A New Generalized-upper Record Values-G Family of Lifetime Distributions

A new family of lifetime distributions is introduced via distribution of the upper record values, the well-known concept in survival analysis and reliability engineering. Some important properties of the proposed model including quantile function, hazard function, order statistics are obtained in a general setting. A special case of this new family is proposed by considering the exponential and Weibull distribution as the parent distributions. In addition estimating unknown parameters of specialized distribution is examined from the perspective of the traditional statistics. A simulation study is presented to investigate the bias and mean square error of the maximum likelihood estimators. Moreover, one example of real data set is studied; point and interval estimations of all parameters are obtained by maximum likelihood and bootstrap (parametric and non-parametric) procedures. Finally, the superiority of the proposed model in terms of the parent exponential distribution over other known distributions is shown via the example of real observations.


Introduction
The statistical distribution theory has been widely explored by researchers in recent years. Given the fact that the data from our surrounding environment follow various statistical models, it is necessary to extract and develop appropriate highquality models. In addition, sometimes it is necessary to provide applications from existing models. For more details, see the Samuel et al. (2018) and Ababneh et al. (2018).
Recently, Alzaatreh et al. (2013) have introduced a new model of lifetime distributions, which the researchers refer to its special case as generalized G distribution. It is based on the combination of one arbitrary CDF F of a continuous random variable X with the baseline CDF G. The integration form of new CDF H is stated as (1) where f is the corresponding density function of F and F (1) = P (X 1). This interesting method attracted the attention of some researchers. Generating new model based on this method resulted in creating very ‡exible statistical modeling.
The upper and lower record values, in a sequence of independent and identically distributed (iid) random variables X 1 ; X 2 ; :::, have applications in di¤erent areas of applied probability and reliability engineering. Let X i 's have a common absolutely continuous distribution G with survival function G. De…ne a sequence of record times U (n); n = 1; 2; :::, as follows: U (n + 1) = min fj : j > U (n); X j > Xg; n 1; with U (1) = 1. Then, the sequence of upper record values fR n ; n 1g is de…ned by R n = X U (n); n 1, where R 1 = X 1 . The survival function of R n is given by [ log G(t)] x x! ; t 0; n = 1; 2:::: The corresponding CDF of the random variable R n is [ log G(t)] x x! ; t 0; n = 1; 2:::: Here, we introduce a new family of lifetime distributions by compounding CDF of upper record values G U n (t) of a parent distribution G and an arbitrary CDF F with P DF f .
This new model will be denote by G U R G(or GU RG) distribution. One of our main motivation to introduce this new category of distributions is to provide more ‡exibility for …tting real datasets in comparing with other well-known classic statistical distributions.
We …rst derive the fundamental and statistical properties of GU RG in a general setting and then we propose a special case of this model by considering Weibull distribution instead of the parent distribution G and exponential distribution instead of the parent distribution F for …xed value n = 2. It is referred to as GU RW E distribution. We provide a comprehensive discussion about the statistical and reliability properties of the new GU RW E model. Furthermore, we consider Maximum A NEW G ENERALIZED-UPPER RECO RD VALUES 997 likelihood and bootstrap estimation procedures to estimate the unknown parameters of the new model for complete data set. In addition, the asymptotic con…dence intervals and parametric and non-parametric bootstrap con…dence intervals are calculated.

New general model and its properties
In this section, we provide the structure of our new model and some of its main properties in a general setting. Motivated by the idea of Alzaatreh et al. (2013), a new class of statistical distributions is proposed. The new model is constructed by implementing Alzaattreh idea to the upper record value distribution G U n (t). Let the non-negative random variable X have CDF and P DF F and f , respectively. In view of (1), the CDF of new general class of lifetime distributions is de…ned as: ; x 0; n = 1; 2::: ); x > 0; n = 1; 2::: where g U n (x) is the P DF of the n upper record value distribution and [ log G(x)] n 1 (n 1)! ; x > 0; n = 1; 2::: Using (2) and (4), the survival H(x; n) and the hazard rate r(x; n) functions for GU RG distribution are given, respectively, by: ; x > 0; n = 1; 2:::: The pth quantile x p of the GU RG distribution can be obtained from where G U 1 n is the inverse function of CDF G U n :

Special case based on the parent Weibull and Exponential Distributions
Let G(x) = 1 e x , F (x) = 1 e x and n = 2. From (2) we have: The corresponding P DF is : where x > 0; ; ; > 0.  3.1. Some properties of the GU REW distribution. In this section, we obtain some properties of the GU REW distribution, such as quantiles, moments, moment generating function and order statistics distribution. The characterizations of GU REW distribution are presented in subsection 3.5.

3.2.
Quantiles. For the GU REW distribution, the pth quantile x p is the solution of H(x p ) = p, hence which is the base of generating GUREW random variates, where W 1 denotes the negative branch of the Lambert function.
where X W W eibull( (t + 1); ) and E X W [X r+(j+1) ] = (1+ r+j +j ) ( (t+1)) r+(j+1) . The variance, CV , CS, and CK are given by and respectively. Table 1 lists the …rst six moments of the GU REW distribution for selected values of the parameters, when = 3. Table 2 lists the …rst six moments of the GU REW distribution for selected values of the parameters, when = 0:5. These values can be determined numerically using R.
The moment generating function of the GU REW distribution is given by where X G Gamma (j + 2; (t + 1)).   In this subsection, we present the distribution of the ith order statistic from the GU REW distribution. The P DF of the ith order statistic from the GU REW P DF , f GU REW (x), is given by Using the binomial expansion we have 3.5. Characterization Results. This section is devoted to the characterizations of the GU REW distribution in di¤erent directions: (i) based on the ratio of two truncated moments; (ii) in terms of the reverse hazard function and (iii) based on the conditional expectation of certain function of the random variable. Note that (i) can be employed also when the cdf does not have a closed form. We would also like to mention that due to the nature of GU REW distribution, our characterizations may be the only possible ones. We present our characterizations (i) (iii) in three subsections.
3.5.1. Characterizations based on two truncated moments. This subsection deals with the characterizations of GU REW distribution based on the ratio of two truncated moments. Our …rst characterization employs a theorem due to Glänzel (1987); see Theorem 1 of Appendix A . The result, however, holds also when the interval H is not closed, since the condition of the Theorem is on the interior of H. Proposition 3.5.1. Let X : ! (0; 1) be a continuous random variable The random variable X has PDF (6) if and only if the function de…ned in Theorem 1 is of the form Proof. Suppose the random variable X has PDF (6), then and Conversely, if is of the above form, then and consequently s (x) = x ; x > 0: Now, according to Theorem 1, X has density (6) : Corollary 3.5.1. Let X : ! (0; 1) be a continuous random variable and let q 1 (x) be as in Proposition A.1. The random variable X has PDF (6) if and only if there exist functions q 2 and de…ned in Theorem 1 satisfying the following di¤erential equation where D is a constant. We like to point out that one set of functions satisfying the above di¤erential equation is given in Proposition 3.5.1 with D = 0: Clearly, there are other triplets (q 1 ; q 2 ; ) which satisfy conditions of Theorem1.
3.5.2. Characterization in terms of reverse hazard function. The reverse hazard function, r F , of a twice di¤erentiable distribution function, F , is de…ned as ; x 2 support of F: In this subsection we present a characterization of GUREW distribution in terms of the reverse hazard function.
Proposition 3.5.2. Let X : ! (0; 1) be a continuous random variable. The random variable X has PDF (6) if and only if its reverse hazard function r F (x) satis…es the following di¤erential equation Proof. If X has PDF (6) , the clearly the above di¤erential equation holds. Now, if this equation holds, the d dx from which we obtain the reverse hazard function corresponding to the PDF (6).

3.5.3.
Characterization based on the conditional expectation of certain function of the random variable. In this subsection we employ a single function of X and characterize the distribution of X in terms of the truncated moment of (X) : The following proposition has already appeared in Hamedani's previous work (2013), so we will just state it here which can be used to characterize GUREW distribution.

Inference procedure
In this section, we consider estimation of the unknown parameters of the GU REW ( ; ; ) distribution via maximum likelihood method and bootstrap estimation.
4.1. Maximum likelihood estimation. Let x 1 ; : : : ; x n be a random sample from the GU REW distribution and = ( ; ; ) be the vector of parameters. The loglikelihood function is given by The elements of the score vector are given by respectively.
The maximum likelihood estimate,^ of = ( ; ; ) is obtained by solving the nonlinear equations dL d = 0; dL d = 0; dL d = 0 simultaneously. These equations do not have closed forms so, the values of the parameters , and must be found using iterative methods. Therefore, the maximum likelihood estimate,^ of = ( ; ; ) can be determined using an iterative method such as the Newton-Raphson procedure.

Bootstrap estimation.
The parameters of the …tted distribution can be estimated by parametric (resampling from the …tted distribution) or non-parametric (resampling with replacement from the original data set) bootstraps resampling (see Efron and Tibshirani, 1994). These two parametric and nonparametric bootstrap procedures are described as below.

Parametric bootstrap procedure:
(1) Estimate (vector of unknown parameters), say^ , by using the M LE procedure based on a random sample. (2) Generate a bootstrap sample fX 1 ; : : : ; X m g using^ and obtain the bootstrap estimate of , say b , from the bootstrap sample based on the M LE procedure.

Nonparametric bootstrap procedure
(1) Generate a bootstrap sample fX 1 ; : : : ; X m g , with replacement from the original data set. . Then obtain -quantiles and 100(1 )% con…dence intervals for the parameters. In the case of GU REW distribution, the nonparametric bootstrap estimators (NPBs) of ; and , are^ N P B ;^ N P B and^ N P B , respectively.

Algorithm and a simulation study
In this section, we give an algorithm for generating the random data x 1 ; : : : ; x n from the GUREW distribution and hence a simulation study is done to evaluate the performance of the MLEs.

5.1.
Algorithm. Here, we obtain an algorithm for generating the random data x 1 ; : : : ; x n from the GUREW distribution as follows.
The algorithm is based on generating random data from the inverse CDF of the GUREW distribution.

Monte Carlo simulation study.
Here, we assess the performance of the MLE's of the parameters with respect to the sample size n for the GU REW distribution. The assessment of the performance is based on a simulation study via Monte Carlo method. Let^ ;^ and^ be the MLEs of the parameters , and , respectively. We calculate the mean square error (MSE) and bias of the MLE's of the parameters ; and based on the simulation results of 2000 independent replications. Results are summarized in Table 3 for di¤erent values of ; and . From Table 3 the results verify that MSE of the MLE's of the parameters decrease Table 3. MSEs and Average biases(values in parentheses) of the simulated estimates. with respect to sample size n for all the parameters. So, the MLEs of ; and are consistent estimators.

Practical data application
In this section, we present an application of the GU REW distribution to a practical data set to illustrate its ‡exibility among a set of competitive models. In order to achieve this goal, we consider a real data set corresponding to the remission times (in months) of a random sample of 128 bladder cancer patients. These data were previously studied by Lee and Wang (2003). This data set consists of the following observations: Graphical measure: The total time test (T T T ) plot due to Aarset (1987) is an important graphical approach to verify whether the data can be applied to a speci…c distribution or not. According to Aarset (1987), the empirical version of the T T T plot is given by plotting T (r=n) = [ P r i=1 y i:n + (n r)y r:n ]= P n i=1 y i:n against r=n, where r = 1; : : : ; n and y i:n (i = 1; : : : ; n) are the order statistics of the sample. Aarset (1987) showed that the hazard function is constant if the T T T plot is graphically presented as a straight diagonal, the hazard function is increasing (or decreasing) if the T T T plot is concave (or convex). The hazard function is U-shaped if the T T T plot is convex and then concave, if not, the hazard function is unimodal. The T T T plots for data set is presented in Fig 3. These plots indicate that the empirical hazard rate functions of the data set is upside-down bathtub shapes. Therefore, the GU REW distribution is appropriate to …t this data set.
6.1. Bootstrap inference for GU REW parameters. In this section, we obtain point and 95% con…dence interval (CI) estimation of the GU REW parameters by parametric and non-parametric bootstrap methods. We provide results of bootstrap estimation in Table 4 for the complete data set. It is interesting to observe the joint distribution of the bootstrapped values in a scatter plot in order to understand the potential structural correlation between the parameters. The corresponding plots of the bootstrap estimation are shown in   for the data set. The GU REW distribution provides the best …t for the data set as it shows the lowest AIC, A and W than other considered models. The relative histograms, …tted GU REW , Lindley, GL, GaL,EXP , P L, EL, gamma, generalized exponential and Weibull P DF s for data are plotted in Fig 5. The plots of the empirical and …tted survival functions, P P plots and Q Q plots for the GU REW and other …tted distributions are displayed in Fig 5 and Fig 6 respectively. These plots also support the results in Table 5. We compare the GU REW model with a set of competitive models, namely: (i) Lindley distribution (Lindley, 1958). The one-parameter Lindley density function is given by where > 0: where > 0 and > 0.
(vi) The two-parameter Weibull distribution is given by where > 0 and > 0: (vii) The two-parameter Gamma distribution is given by where > 0 and > 0 and ( ) = R 1 0 t 1 e t dt. (viii) The one parameter Exponential distribution is given by The two-parameter generalized exponential (GE) distribution is given by where > 0 and > 0:

Conclusion
In this article, a new model for the lifetime distributions is introduced and its main properties are discussed. A special submodel of this family is taken up by considering exponential distributions in place of the parent distribution F and Weibull distribution in place of the parent distribution G. We show that the proposed distribution has variability of hazard rate shapes such as increasing, decreasing and upside-down bathtub shapes. From a practical point of view, we show that the proposed distribution is more ‡exible than some commonly known statistical distributions for a given data set.  is de…ned with some real function . Assume that q 1 ; q 2 2 C 1 (H), 2 C 2 (H) and F is twice continuously di¤erentiable and strictly monotone function on the set H. Finally, assume that the equation q 1 = q 2 has no real solution in the interior of H. Then F is uniquely determined by the functions q 1 ; q 2 and , particularly where the function s is a solution of the di¤erential equation s 0 = 0 q1 q1 q2 and C is the normalization constant, such that R H dF = 1. References