Some properties and applications of half-Cauchy exponential extension distribution

In this section we have taken the half-Cauchy distribution which is the special case of the Cauchy distribution by breakdown of curve on origin such that it will consider only non-negative values. [Shaw (1995)] has used the half Cauchy distribution having a heavy-tailed, as an alternative to model spreading distances, since it can forecast more recurrent long-distance spreading events. In addition, the half-Cauchy distribution is also used by [Paradis et al. 2002] [21] to model ringing data on tits having two species in Ireland and Britain. Let us consider that a no-negative random variable X follows half-Cauchy distribution and its cumulative distribution function (CDF) is expressed as

Here, we are interested to generate new model based on half-Cauchy family. The generating family of distribution was given by [Zografas & Balakrishnan, 2009] and CDF of family of distribution was obtained as

Here () Gx is the CDF of any baseline distribution and () rt is the PDF of any distribution. The family of half-Cauchy with CDF is defined taking () rt which is the PDF of earlier defined half-Cauchy distribution defined in (2) as,

The PDF corresponding to (4) can be expressed as,   1 2

2

( ) 1 ( ) 1 log 1 ( ) ; 0, 0 1 ( )

The rest part of study is organized in different sections. In Section 2, model analysis where, the half Cauchy exponential extension distribution is defined and also we have presented the properties of the proposed model like survival, probability density, hazard rate, cumulative distribution, and cumulative hazard functions. Quantiles, measures of variability, measure of symmetry of the data based on quantiles along with measure of normality based on octiles. In Section 3, parameter estimation techniques are used.

The application of the proposed model to real data set is presented in Section 4. In section 5 model comparisons is discussed and finally some concluding explanations are entered in Section 6.

Model Analysis The Half Cauchy Exponential Extension (HCEE) distribution:

The extension of the exponential distribution was defined by Nadarajah & Haghighi (2011) [19] . We named it as exponential extension distribution. The CDF of exponential extension distribution is defined as

The PDF corresponding to ( 6) is

Substituting ( 6) and ( 7) in ( 4) and ( 5) we obtained CDF of half-Cauchy exponential extension distribution, which is defined as,

And the PDF of half-Cauchy exponential extension can be expressed as

Survival function

The survival function of HCEE is,

Reverse hazard function of HCEE

The reverse hazard function of HCEE is as

Cumulative Hazard Function:

The chf of the HCEE   ,,

The various shapes of pdf and hazard rate function of HCEE   ,,    with various possible parameter values are displayed in Figure 1.

Quantile function

Let X is a non-negative random variable with CDF   X Fx then quantile function can be defined as,

The random deviate generation for the HCEE   ,,

Skewness and Kurtosis

The Quantile coefficient of measure of symmetry is,

and Expression for testing normality by using octiles of the data is

Parameter Estimation 3.1. Maximum Likelihood Estimation

In this subsection, we have presented MLE method of the HCEE model to estimate the parameters. Let

First order derivative of ( 16) with respect to β, λ and θ are

and solving simultaneously for the β, λ and θ we can obtain the MLE of the HCEE   ,,

Normally, it is not possible to solve above non-linear equations, so with the aid of suitable programming language, we can solve them easily. Let ˆˆ( , , )

    then resulting asymptotic normality will be

and corresponding Fisher's information matrix is,

Since  is unknown so it is worthless that the MLE has an asymptotic variance    

By using Newton-Raphson for optimization of the log-likelihood function we can get variance and covariance matrix as,

Var( ) Cov( , ) ˆˆˆˆĈ ov( , ) Cov( , )

Least-Square Estimation

Use of least-square estimation for estimating the unknown parameters β, λ and θ of HCEE distribution minimizes the function

with respect to β, λ and θ.

be a random sample with n units from ordered random variable      

Differentiating the relation (3.3.4) with respect to β, λ and θ as,

We can minimize following relation to get weighted least square estimators as

where w i are weight and is given as

By minimizing relation (20) we can estimate unknown parameters β, λ and θ using weighted least square method with respect to β, λ and θ.

Cramer-Von-Mises estimation

By minimizing relation (21) we can get estimated values of unknown parameters β, λ and θ using Cramer-Von-Mises technique of estimation

Differentiation of ( 21) with respect to β, λ and θ will give following results, 1) log(1 ) 1 1 (1 )

After solving above non-linear equations = 0, = 0 and 0

we can get the CVM estimators of the parameters.

Application to Real Dataset

In this part of the study, applicability and validity of HCEE model taking a real data set is explained. The data is the breaking stress of carbon fibers having 50 mm length (GPa). The data was used by (Nichols & Padgett, 2006) [20] . The data is used previously in many researchers. The dataset is given as 0.39, 0.85, 1.08, 1. 2 and the profile graphs show that ML estimates can be uniquely calculated. [14] . We got the value of Log-Likelihood as l = -85.3059. Estimated parameters value with their standard errors (SE) for β, λ and θ and 95% asymptotic confidence interval (ACI) in Table 1. We have also plotted the P-P plot (left) and Q-Q plot (right) for testing the normality of model Using above estimated parameter values by all three methods we have calculated log-likelihood and AIC criterion and are presented in Table 2. For testing the goodness of fit of the model, the KS, W and A 2 statistic along with respective p-value of MLE, LSE and CVE estimates are presented in Table 3 Table 3: The KS, W and A 2 statistics and p-value

Method

KS(p-value) W(p-value) A 2 (p-value

Model Comparison

Here we have compared the applicability of the model HCEE with other models used by researchers using the same data set. Here we have chosen five distributions to compare the potentiality of the HCEE. Five distributions considered are; Exponentiated Exponential Poisson (EEP), Generalized Exponential Extension, (GEE), Power Cauchy (PC) distribution, Generalized gompertz distribution, and Exponentiated Weibull Distribution (EW). We also displayed the histogram of the dataset, graph for goodness-of-fit of HCEE model along with models taken in considerations in left panel of Figure 5. In right panel of the graph, empirical CDF and fitted CDF are shown. From plotting it is observed that HCEE distribution fits the dataset better than other models taken in consideration. We have illustrated the Akaike information criterion (AIC), Bayesian information criterion (BIC), Corrected Akaike information criterion (CAIC), and Hannan-Quinn information criterion (HQIC) for the evaluation of the applicability of HCEE distribution. These values are presented in Table 4. Validation criteria obtained clearly show that model HCEE has fewer values than the value of other considered models. This suggests that proposed model fit data better than other models taken in considerations.

Models for comparison

Following are the PDF of the models taken in consideration for comparisons of the proposed models

(i). Exponentiated Exponential Poisson (EEP)

The probability density function of EEP (Ristić & Nadarajah, 2014) [22] is

(ii). Generalized Exponential Extension (GEE)

The pdf of GEE was introduced by (Lemonte, 2013) [13] . Distribution has upside down bathtub shaped hazard rate function having three parameters as

Generalized gompertz distribution

The probability density function of Gompertz distribution (El-Gohary et al., 2013) [9] having three parameters as α, λ and θ is

Exponentiated Weibull (EW)Distribution

A new family of distributions, namely the Exponentiated Exponential Distribution was introduced by Gupta et al. (1998) [26] . EW is a generalization of the Exponentiated Exponential Family as well as the Weibull Family. EW distribution also has a very nice physical interpretation. We have also plotted the graph of the empirical CDF and the estimated fitted CDF to illustrate the goodness of fit of the HCEE model in figure 6.

Fig 6:

The plot of empirical CDF and estimated CDF of HCEE model.

Conclusion

Here, a new distribution named half Cauchy exponential extension distribution is presented. A detailed analysis of some basic statistical characteristics of new model is done. We have derived some important expression for the hazard rate, survival, the quantile function etc. We have also calculated summary of the data as mean, median skewness and kurtosis etc and are For estimating the unknown parameters, three important and more effective methods of estimation as MLE, LSE and CVME are used. Here we found that the CVM method is relatively better than MLE and LSE methods. PDF curve of the proposed model HCEE has different shaped density curve for different values of parameters showing that curve is positively skewed having increasing-decreasing hazard rate curve. Here we have also considered a real dataset and compared the applicability and suitability of the proposed model with some other models introduced by the earlier researchers using same dataset and same methods of parameter estimation. It is found that the proposed model fits data more adequately than the other models taken in considerations. We also considered different methods of model validation techniques to test the validity showing that proposed model has lesser values of information criteria than competing models. So we can conclude that model HCEE will play significant role in data analysis and new model formulations.

References

1. Abdulkabir M, Ipinyomi RA. Type ii half logistic exponentiated exponential distribution: properties and applications. Pakistan Journal of Statistics. 2020;36(1).
2. Abouammoh AM, Alshingiti AM. Reliability estimation of generalized inverted exponential distribution. Journal of Statistical Computation and Simulation. 2009;79(11):1301-1315.
3. Afify AZ, Cordeiro GM, Yousof HM, Alzaatreh A, Nofal ZM. The Kumaraswamy transmuted-G family of distributions: Properties and applications. Journal of Data Science. 2016;14(2):245-270.
4. Barreto-Souza W, Santos AH, Cordeiro GM. The beta generalized exponential distribution. Journal of statistical Computation and Simulation. 2010;80(2):159-172.
5. Chaudhary AK, Sapkota LP, Kumar V. Truncated Cauchy power-exponential distribution: Theory and Applications. IOSR Journal of Mathematics (IOSR-JM). 2020;16(6):44-52.
6. Cordeiro GM, de Castro M. A new family of generalized distributions. Journal of statistical computation and simulation. 2011;81(7):883-898.
7. Cordeiro GM, Alizadeh M, Ortega EMM. The exponentiated half-logistic family of distributions: Properties and applications, J. Probab. Stat, 2014, 21.
8. El-Gohary A, Alshamrani A, Al-Otaibi AN. The generalized Gompertz distribution. Applied Mathematical Modelling. 2013;37(1-2):13-24.
9. Gomez YM, Bolfarine H, Gomez HW. A new extension of exponential Distribution. Rev. Colombiana do Estadistic. 2014;37(1):25-34.
10. Gupta RD, Kundu D. Generalized exponential distribution: Existing results and some recent developments. Journal of Statistical planning and inference. 2007;137(11):3537-3547.
11. Kumar V. Bayesian analysis of exponential extension model. J Nat. Acad. Math. 2010;24:109-128.
12. Lemonte AJ. A new exponential-type distribution with constant, decreasing, increasing, upside-down bathtub and bathtubshaped failure rate function. Computational Statistics & Data Analysis. 2013;62:149-170.
13. Louzada F, Marchi V, Roman M. The exponentiated exponential geometric distribution: a distribution with decreasing, increasing and unimodal failure rate. Statistics: A Journal of Theoretical and Applied Statistics. 2014;48(1):167-181.
14. Mailund T. Functional Programming in R: Advanced Statistical Programming for Data Science, Analysis and Finance. Apress, Aarhus N, Denmark; 2017. ISBN-13 (pbk). 978-1-4842-2745-9 ISBN-13 (electronic): 978-1-4842-2746-6 DOI 10.1007/978-1-4842-2746-6
15. Merovci F. Transmuted exponentiated exponential distribution. Mathematical Sciences And Applications E-Notes. 2013;1(2):112-122.
16. Moors J. A quantile alternative for kurtosis. The Statistician. 1988;37(1):25-32.
17. Nadarajah S, Kotz S. The beta exponential distribution. Reliability Engineering and System Safety. 2006;91(6):689-697.
18. Nadarajah S, Haghighi F. An extension of the exponential distribution. Statistics. 2011;45(6):543-558.
19. Nichols MD, Padgett WJ. A bootstrap control chart for Weibull percentiles. Quality and reliability engineering international. 2006;22(2):141-151.
20. Paradis E, Baillie SR, Sutherland WJ. Modeling large-scale dispersal distances. Ecological Modelling. 2002;151(2-3):279292.
21. Ristić MM, Balakrishnan N. The gamma-exponentiated exponential distribution. Journal of statistical computation and simulation. 2012;82(8):1191-1206.
22. Ristić MM, Nadarajah S. A new lifetime distribution. Journal of Statistical Computation and Simulation. 2014;84(1):135-150.
23. Shaw MW. Simulation of population expansion and spatial pattern when individual dispersal distributions do not decline exponentially with distance. Proceedings of the Royal Society B: Biological Sciences. 1995;259(1356):243-248.
24. Gupta RC, Gupta PL, Gupta RD. Modeling failure time data by Lehman alternatives. Communications in Statistics-Theory and methods. 1998;27(4):887-904.