On the Burr XII-Power Cauchy Distribution: Properties and Applications

We propose a new four-parameter lifetime model with flexible hazard rate called the Burr XII Power Cauchy (BXII-PC) distribution. We derive the BXII-PC distribution via (i) the T-X family technique and (ii) nexus between the exponential and gamma variables. The new proposed distribution is flexible as it has famous sub-models such as Burr XII-half Cauchy, Lomax-power Cauchy, Lomaxhalf Cauchy, Log-logistic-power Cauchy, log-logistic-half Cauchy. The failure rate function for the BXII-PC distribution is flexible as it can accommodate various shapes such as the modified bathtub, inverted bathtub, increasing, decreasing; increasing-decreasing and decreasing-increasing-decreasing. Its density function can take shapes such as exponential, J, reverse-J, left-skewed, right-skewed and symmetrical. To illustrate the importance of the BXII-PC distribution, we establish various mathematical properties such as random number generator, moments, inequality measures, reliability measures and characterization. Six estimation methods are used to estimate the unknown parameters of the proposed distribution. We perform a simulation study on the basis of the graphical results to demonstrate the performance of the maximum likelihood, maximum product spacings, least squares, weighted least squares, Cramer-von Mises and Anderson-Darling estimators of the parameters of the BXII-PC distribution. We consider an application to a real data set to prove empirically the potentiality of the proposed model.


Introduction
Data analysis is imperative in every aspect of statistical analysis. The statistical characteristics such as skewness, kurtosis, bimodality, monotonic and non-monotonic failure rates are obtained from datasets. The selection of a suitable model for data analysis is challenging task because it depends on the nature of the dataset. However, if a wrong model is applied to analyze the dataset it leads to loss of information and invalid inferences. It is necessary to search and identify the most suitable model for the given dataset.
measures, conditional moments, reliability measures and characterization. Section 4 is devoted to parameter estimation methods. Section 5 presents simulation studies on the basis of graphical results to see the performance of maximum likelihood, maximum product spacings, least squares, weighted least squares, Cramer-von Mises and Anderson-Darling estimators of the BXII-PC distribution. In Section 6, we consider an application to illustrate the potentiality and utility of the BXII-PC model. We test the competency of the BXII-PC model via various model selection criteria. In Section 7, we offer some conclusions.

The BXII-PC distribution
We derive the BXII-PC distribution from the T-X family technique. We also obtain this model by linking the exponential and gamma variables. We discuss basic structural properties. We highlight the shapes of the density and failure rate functions.
The cumulative hazard rate function of the Power Cauchy distribution is The cdf of the T-X family [19] of distributions has the form where r (t) is the pdf of the random variable (rv) T , where T ∈ [a, b] for −∞ ≤ a < b < ∞ and W G (x; ξ) is a function of the baseline cdf of a rv X with the vector parameter ξ, which satisfies the conditions: i) W G (x; ξ) ∈ [a, b] , ii) W G (x; ξ) is differentiable and monotonically non-decreasing and iii) lim The pdf of the T-X family can be expressed as We derive the cdf of the BXII-PC distribution from the T-X family technique by setting The cdf of the BXII-PC distribution takes the form 1 β 1 κ .

Shapes of the BXII-PC density and hazard rate functions
We plot the density and failure rate functions of the BXII-PC distribution for selected parameter values. The BXII-PC density can display numerous shapes such as symmetrical, right-skewed, leftskewed, J, reverse-J and exponential (as Figure 1). The failure rate function can highlight shapes as modified bathtub, inverted bathtub, increasing, decreasing; increasing-decreasing and decreasingincreasing-decreasing (as Figure 2). Therefore, the BXII-PC distribution is quite flexible and can be applied to numerous data sets.

Mathematical properties
Here, we present certain mathematical and statistical properties such as the ordinary moments, incomplete moments, inequality measures, conditional moments, reliability measures and characterization.

Moments
The moments are significant tools for statistical analysis in pragmatic sciences. The r th ordinary moment of X, say µ r = E (X r ), can be expressed from (2.4) as we have The following power series ( [21]) can be obtained from Mathematica where a 0 (s) = 1, a 1 (s) = −s/3, a 2 (s) = s (5s − 7) /90, etc.
Using this expression we have Letting w β = y, w = y where αβ > j and B (., .) is the beta function.

Conditional moments
Life expectancy, mean waiting time and inequality measures can be obtained from the incomplete moments. The sth incomplete moment for the BXII-PC distribution is The mean deviation about the mean (δ 1 = E |X − µ|) and about the median (δ 2 = E |X −μ|) can be written as The quantities M 1 (µ) and M 1 (μ) can be obtained from (3.2). For specific probability p, Lorenz and Bonferroni curves are computed as L(p) =
and for X 2 ∼ BXII − PC (α 2 , β, κ, θ), Thus Therefore for the BXII-PC distribution, random variable X 1 is said to be smaller than a random variable X 2 in likelihood ratio order

Reliability estimation of multicomponent stress-strength model
Consider a system that has m identical components out of which s components are functioning. The strengths of m components are X i , i = 1, 2, ..., m with common cdf F while, the stress Y imposed on the components has cdf G. The strengths X i , i = 1, 2, ..., m and stress Y are i.i.d. The probability that the system operates properly is reliability of the system i.e. Let X ∼ BXII − PC (α 1 , β, κ, θ) and Y ∼ BXII − PC (α 2 , β, κ, θ) with common parameters β, κ, θ and unknown shape parameters α 1 and α 2 . The reliability that the system operates properly in multicomponent stress-strength for the BXII-PC distribution is which is independent of the parameters β, κ and θ.

Characterizations based on truncated moment of a function of the random variable
In this subsection, we first present a characterization of the BXII-PC distribution in terms of a simple relationship between truncated moment of a function of X and another function. This characterization result employs a version of a theorem due to [24]; see Theorem 7.1 of Appendix A. Note that the result holds also when the interval H is not closed. Moreover, as mentioned above, it could be also applied when the cdf F does not have a closed form. As shown in [25], this characterization is stable in the sense of weak convergence.
where D is a constant. Note that a set of functions satisfying the above differential equation is given in Proposition 3.5.1 with D=0. However, it should also be noted that there are other pairs (η, q) satisfying conditions of Theorem 7.1.

Different estimation methods
In this section, we propose various estimators for estimating the unknown parameters of the BXII-PC distribution. We discuss maximum likelihood, maximum product spacings, least squares, weighted least squares, Cramer-von Mises and Anderson-Darling estimation methods and compare their performances on the basis of a simulated sample from the BXII-PC distribution. The details are as follows.

Maximum product spacing estimates
The maximum product spacing (MPS) method is an alternative method to MLE for parameter estimation. This method was proposed by [26,27] as well as independently developed by [28] as an approximation to the Kullback-Leibler measure of information. This method is based on the idea that differences (spacings) between the values of the cdf at consecutive data points should be identically distributed. Let X (1) , X (2) , . . . , X (n) be ordered sample of size n from the BXII-PC distribution. The geometric mean of the differences is given by where, the difference D i is defined as The maximum product spacing (MPS) estimates, sayα MPS ,β MPS ,θ MPS andκ MPS , of α, β, θ and κ are obtained by maximizing the geometric mean of the differences. Substituting cdf of BXII-PC distribution in Eq (4.2) and taking logarithm of the above expression, we have where, F x (0) = 0 and F x (n+1) = 1. The MPSEsα MPS ,β MPS ,θ MPS andκ MPS are obtained by maximizing MPS (ξ).

Least squares estimates
Let X (1) , X (2) , . . . , X (n) be ordered sample of size n from the BXII-PC distribution. Then, the expectation of the empirical cumulative distribution function is defined as The least square estimates (LSEs) say,α LSE ,β LS E ,θ LS E andκ LS E , of α, β, θ and κ are obtained by minimizing (4.4)

Anderson-Darling estimation
This estimator is based on Anderson-Darling goodness-of-fits statistics which was introduced by [29]. The Anderson-Darling (AD) minimum distance estimates,α AD ,β AD ,θ AD andκ AD , of α , β , θ and κ are obtained by minimizing

Simulation experiments
In this Section, we perform the simulation studies by using the BXII-PC to see the performance of the above estimators corresponding to this distribution and obtain the graphical results. We generate N=1000 samples of size n=20, 30, . . . , 800 from the BXII-PC distribution with true parameter values α = 15, β = 5, θ = 0.5 and κ = 2. The random numbers generation is obtained by its quantile function. In this simulation study, we calculate the empirical mean, bias and mean square errors (MSEs) and the mean relative estimates (MREs) of all estimators to compare in terms of their biases, MSEs and MREs with varying sample size. The empirical bias, MSE and MRE are calculated by (for h = α, β, θ, κ) respectively. We expect that the empirical means are close to true values. MREs are closer to one when the MSEs and biases are near zero. All results related to estimations were obtained using optim-CG routine in the R programme.
The results of this simulation study are shown in Figures 3-6. These figures show that all estimators are to be consistent, since the MSE and biases decrease with increasing sample size and the values of MREs tend to one as expected. It is clear that the estimates of parameters are asymptotically unbiased. For all parameters estimations, the performances of all estimators are close except the MPS method.

Application of the BXII-PC distribution
We consider an application to successive failures of the air conditioning system [31] for authentication the flexibility, utility and potentiality of the BXII-PC distribution.  Table 2 reports the MLEs, their standard errors (in parentheses) and goodness of fit statistics such as W*, A*, KS (p-values). Table 3 displays the values of −2ˆ , AIC, CAIC, BIC and HQIC.
We infer from the Tables 2 and 3 that BXII-PC distribution is best model, with the smallest values for all criteria of goodness of fit statistics (except BIC). Figure 7 infers that the BXII-PC distribution is best fitted to empirical data.

Conclusions
We propose a new probability distribution, named BXII-PC distribution, based on Cauchy and Burr XII distribution via T-X family method. Its pdf and hrf shapes are seen as very flexible forms. To illustrate the importance of the BXII-PC distribution, we establish various mathematical properties such as random number generator, sub-models, moments related properties, inequality measures, reliability measures and characterizations. We estimate the model parameters by six different methods. We perform a simulation study on the basis of graphical results to evaluate the performance of maximum likelihood, maximum product spacings, least squares, weighted least squares, Cramer-von Mises and Anderson-Darling estimators of the BXII-PC distribution. We demonstrate the potentiality and utility of the BXII-PC distribution by considering an application to successive failures of the air conditioning system. We apply various model selection criteria and graphical tools to examine the adequacy of the proposed distribution. We infer that the BXII-PC model is empirically suitable for the lifetime applications (successive failures analysis). Therefore, the BXII-PC model is a flexible, reasonable and parsimonious to other existing distributions. Hence it should be included in the distribution theory to assist the researchers. Further, as perspective of future projects, we may consider several intensive subjects (i) unit BXII-PC; (ii) Burr III-PC; (iii) log-Burr XII-Power Cauchy regression; (iv) various characteristics of the bivariate and the multivariate extensions of the BXII-PC; (v) Bayesian estimation of the BXII-PC parameters via complete and censored samples under different loss functions and (vi) the study of the complexity of the BXII-PC via Bayesian methods.
Appendix A Theorem 7.1. Let (Ω, F, P) be a given probability space and let H = [a 1 , a 2 ] be an interval with a 1 < a 2 (a 1 = −∞, a 2 = ∞). Let X : Ω → [a 1 , a 2 ] be a continuous random variable with distribution function F and Let g (x) be a real function defined on H = [a 1 , a 2 ] such that E[g (X)| X ≥ x] = h (x) for x ∈ H is defined with some real function h (x) should be in simple form. Assume that g (x) ε C ([a 1 , a 2 ]), h (x) ε C 2 ([a 1 , a 2 ]) and F is twofold continuously differentiable and strictly monotone function on the set [a 1 , a 2 ]. We conclude, assuming that the equation g (x) = h (x) has no real solution in the inside of [a 1 , a 2 ]. Then F is obtained from the functions g (x) and h (x) as F (x) =