The Burr X Nadarajah Haghighi Distribution: Statistical Properties and Application to the Exceedances of Flood Peaks Data

Corresponding Authors: Hisham A.H. Elsayed Department of Statistics, Mathematics and Insurance, Ain Shams University, Egypt E-mail: dr.hisham@commerce.asu.edu.eg Abstract: In this study, new version from the Nadarajah Haghighi model is proposed. The introduced model has a failure rate function that may change in different directions. Statistical properties of the new density function are derived along with the analysis of variance, Skewness and Kurtosis. Parameter estimates are obtained by the method of maximum likelihood. Illustration of real data set was employed to measure flexibility of the new model. A simulation study was executed to test performance of the proposed version.


Introduction
Recently, a new generalization of the Exponential (E) distribution as an alternative model to the gamma (Ga), Weibull (W) and Exponentiated-Exponential (EE) distributions was proposed by Nadarajah and Haghighi (2011). The Cumulative Distribution Function (CDF) of Nadarajah and Haghighi (NH)  where,  and  are the shape and scale parameters, respectively, which are both greater than zero. Clearly, when  = 1, we have the standard Exponential (E) model. Nadarajah and Haghighi (2011) pointed out that the density function (g(x)) has the attractive feature of always having zero mode. They also showed that larger values of  in (g (x)) will lead to faster decay of the upper tail. In this study, we will refer to the proposed distribution as the Burr X Nadarajah Haghighi (BXNH) model. According to Yousof et al. (2017a), the CDF and the PDF of the BX-G family of distributions can be expressed as: with corresponding PDF:   Figure 1 shows that the new density function can take a unimodal, symmetric and right skewed shapes. Figure 2 shows that the HRF may be increasing or upside-down or decreasing or bathtub (U) or increasing then constant or constant shaped failure rate function.
Consider the series expansion: Applying the expansion in (8) to (7) where: where: Is the CDF of the ENH density with power parameter   2 2 .

Moments and Moment Generating Function
The r th ordinary moment of X is given by: Then we obtain: where: Setting r = 1 in (11), we have the mean of X: denotes the complementary incomplete gamma function, which can be evaluated in MATHEMATICA, R, etc. The variance Var(X), skewness Ske(X) and kurtosis Ku(X) measures can be calculated from the ordinary moments using well-known relationships (see subsection 3.7 ).The variance (Var(X)), skewness (Ske(X)) and kurtosis (Ku(X)) can also be calculated from the ordinary moments using well-known relationships. Here, we provide a formulae for the moment generating function (MGF) MX (t) = E (e tX ) of X. Clearly, the MGF can be derived from (9) as:

Incomplete Moments
The s th incomplete moment, say s (t), of X can be expressed from (9) as: And: The mean deviations about the mean and about the median [2 = E (|XM|)] of X are given by: (3) and 1(t) is the first incomplete moment given by (12) with s = 1. The 1(t) can be derived from (12) as: and:

Probability Weighted Moments (PWMs)
The (s, r)th PWM of X following the Burr type X generator, say s,r, is formally defined by:

Residual and Reversed Residual Life
The n th moment of the residual life, say: and it represents the waiting time elapsed since the failure of an item on condition had occurred in (0,t). The MIT of the BXNH distribution can be obtained easily by setting n = 1 in the above equation of An (t).

Stress-Strength Reliability Model
Stress-strength reliability model is the most widely approach used for reliability estimation. The stressstrength reliability model is used in many applications of physics and engineering such as system collapse and strength failure. In stress strength reliability modeling, RX2<X1 = Pr(X2<X1) is a measure of reliability of the system when it is subjected to random stress X2 and has strength X1.
The system only fails when the applied stress exceeds its strength. This means that component will be satisfied for X1>X2. Hence the performance of a system can be considered as RX2<X1 and naturally arise in electrical and electronic systems. The reliability, RX2<X1, can also be explained as the probability that the system is strong enough to defeat the stress imposed on it.

Order Statistics
Let X1,…,Xn be a Random Sample (RS) from the BXNH of distribution and let X(1),…,X(n) be the corresponding order statistics. The PDF of i th order statistic, say Xi:n, can be written as: where, B(.,.) is the beta function. Using (3), (4), (9) and (10) in equation (13) we get: The PDF of Xi:n can be expressed as: Then, the density function of the BXNH order statistics is a mixture of ENH density. Based on (14), the moments of Xi:n can be expressed as: , , 0 0 0 2 2, , 1 ,1 1,1 .

Numerical Analysis for the E(X); Var(X), Ske(X) and Ku(X) Measures
Numerical analysis for the E(X), Var(X), Ske(X) and Ku(X) are calculated in Table 1 using (10) and wellknown relationships for some selected values of parameter ,  and  using the R software. Based on Table  1 we note that: 1. The skewness of the BXNH distribution is always positive 2. The kurtosis of the BXNH distribution can be only more than 3 3. The parameter  has a xed e⁄ect on the Ske(X) and Ku(X) for all di⁄erent values of and : When = 5 and = 0:25; Ske(X) = 0:7646761 and Ku(X) = 3:892269 for any value of the parameter  . when = 2 and = 0:15; Ske(X) = 1:799314 and Ku(X) = 8:140326 for any value of the parameter  . 4. The mean of the BXNH distribution increases as increases 5. The mean of the proposed model decreases as  and  increases

Maximum Likelihood Estimation
Let x1,…,xn be a rs from BXNH distribution with parameter vector  =   ,,     . The log-likelihood function for,  say (  ), is given by:    The model selection is applied using the estimated log-likelihood 155  This data also have been applied by Lemonte (2013) for the ENH distribution. In the applications, the information about the hazard shape can help in selecting a particular model. For this aim, an important tool called the TTT plot (see Aarset (1987)) is useful. The TTT plot for the exceedances of flood peaks data in Fig. 3 denotes that the failure rate function of these data is a bathtub-shaped (U) function.
All results of this application are listed in Table 3 and 4. These results show that the OLNH distribution has the lowest values for AIC, CAIC, BIC, HQIC and K-S values and also has the biggest estimated log-likelihood and p-value for the K-S statistics among all the fitted models. Thus, it could be chosen as the best model under these criteria and compared to the other fitted models.
Based on the estimated values of parameters given in Table 3 we note that the E(X) = 12:03718; Var(X) = 155.4608, Ske(X) = 1.741001 and Ku(X) = 6.801245. Finally, we plot estimated functions for the density, CDF, P-P, Kaplan-Meier survival plots of the BXNH for the exceedances of flood peaks data in Fig. 4. Clearly, the BXNH distribution provides a closer fit to the empirical PDF and CDF. Also, from these figures, we get a bathtubshaped (U-shaped) for the estimated HRF for the exceedances of flood peaks data, which is coincide with the TTT plot given is Fig. 3.

Conclusion
In this article, a new three-parameter version of the Nadarajah Highlight (NH) model is introduced and studied. The new density can be expressed as a straightforward linear mixture of exponentiated Nadarajah Haghighi (ENH) density. It was shown that failure rate

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function of the new model can be increasing, upsidedown, decreasing, bathtub, increasing then constant and constant. Some of its statistical properties including the ordinary moments, incomplete moments, moment generating function, probability weighted moments, order statistics, moment of residual life and reversed residual life have been derived. Measures of variance, skewness and kurtosis were given by a numerical analysis. A Monte Carlo simulation study is conducted to assess the performance of the maximum likelihood method. The flexibility of the new model is illustrated by a real data set. We hope that the new distribution attract wider applications in areas such as economics (income inequality), survival and lifetime data analysis, hydrology, engineering, meteorology and others.