Marshall-Olkin Alpha Power Rayleigh Distribution: Properties, Characterizations, Estimation and Engineering applications

In this paper, we introduce a new there-parameter Rayleigh distribution, called the Marshall-Olkin alpha power Rayleigh (MOAPR) distribution. Some statistical properties of the MOAPR distribution are obtained. The proposed model is characterized based on truncated moments and reverse hazard function. The maximum likelihood and bootstrap estimation methods are considered to estimate the MOPAR parameters. A Monte Carlo simulation study is performed to compare the maximum likelihood and bootstrap estimation methods. Superiority of the MOAPR distribution over some well-known distributions is illustrated by means of two real data sets.


Background
In many applied sciences such as medicine, insurance and engineering, among others, modeling and analyzing life testing of data is crucial. The statistical distributions have been used to describe real life phenomena and considerable effort has been expended in the development of the large classes of standard probability distributions and generating new flexible distributions. Many classes of distributions have been developed to describe different phenomena. The cumulative distribution function (cdf) and probability density function (pdf) of Rayleigh distribution are given by G(y; λ) = 1 − e − y 2 2λ 2 , and g(y; λ) = y λ 2 e − y 2 2λ 2 , respectively. Marshall and Olkin (1997) have suggested a new wider class of distributions called the extended Marshall-Olkin generated (MO-G) family. For any baseline cdf, F(y), the cdf and pdf of the MO-G have the forms F (y; θ) = G(y) θ + (1 − θ)G(y) , , θ > 0, y ∈ , and f (y; θ) = θg(y) respectively. The MO-G family offer a wide range of behavior than basic distribution from which they are derived. For more details, see Ghitany (2005), Ghitany and f (y; α) = ln(α) α − 1 α G(y) g(y); α = 1, α > 0, y ∈ , and f (y; α, θ) = θ ln(α) α − 1 α G(y) g(y) In this paper, we study a new three-parameter distribution, called Marshall-Olkin alpha power Rayleigh (MOAPR) distribution which extends the Rayleigh distribution and provides more flexibility in modeling engineering data. The rest of this paper is organized as follows: The MOAPR distribution is defined in Section 2. Some of its statistical properties are given in Section 3. We provide some characterizations of the MOAPR model in Section 4. The model parameters are estimated via the maximum likelihood and bootstrap methods in Section 5. In Section 6, the potentiality of the estimation approaches is assessed via simulation results. In Section 7, two applications to real data are discussed. Finally, some remarks are offered in Section 8.

The MOAPR Distribution
In this section, we will introduce a new extension of Rayleigh distribution by using the MOAP-G family and some of its special sub-models. The MOAP-G family and Rayleigh distribution have been used to generate the MOAPR distribution. It is represented by the random variable Y ∼ M OAP R(α, θ, λ). From Equations (7, 8, 1 and 2), the cdf of the MOAPR is given by Its pdf takes the form  The MOAPR distribution contains some special cases which are listed in Table 1. The survival function of the MOAPR distribution reduces to S(y; α, θ, λ) =   The failure rate function of the MOAPR distribution has the form f r(y; α, θ, λ) = (α − 1) ln(α)α 1−e − y 2 2λ 2 y

Statistical Properties
In this section, we discuss some statistical properties of the MOAPR distribution such as quantile function, moments, moment generating function and related measures, and stress-strength model.

Quantiles Function
By inverting Equation (9), we have the quantile function of the MOAPR model From Equation (13), we can obtain the median (M) function or second quartile of the MOAPR distribution when q=0.5 as follow From Equations (14 and 13), we can obtain the Galton skewness (SK).

Moments
If the random variable Y is distributed as MOAPR distribution, then its r th moments around zero can be expressed as follows where From Equation (15) with r = 1, we obtain The mean of a MOAPR random variable is: The variance of the MOAPR random variable is:

Moment Generating Function
The moment generating function of the MOAPR distribution is given by where erf (y) is the Gauss error function: erf (y) = 2 π y 0 e −t 2 dt. The quantile function and moments are used to calculate the mean, median, variance, skewness and kurtosis of Y for different values of the parameters of the MOAPR distribution. The numerical values of these measures are computed by using the R program and are displayed in Table 2.  Table 2 shows that for fixed λ and θ, the mean, median and variance are increasing functions in α, while the skewness and kurtosis are decreasing functions in α. Also, for fixed α and θ, the mean, median and variance are increasing functions in λ, while the skewness and kurtosis are decreasing functions in λ. Further, for fixed λ and α, the mean, median and variance are increasing functions in θ, while the skewness and kurtosis are decreasing functions in θ.

Stress-Strength Model
Let X and Y be the independent strength and stress random variables observed from the MOAPR distribution, then the stress-strength reliability R is

Characterization Results
This section is devoted to the characterizations of the MOAPR distribution in two directions : (i) based on a relationship between two truncated moments and (ii) in terms of the reverse hazard function. We present our characterizations (i) and (ii) in two subsections.

Characterizations Based on Truncated Moments
In this subsection we present characterizations of MOAPR distribution in terms of a simple relationship between two truncated moments. The first characterization result employs a theorem due to Glänzel [1] , see Theorem 1 of Appendix A. Note that the result holds also when the interval H is not closed. Moreover, it could be also applied when the cdf F does not have a closed form. As shown in [2], this characterization is stable in the sense of weak convergence. Due to the nature of the cdf of MOAPR, our characterizations may be the only possible ones.  (10) if and only if the function ξ defined in Theorem 1 has the form Proof. Let Y be a random variable with pdf (10), then and finally Conversely, if ξ is given as above, then and hence s (y) = y 2 2λ 2 , y > 0. Now, in view of Theorem 1, Y has density (10) .
where D is a constant. Note that a set of functions satisfying the above differential equation is given in Proposition 4.1.1 with D = 0. It should, however, be noted that there are other triplets (h, g, ξ) satisfying the conditions of Theorem 1.

Characterization in Terms of Reverse Hazard Function
The reverse hazard function, f r , of a twice differentiable distribution function, F , is defined as In this subsection we present a characterization of the MOAPR distribution, for θ = 1, in terms of the reverse hazard function. Proof. If Y has pdf (10), then clearly the above differential equation holds. Now, if the differential equation holds, then from which we arrive at which is the reverse hazard function corresponding to the pdf (10) for θ = 1. Remark 4.2.1. We have used the fact that the reverse hazard function of a random variable Y uniquely determine the distribution of Y.

Parameter Estimation
In this section, we investigate the maximum likelihood estimation (MLE) of the unknown parameters of MOAPR model for a complete sample. Furthermore, we discuss bootstrap method to obtain the estimates of the MOAPR parameter by using MLE method. Let y 1 , ..., y n be a random sample of size n from the MOAPR distribution. Then, the likelihood function of the MOAPR distribution follows as The log-likelihood function reduces to where Ω is a vector of the MOAPR parameters. The estimators of α, θ and λ are obtained by differentiating the log-likelihood equation (18) with respect to each parameter separately, as follows .
The bootstrap is a resampling method for statistical inference. We consider parametric bootstrap estimation based on the MLE method.

Simulation Study
In this section, we obtain the Monte Carlo simulation results that is performed to see the effectiveness of maximum likelihood estimators (MLEs) of the MOAPR parameters by R language. We mainly compare between MLEs and bootstrapped MLEs in terms of bias vales, the mean squared errors (MSE). We compute MLEs and bootstap MLEs using the Newton-Raphson method. We start by building our model with generate all simulation controls. In this stage, we must do the following steps by order: • Step 1: Suppose the following values for the parameter vector of the MOAPR distribution: α = 0.5, 1.5, 3, θ = 0.5, 1.5, 3, λ = 0.5, 3.
• Step 3: Solve differential equations for both estimation methods, to obtain the estimators of the MOAPR parameters.
• Step 4: Repeat this experiment (L) times. In each experiment use the same values of the parameters. It is certain that, the values of generating random are varying from experiment to experiment even though sample size (n) does not change.
Finally, we have L-values of bias and MSE, we compute the average biases and average MSEs over 10,000 runs. This number of runs will give the accuracy in the orderÂ±0.01 (see Karian and Dudewicz (1998)). The simulated results are presented in Tables 3, 4 Tables 6 and 7 show that the MOAPR provides close fit to both data sets that the MOAPE, MOR, APR and R distributions. The fitted pdf, cdf, sf, PP plots of the MOAPR model are displayed in Figure 6, for the two data sets.

Conclusion
In this paper, we propose and study a new three-parameter model, called the Marshall-Olkin alpha power-Rayleigh (MOAPR) distribution to extend the Rayleigh distribution and provide more flexibility to analyze positive real data using the generated model. Some mathematical properties of the MOAPR are provided. The MOAPR generalizes the Marshall-Olkin Rayleigh, alpha power Rayleigh, Rayleigh and some other new models. The maximum likelihood is used to estimate the MOAPR along with parameters. Further, the bootstrap estimates are obtained. the MOAPR parameter by using MLE method. Simulation results are provided to assess the performance of the proposed maximum likelihood and bootstrap methods. Two real data applications of the MOAPR distribution are conducted to illustrate the flexibility of the proposed model.
is defined with some real function ξ. Assume that h, g ∈ C 1 (H), ξ ∈ C 2 (H) and F is twice continuously differentiable and strictly monotone function on the set H. Finally, assume that the equation ξh = g has no real solution in the interior of H. Then F is uniquely determined by the functions h, g and ξ , particularly where the function s is a solution of the differential equation s = ξ h ξh−g and C is the normalization constant, such that H dF = 1.
We like to mention that this kind of characterization based on the ratio of truncated moments is stable in the sense of weak convergence (see, Glänzel [2]), in particular, let us assume that there is a sequence {Y n } of random variables with distribution functions {F n } such that the functions h n , g n and ξ n (n ∈ N) satisfy the conditions of Theorem 1 and let h n → h , g n → g for some continuously differentiable real functions h and g. Let, finally, X be a random variable with distribution F . Under the condition that h n (Y ) and g n (Y ) are uniformly integrable and the family {F n } is relatively compact, the sequence Y n converges to Y in distribution if and only if ξ n converges to ξ , where This stability theorem makes sure that the convergence of distribution functions is reflected by corresponding convergence of the functions h, g and ξ , respectively. It guarantees, for instance, the 'convergence' of characterization of the Wald distribution to that of the Lévy-Smirnov distribution if α → ∞.
A further consequence of the stability property of Theorem 1 is the application of this theorem to special tasks in statistical practice such as the estimation of the parameters of discrete distributions. For such purpose, the functions h, g and, specially, ξ should be as simple as possible. Since the function triplet is not uniquely determined it is often possible to choose ξ as a linear function. Therefore, it is worth analyzing some special cases which helps to find new characterizations reflecting the relationship between individual continuous univariate distributions and appropriate in other areas of statistics.