A Two Parameters Rani Distribution: Estimation and Tests for Right Censoring Data with an Application

In this paper, we developed a new distribution, namely the two parameters Rani distribution (TPRD). Some statistical properties of the proposed distribution are derived including the moments, moment-generating function, reliability function, hazard function, reversed hazard function, odds function, the density function of order statistics, stochastically ordering, and the entropies. The maximum likelihood method is used for model parameters estimation. Following the same approach suggested by Bagdonavicius and Nikulin (2011), modified chi squared goodness-of-fit tests are constructed for right censored data and some tests for right data is considered. An application study is presented to illustrate the ability of the suggested model in fitting aluminum reduction cells sets and the strength data of glass of the aircraft window.


Introduction
proposed a distribution of one parameter called as a Rani distribution (RD) with probability density function (pdf) given by ( ) , 24 and the cumulative distribution function (cdf) of the RD is defined as 3 3 2 2 5 ( 4 12 24 1 e , 0, 0. 2 ( ; ) 1 4 Shanker showed the flexibility of the RD in modeling real life time data. In the literature, many authors suggested different new distributions for modeling and fitting various lifetime data in several fields such as medicine, industry, biology, nursing, agriculture, insurance and other fields. These distributions are generated using various methods. As an example, recently, Alhyasat et al. (2020) proposed power size biased two-parameter Akash distribution. Alsmairan and Al-Omari (2020) introduced weighted Suja distribution. Al-Omari and Garaibah (2018) proposed Topp-Leone Mukherjee-Islam distribution. Al-Omari, and Alsmairan (2019) proposed length-biased Suja distribution. Usman et al. (2019) introduced the Marshall-Olkin Length biased exponential distribution.  suggested a size biased Ishita distribution. Garaibah and Al-Omari(2019) proposed the transmuted Ishita distribution .
In this paper, we developed the RD by adding a new parameter to the RD in order to improve its flexibility in fitting real data. Also, we investigated the problem of right censored data with some well-known tests for the TPRD. Aidi and Seddik-Ameur (2016) considered Chi-square tests for generalized exponential AFT distributions with censored data.
The maximum likelihood estimates are investigated in the case of censored data and also, the estimated information matrix is presented. The structure of a modified chi-square goodness-of-fit test for the new TPRD when the data are right censored is proposed. For these purposes, we used the criteria tests considered by Bagdonavicius and Nikulin (2011), Bagdonavicius et al. (2013). This statistic test is based on the maximum likelihood estimators on initial data, and follows by chi-square distribution. In order to confirm the usefulness of the proposed goodnessof-fit test, and the new model, an important simulation study was carried out. Theoretical results obtained from this study are applied to a real data set from reliability.
The rest of this paper is organized as follows: In Section 2, the two parameters Rani distribution is presented with its pdf and cdf and some plots are involved. Some statistical properties of the TPRD including the moments, variance, skewness kurtosis, the rth moment and the moment generating function are given in Section 3, also some simulations are considered. Section 4, is devoted to the distribution reliability analysis. The order statistics and stochastic ordering are presented in Section 5. In Section 6, the entropies of the distribution are derived. The TPRD parameters estimation is considered in Section 7 and the statistic for right censored data is given in Section 8. Criteria test for TPRD distribution are presented in Section 9. A simulation study and an application are given in Section 10 and 11, respectively. Finally, the paper is concluded in Section 12.

The suggested model
In this section, we introduce the pdf of the two parameters Rani distribution (TPRD) as ( ) It is of interest to note here that Also, it can be noted that 12 ( ( ; ) , ( ; ) 1 ) ( ; ) 2 ) Plots of the pdf and cdf of the TPRD distribution are presented in Figures (1) and (2), respectively, for various distribution parameters on the interval [0,3] x  . .
Figure (1) shows that the TPRD distribution is non symmetric and skewed to the right. Also, the skewness of the distribution depends on the values of the parameters. Pak.j.stat.oper.res. Vol.17 No. 4 2021pp 1037-1049

Moments of the TPRD distribution
In this section, we presented some statistical properties of the TPRD as the moments, the rth moment, coefficient of variation, coefficient of skewness and coefficient of kurtosis.
The coefficient of skewness of the suggested TPRD distribution is given by

Reliability
The reliability (survival) and hazard rate functions of the TPRD distribution are, respectively given by  fx Hx The cumulative hazard rate function of the TPRD is . ln 1 24 (3) and (4) Figures (5) and (6)

Order Statistics and Stochastic Ordering
Let (1: ) (2: ) , ,..., n n n n X X X be the order statistics of the random sample 12 , ,..., n X X X chosen from a pdf () fx and a cdf () Fx. The pdf of the jth order statistics say ( : ) in X , is given by Substituting the pdf and cdf of the TPRD in Equation (16) to obtain the pdf of ( : ) jn X as The stochastic ordering can be used to compare two positive continuous random variables. A random variable X is smaller than a random variable Y in 1) Mean residual life order denoted by 2) Likelihood ratio order denoted by

Maximum likelihood estimation
Here, the parameters of the TPRD distribution are estimated using the method of maximum likelihood. Let 1 , 2 , … be a random sample distributed according to the TPRD distribution, the likelihood function is obtained by The logarithm of Equation (22) The maximum likelihood estimators ̂ and ̂ of the unknown parameters α and are derived from the nonlinear following score equations: (23)

Estimation under right-censored data
The hypothesizing test will be discussed under complete and censored data, however, the TPRD is only defined for complete data. Since the MLE is usually considered for right-censored data, let us consider 1 , 2 , . . . , be a random right censored sample obtained from the TPRD distribution with the parameter vector = ( , ) . The censoring time τ is fixed. So, the observation is equal to = ( , ) where Monte Carlo technique or other iterative methods can be used to determine the values of̂ and ̂.

Test statistic for right censored data
Let ₁, . . . , be n i.i.d. random variables grouped into classes . To assess the adequacy of a parametric model  Pak.j.stat.oper.res. Vol.17 No. 4 where ̂ is the maximum likelihood estimator of on initial non-grouped data.
Under the null hypothesis H0 , the limit distribution of the statistic Y² is a chi-square with = ( ) degrees of freedom. The description and applications of modified chi-square tests are discussed in Voinov et al. (2013). The interval limits for grouping data into j classes are considered as data functions and defined by where Λ( , ) is the cumulative hazard function, such as the expected failure times to fall into these intervals are = for any j, with = ∑ Λ( , ) =1 . The distribution of this statistic test 2 is chi-square (see Voinov et al., 2013). For goodness of fit tests one can see Zamanzade (2017, 2016) for goodness of fit-tests for Laplace and Rayleigh distributions using ranked set sampling.

Criteria test for TPRD distribution
For testing the null hypothesis H₀ that data belong to the TPRD model, we construct a modified chi-squared type goodness-of-fit test based on the statistic Y². Suppose that τ is a finite time, and observed data are grouped into > sub-intervals = (

Simulations
In this section, a simulation study is conducted to investigate the efficiency of several estimators considered in this study.

Maximum likelihood estimation
We generated = 10,000 right censored samples with different sizes ( = 25, 50, 130, 350, 500) from the TPRD model with parameters = 0.7 = 1.5. Using R statistical software and the Barzilai-Borwein (BB) algorithm (Ravi, 2009), we calculate the maximum likelihood estimators of the unknown parameters and their mean squared errors (MSE). The results are given in Table 2. The maximum likelihood estimated parameter values, presented in Table 2, agree closely with the true parameter values. Also, as the sample sizes are increasing the estimation of the parameters be more efficient.

Criteria test
For testing the null hypothesis H0 that right censored data become from TPRD model, we compute the criteria statistic 2 ( ) as defined above for 10,000 simulated samples from the hypothesized distribution with different sizes (30,50,150,350,500).Then, we calculate empirical levels of significance, when 22 ( ), Yr    corresponding to theoretical levels of significance ( = 0.10, = 0.05, = 0.01) We choose = 4. The results are reported in Table  3. The null hypothesis H0 for which simulated samples are fitted by TPRD distribution, is widely validated for the different levels of significance. Therefore, the test proposed in this work, can be used to fit data from this new distribution.

Applications
To show the flexibility of the proposed distribution and the usefulness of the criteria test 2 , we analyze censored and uncensored real data sets. Using model selection criteria (NLL, AIC, CAIC and BIC) and 2 , we show that the TPRD distribution fits data better than some alternatives such as the lognormal, gamma and Weibull distributions. Pak.j.stat.oper.res. Vol.17 No. 4  are grouped into = 4 intervals . We give the necessary calculus in the following Table 4. For significance level ε = 0.05, the critical value 4 2 = 9.4877 is superior than the value of 2 = 7.263, so we can say that the proposed model TPRD fits these data.
After calculating the maximum likelihood estimators of the unknown parameters, we use classical criteria (NLL, AIC, CAIC, BIC ) to select the best model which describes these data. From the results given in Table 5, we can see that values of the different criteria for the TPRD distribution are the smallest. So we can say that the TPRD distribution fits these data better than the lognormal, gamma and Weibull distributions.

Conclusions
A new continuous two-parameter lifetime distribution called as the two parameters Rani (TPRD) distribution. Some statistical properties includes the mean, variance, moment generating function, the rth moment, the coefficients of variation, skewness and kurtosis are obtained. Also, the distributions of order statistics and the stochastic ordering are studied, reliability analysis, maximum likelihood estimation and goodness of fit tests of the distribution are proved based on right censoring data. The Rényi entropy and q-entropy are derived. Real data sets are considered for illustration.