Order Statistics

Let X1,X2,...,Xn be a random sample from HCE(9,t) . Also, let X(1),X(2),...,X(n), denote the corresponding order statistics. Then the pdf and cdf of kth order statistics, are given by

fx(x) =

n!

[G(x)]k-1 [1 - G(x)]n-kg(x)

(k - 1)!(n - k)!

(k - 1)!(n - k)!

2 (- ln(1 - e-ex)xnk-1

1--arctan

n

(j

2 (- ln(1 - e-ex)

— arctan ---1

n \ a

2e e-ex na 1 - e-ex

1 +

- ln(1 - e-ex )

u

1

(17)

and

Fx (x) = Y , j=k\ 1

Y

j=k

n!

j J (k - 1)!(n - k)!

1

2

— arctan

n

[G(x)]j[1 - G(x)]n-j - ln(1 - e-ex)

(j

2 (- ln(1 - e-ex)

— arctan ----

n \ a

n-j

(18)

respectively.

The pdf of the minimum is,

fX(!)(x)= n

2e e-ex na 1 - e-ex

2 (- ln(1 - e-ex)

— arctan ---1

n \ a

n-1

1 +

- ln(1 - e-ex )

(j

(19)

and the pdf of the maximum is,

fx(n)(x)

2e e

ex

na 1 - e-ex

1

2 (- ln(1 - e-ex)

— arctan ---1

n V a

n1

1 +

- ln(1 - e-ex )

a

n

n

n

1

1

4. Parameter Estimation

In this section, we describe the maximum likelihood estimation procedure, method of least squares, method of Cramer-von-Mises, Anderson-Darling methods, and Method of maximum product of spacings to estimate the parameters 9 and a, in the HCE distribution. We assume throughout that xi, x2,..., xn is a random sample of size n from the HCE distribution both

parameters 9 and a are unknown.

4.1. Maximum Likelihood Estimation(MLE)

Here, we consider the estimation of the unknown parameters of the new distribution by the maximum likelihood method. Consider a random sample (xi, x2,..., xn) of size n, from the HCE(9, a) distribution. Then, the log likelihood function is given by,

n n n ( /_ln(l _ e-9xV

log L = n log(29) - n log(na) - 9 £ xi - £ ln(1 - e-9xi) - £ 1 + ' (1 e )

i=1 i=1 i=1 V

The likelihood equations are,

d log L _n_ £ _ £ xie-9 _ £ xie-9xi ln(1 - e-9x) ä £ xi £ /1 _ a-9xA 2 £

and

d9 0 t[ ' =1 (1 - e-dxi ) f~{cr2 (1 - e-0xi ) (1 + - ln(l-e-0x) )2

= 0, (21)

dlogL _ _n_ £ ln(1 - e-0x)(1 - e-0xi)

0. (22)

These equations do not have explicit solutions and they have to be obtained numerically using statistical software like the nlm package in R programming.

If the parameter vector of HCE(6, a) be © = (6, a) and the associated MLE for © is © = (66, cr) , then the resulting asymptotic normality is (© - ©) ^ N(0, (I(©))-). Where the observed Fisher's information matrix (I(©)) is given by,

I (©)

_E( d2l°g L ) _E( d2 log L ) E( d02 ) E( Beda )

-E( ^ ) -E( ^ )

dOda > da2

and hence the variance covariance matrix would be I-1 (©).

As a result of MLEs' asymptotic normality, we may construct approximate 100(1 — a)% confidence intervals for O and a of HCE(O, a) as below:

O ± Za SE(O),a ± Za SE(a)

Theorem 3. Let g1(O;a, x) denote the function on the right-hand side (RHS) of equation (21), where a is the true value of the parameter. Then there exists a unique solution for g1(O; a, x) = 0, for Oe(0, <x>).

Proof. We have

n £ £ xte-e £ xie-0xi ln(1 - e-0x)

g1(0; a, x) = - - Y xi - Y —- 2 Y . ' ' ; 0 H H (1- e-exi) M a2

U i =1 (1 - e-exi) i=1 a2(1 - e-0xi) (1 + -ln(1-e-0x) )2

Now

lim g1(0;r, x) = to,

0—0

On the other hand

lim gi(0;r,, x) < 0.

0—to

Therefore there exist at least one root, say 0e(0, to) such that g1(0; r, x) = 0 To show uniqueness, the first derivative of g1 (0; r, x) = 0 is

dgi (0;<r^, x) < 0 d0 <0.

Hence there exist a solution for g1 (0; rr,, x) = 0, and root, 0 is unique. ■

Theorem 4. Let g2 (r, 0, x) = 0 denote the function on the right hand side (RHS) of equation (22), where 0 is the true value of the parameter. Then there exists a unique solution for g2(r, 0, x) = 0, for ae(0, to).

Proof. We have

n " ln(1 - e-0x)(1 - e-0xi)

Now

g2 C, 0 x) = 2 E- + )2

lim g2( a, d, x) = -to,

cr^Q

On the other hand

lim g2 (r, 0, x) > 0.

r —Yto

Therefore there exist atleast one root, say cre(0, to) such that g2(r, 0, x) = 0 To show uniqueness, the first derivative of g2(r, 0, x) = 0 is

dg2(r 0, x) < 0

dr <0.

Hence there exist a solution for g2(r, 0, x) = 0, and root, a is unique. ■

4.2. Method of Cramer-von Mises

Cramer-von-Mises type minimum distance estimators are based on minimizing the distance between the theoretical and empirical cumulative distribution functions. In [7] provided empirical evidence that the bias of these estimators is smaller than the bias of other minimum distance estimators. The Cramer-von-Mises estimators, Qqme and frcME are the values of 0 and r minimizing

1 n

a) = Tin + L

12n i=i

G(ti | 0,a) -

2i - 1 2n

2

Differentiating the above equation partially, with respect to the parameters 0 and a respectively and equating them to zero, we get the normal equations. Since the normal equations are non-linear, we can use iterative method to obtain the solution.

K.Jayakumar, Fasna K. RT&A, No.3 (74) HALF CAUCHY - EXPONENTIAL DISTRIBUTION_Volume 18, September 2023

4.3. Method of Anderson-Darling

The method of Anderson-Darling test was developed by [9] as an alternative to statistical tests for detecting sample distributions departure from normality.

The Anderson-Darling estimators 9ade and &ade are the values of 9 and r minimizes

1 n

A(9,c) = -n - - £(2i - l){logG(h | 9,r,9) - logG(tn+i-i | 9,r)}. n i=i

Differentiating the above equation partially, with respect to the parameters 9 and r respectively and them equating to zero, we get the normal equations. Since the normal equations are non-linear, we can use the iterative method to obtain the solution.

4.4. Method of Least-Square Estimation

The least-square estimators were proposed by [2] to estimate the parameters of Beta distributions. Here, we apply the same technique for the HCE distribution. The least-square estimators of the unknown parameters 0 and r of HCE distribution can be obtained by minimizing

E

i=1

G(ti | 9,a) -

n +1

with respect to unknown parameters 9 and a.

2

n

5. Applications

In this section, we have taken two real life data set to illustrate the importance of the proposed distribution. For each data set, we estimate the unknown parameters of each distribution by the maximum-likelihood method, method of least squares, method of Cramer-von-Mises, Anderson-Darling methods, and Method of maximum product of spacings.With these obtained estimates, we obtain the values of the Akaike information criterion (AIC) and Bayesian information criterion (BIC) as well as Kolmogorov-Smirnov statistic and the corresponding p-value. Here, AIC = —2 ln(L) + 2k and BIC = —2 ln(L) + k ln(n); where L is the likelihood function evaluated at the maximum likelihood estimates, k is the number of parameters and n is the sample size. The K-S distance Dn = supx|F(x) — Fn(x) |,where,Fn(x) is the empirical distribution. Kolmogorov-Smirnov (K-S) statistics is computed to compare the fitted models. The required computations are carried out in the R-language introduced by [10].

5.1. Data set I

We consider the corona-virus cases distribution among the fifteen countries viz.,France, Italy, Spain, US, Germony, UK, Turkey, Iran, Russia, China, Brazil, Canada, Belgium, Netherlands and Switzerland. Data has taken from a website and URL is https://www.worldometers.info/coronavirus/coronavirus-cases/.

Data is given in percentage and the observations are:

5.37,6.56,7.61,32.83,5.24,5.06,3.65,3.03 2.89,2.74,2.10,1.57,1.55,1.27,0.97.

The data is skewed-to-the right with skewness =3.0901 and kurtosis =8.4119

The descriptive statistics of the above data set are given in Table 1. The MLEs for 0 and are

listed in Table 2 along with their standard errors (S.E.). The values in Table 3 show that the HCE

distribution leads to a better fit for the other three models. Based on the values of the AIC and

BIC criteria as well as the value of the KS-statistic and the corresponding p-value, we observe that

the HCE distribution provides the best fit for these data among all the models considered. Table

3, it has been observed that the proposed model is best fit as compared to xgamma distribution

Table 1: The descriptive statistics of Data set.

Min 1st Q Median Mean 3rd Q Max

0.970 1.835 3.030 5.496 5.305 32.830

Table 2: S.E., MLEfor 0 and a

Parameters MLE S.E.

0 0.2359 0.0799

a 0.5618 0.5618

(XGD) by [11], Akash distribution (AKD) by [21], and exponential power distribution (EPD) by [17].

Figure 5 shows the fitted density curves, Empirical and the fitted cumulative distribution functions for the Data set I.

The goodness-of-fit of the CVME, MLE, LSE, and ADE methods are observed by the test statistic

0 10 20 30 40 0 10 20 30 40

(a) Fitted density curves for the data set I (b) Empirical and the fitted cumulative distribution functions

for the data set I

Figure 5: Histogram with fitted pdf's (left) and Empirical cdf with fitted cdf's (right) for the data set I.

values and their p-values for CVM (Cramer-Von Mises), KS (Kolmogorov-Simnorov), and AD (Anderson-Darling) for the dataset I which are displayed in Table 4.

Figure 6 shows fitted distribution's histogram and the density function having CVME, MLE, LSE, and ADE for the data set I of HCE distribution.

5.2. Data set II

The following data comes from a 59-conductor accelerated life test by [16]. Atomic movement in the circuit's conductors can create failures in microcircuits, which is known as electromigration.

Table 3: Goodness of fit for various models fitted for the Data set I.

Model -LL AIC BIC K-S(p-value)

XGD -43.555 89.111 89.819 0.2501(0.2585)

AKD -44.565 91.131 91.839 0.2687(0.1905)

EPD -42.940 89.881 91.297 0.2471(0.2709)

HCE -42.435 88.871 90.286 0.2166(0.4224)

K.Jayakumar, Fasna K. RT&A, No.3 (74) HALF CAUCHY - EXPONENTIAL DISTRIBUTION_Volume 18, September 2023

Table 4: Statistics values and their associated p-valuesfor the dataset I.

Estimation method Estimates K-S(p-value) CVM(p-value) AD(p-value)

CVME 0.4402 0.2360 0.1398(0.8924) 0.0446(0.9144) 0.9312(0.3937)

MLE 0.2359 0.5618 0.2166(0.4224) 0.1210(0.4962) 1.0133(0.349)

ADE 0.3529 0.3219 0.1619(0.7698) 0.0570(0.84) 0.8506(0.4438)

LSE 0.3918 0.2866 0.1553(0.8101) 0.0480(0.8951) 0.8731(0.4292)

Figure 6: fitted distribution's histogram and the density function having CVME, MLE, LSE, and ADEfor the data set I.

There are no censored observations, and the failure times are in hours.

5.923, 4.288, 6.522, 4.137, 6.071, 7.495, 6.573, 6.538, 5.589, 6.087, 5.807, 6.725, 8.532, 9.663, 6.545, 10.491, 7.543, 6.956, 6.492, 5.459, 8.120, 4.706, 8.687, 2.997, 8.591, 6.129,11.038, 5.381,10.092, 7.496, 4.531, 7.974, 8.799, 7.683, 7.224, 7.365, 6.923, 5.640, 5.434, 7.937, 6.515, 6.476, 6.369, 7.024, 8.336, 9.218, 7.945, 6.869, 6.352, 4.700, 6.948, 9.254, 5.009, 7.489, 7.398, 6.033,7.459, 9.289 , 6.958. The data is skewed-to-the right with skewness =0.1932 and kurtosis =0.0874 The descriptive statistics of the above data set are given in Table 5. The MLEs for d and a are listed in Table 6 along with their standard errors (S.E.). The values in Table 7 show that the HCE distribution leads to a better fit for the other five models. Based on the values of the AIC and BIC criteria as well as the value of the KS-statistic and the corresponding p-value, we observe that the HCE distribution provides the best fit for these data among all the models considered. Table 7, it has been observed that the proposed model is best fit as compared to Lindley-Exponential (LE) model by [13], generalized exponential (GE) model by [14], modified Weibull (MW) model by [15], exponential power (EP) model by [17], and Weibull extension (WE) model by [18].

Figure 7 shows the fitted density curves, Empirical and the fitted cumulative distribution functions for the Data set II.

The goodness-of-fit of the CVME, MLE, LSE, and ADE methods are observed by the test statistics values and their p-values for CVM (Cramer-Von Mises), KS (Kolmogorov-Simnorov), and AD (Anderson-Darling) for the dataset II which are displayed in Table 8.

Table 5: The descriptive statistics of Data set.

Min 1st Q Median Mean 3rd Q Max

2.997 6.052 6.923 6.980 7.941 11.038

Table 6: S.E., MLEfor 0 and a

Parameters MLE S.E.

0 0.9352 0.1117

a 0.0015 0.0012

Table 7: Goodness of fit for various models fitted for the Data set.

Model -LL AIC BIC K-S p-value

EP -116.5015 237.0029 237.2098 0.1042 0.5103

LE -114.9528 233.9055 234.1198 0.1042 0.5099

GE -114.9473 233.8946 234.1098 0.1365 0.2021

WE -113.6745 233.3491 233.7855 0.1067 0.4796

MW -112.5218 231.0435 231.4799 0.0914 0.6738

HCE -111.7792 227.5584 231.713 0.05806 0.9819

(a) Fitted density curves for the data set II (b) Empirical and the fitted cumulative distribution functions

for the data set II

Figure 7: Histogram with fitted pdf's (left) and Empirical cdf with fitted cdf's (right) for the data set II.

Figure 8 shows fitted distribution's histogram and the density function having CVME, MLE, LSE, and ADE for the data set II of HCE distribution.

6. Concluding remarks

In this article, we have introduced and studied a new family of distributions called the Half Cauchy exponential(HCE) distribution. we have provided explicit expressions for the quantiles, hazard rates, mean deviation about median, the stochastic ordering and order statistics. The model parameters are estimated by maximum likelihood, least-squares, Cramer-von Mises, and Anderson-Darling. Our formulas related to the HCE model are manageable, and with the use of modern computer resources with analytic and numerical capabilities, may turn into adequate tools for a certain purpose of statisticians.

The applicability of the model is demonstrated by using two real data set. From Tables 3 and 7, we observed a better performance of our distribution than the existing distributions. Because HCE distribution has the least test statistic value and the highest p value, we can deduce that it has a considerably better fit than the other distributions studied. Based on these findings, the newly suggested model can be considered as a more efficient, flexible, and therefore may be an alternative to other distributions for modeling positive real data sets. Our proposed model may attract wider applications in survival analysis for modeling positive real data sets. Estimation of

Table 8: Statistics values and their associated p-valuesfor the dataset II.

Estimation method Estimates K-S(p-value) CVM(p-value) AD(p-value)

CVME 0.9055 0.0019 0.0563(0.9867) 0.0233(0.9932) 0.1667(0.997)

MLE 0.9352 0.0015 0.0580(0.9819) 0.0246(0.9908) 0.1798(0.995)

ADE 0.9057 0.0018 0.05598(0.9876) 0.0232(0.9932) 0.1672(0.9969)

LSE 0.8797 0.0022 0.0633(0.9596) 0.0279(0.9832) 0.1800(0.995)

Figure 8: fitted distribution's histogram and the density function having CVME, MLE, LSE, and ADEfor the data set II.

the model parameters under the Bayesian paradigm is currently underway.

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