A Modified Chi-square Type Test Statistic for the Double Burr X Model with Applications to Right Censored Medical and Reliability Data

A new modified version of the Bagdonavičius-Nikulin goodness-of-fit test statistic is presented for validity for the right censor case under the double Burr type X distribution. The maximum likelihood estimation method in censored data case is used and applied. Simulations via the algorithm of Barzilai-Borwein is performed for assessing the right censored estimation method. Another simulation study is presented for testing the null hypothesis under the modified version of the Bagdonavičius and Nikulin goodness-of-fit statistical test. Four right censored data sets are analyzed under the new modified test statistic for checking the distributional validation.


Introduction
Statistical methods for testing the validity of a parametric distribution are in increasing developments. Clearly, the presence of censored observations makes those methods unsuitable. For treating these problems there were many attempts. First, Habib and Thomas (1986) and Hollander and Pena (1992) proposed a modified version of the Chisquared test for the randomly censored real data based on the well-known estimators of the Kaplan and Meyer. Then, Galanova (2012) considered some nonparametric modifications for the Anderson-Darling (AD) test statistic, Kolmogorov-Smirnov (KS) test statistic and the Cramer-Von-Mises (KM) test statistic for the accelerate failure models. Recently, Bagdonavičius and Nikulin (2011a,b) presented and applied a new type of Chi-squared goodnessof-fit test statistic for the censored data (right case), (see Bagdonavičius et al. (2013)). Then, the new type of Chisquared goodness-of-fit test statistic of Bagdonavičius-Nikulin is applied for the distributional validation under the right censored schemes. Recentely, the Bagdonavičius-Nikulin test is used for the distributional validation in case of the right censored data (see Mansour et al (2020a,b), Ibrahim e al. (2021) and Yousof et al. (2021a,b)).
In this work, a new modified type Chi-squared goodness-of-fit test statistic which is established based on the wellknown Bagdonavičius-Nikulin test is presented and applied accordingly for distributional validation under the double

Pakistan Journal of Statistics and Operation Research
Burr type X (DBX) distribution model using the right censored case. A comprehensive simulation study under the right censored case via the well-known Barzilai-Borwein (BB) algorithm is performed for assessing and evaluating the right censored estimation method. Following Ravi and Gilbert (2009), we generated 10,000 samples with different size ( = 15,25,50,130,350,500,1000) from the DBX model using some very carefully selected initial values. The mean square errors (MSEs) are used for assessing the performance of the censored maximum likelihood. The new modified Bagdonavičius-Nikulin type test is applied using four right censored real data sets for distributional validity.
A random variable (RV) is said to have the Burr type X (BX) distribution if its cumulative distribution function (CDF) is given by where = ( 1 , 2 ). The probability density function (PDF) of the DBX corresponding to (2) reduces to A RV having PDF in (3) is denoted by ∼ DBX( ). The hazard rate function (HRF) and cumulative hazard rate function (CHRF) can be derived from the well known relationship. The new additional parameter 1 will allow us to study the tail behavior of the new density (3)

Maximum likelihood estimation in censored data case
Suppose that 1 , 2 , . . . , is a random sample with right censoring from the DBX distribution. The observed data = min( , ); = 1,2, . . , are the minimum of the survival time and censoring time for each subject in the sample. So, can be written in the form ( , ∇ ) =1,..., where , ∇ = 1 if is the moment of failure (complete observation) and ∇ = 0 if is the moment of censoring. The right censoring is assumed to be non informative, so the expression of the likelihood function is The log-likelihood function of the DBX distribution is . The maximum likelihood estimates of the unknown parameters can be obtained using various techniques, either software R, the EM algorithm or Newton Raphson method.

Modified chi-squared type test for right censored data
Methods for testing the validity of parametric models are in constant development, but the presence of censorship make them unsuitable. Habib 2013), for parametric models with right censored data. Based on maximum likelihood estimates on non-grouped data, this test statistic is also based on the differences between the numbers of observed failures and the numbers of expected failures in the grouped intervals chosen. For this, random grouping intervals are considered as data functions. The description of the construction of this chi-squared type test is developed in Voinov et al.(2013). The test statistic is defined as follows. Suppose that 1 , 2 , . . . . . , is a random sample with right censoring from a parametric model, and a finite time . The test statistic is defined as follows.
and ( ,̂) is the CHRF of the DBX model distribution. This test statistic , 2 follows a chi-squared distribution.

Choice of random grouping intervals
Suppose that 1 , 2 , . . . . . , is a random sample with right censoring from the double Burr type X (DBX) distribution and a finite time . In our case, the estimated ̂̂( , , ) is obtained as follows where ̂= (̂1,̂2) are the maximum likelihood estimates of the unknown parameters = ( 1 , 2 ) on initial data, and .

Quadratic form of the statistic
Therefore, the estimated matrix ̂ can be deducted from ̂.

Estimated information matrix ̂( )
We need also the estimated information matrix ̂( ) of the DBX distribution with right censoring. After some calculations and some simplifications, we have obtained the elements of the matrix as follows: .
As all the components of the statistic are given explicitly, then we obtain the test statistic for the DBX distribution with unknown parameters and right censored data. This statistic follows a chi-squared distribution with degrees of freedom.

Simulations
An important simulation study is carried out to show the performance of the techniques used and the feasability of the goodness-of-fit test developed in this work. At this end, we generated = 10,000 right censored samples with different sizes 1 = 15, 2 = 25, 3

Maximum likelihood estimation for DBX model
Using statistical software and the BB algorithm, we calculate the maximum likelihood estimates of the unknown parameters and the corresponding MSEs (see Ravi (2009) for more details). The results of simulating the , 2 test under the BB algorithm and the DBX model are given in Table 1.  The null hypothesis 0 for wich simulated samples are fitted by DBX 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
In this section, we introduce four applications of the DBX distribution to four real-life right censored data sets. The first right censored data set is from a study conducted on the effects of ploidy on the prognosis of patients with cancers of the mouth. Patients were selected who had a paraffin-embedded sample of the cancerous tissue taken at the time of surgery. Follow-up survival data was obtained on each patient. The third right censored data set we apply are the results obtained from Crowder et al. (1991). In an experiment to obtain information on the strength of a certain type of braided cord after the weather the forces of 48 pieces of cord having resisted for a determined time were studied. The right censored force values observed are given as foloows:  For the all right censored data, we use the Bagdonavičius-Nikulin test statistic provided above to verify if these data are modeled by the DBX distribution. For the first right censored data, we calculate the maximum likelihood estimates of the unknown parameters ̂= (̂1,̂2) = (2.566,1.346) , where the data are grouped into = 5 intervals. We give the necessary calculations of the ̂̂( , , ), ( , ), ( , ),̂1 ( ) and ̂2 ( ) for the first right censored data in the following Table 3. where the data are grouped into = 4 intervals. The results of the ̂̂( , , ), ( , ), ( , ),̂1 ( ) and ̂2 ( ) for the second right censored data are given in the following Table 4. For the third right censored data, ̂= (̂1,̂2) = (1.8363,1.0634) . Data are grouped into = 4 intervals. Table 5 gives the results of ̂̂( , , ), ( , ), ( , ),̂1 ( ) and ̂2 ( ) for thethird right censored data.  Table 6 lists the results of ̂̂( , , ), ( , ), ( , ),̂1 ( ) and ̂2 ( ) for thethird fourth censored data. .4877 is greater than the value of , 2 = 6.956, so we can say that the proposed model DBX fit these data.

Concluding remarks
In this article, a modified Bagdonavičius-Nikulin goodness-of-fit test statistic is presented and applied for distributional validation under the right censor case and for the double Burr type X distribution. The modified test statistic ( , 2 ) is given along with all its relevant components. For this purpose, the maximum likelihood estimation method in censored data case is used and applied along with a simulation study via the algorithm of Barzilai-Borwein performed for assessing the right censored estimation method. Another simulation study is presented for testing the null hypothesis under the modified version of the Bagdonavičius and Nikulin goodness-of-fit statistical test. Four right censored data sets are analyzed under the new modified test statistic for checking the distributional validation. According to the modified Bagdonavičius-Nikulin goodness-of-fit test statistic, the double Burr type X distribution can be used in modeling the censored medicine and reliability real data sets. Below are results of the modified test statistic under the right censored data sets: • For the death times in weeks under aneuploid tumors data: 5