Modeling Various Survival Distributions using a Nonparametric Hypothesis Testing Based on Laplace Transform Approach with Some Real Applications

We hope to assess the different processing methodologies or the effectiveness of the devices or systems applied in this nonparametric statistical test by observing the failure behavior of the recorded survival data. The proposed second-order approach of new better than used (NBU 2) will be employed, which requires that the test data behaves either like NBU 2 property or exponentially. If the survival data is NBU 2, the proposed treatment method is likely to be beneficial. If the data are exponential, on the other hand, the recommended treatment method has no positive or negative influence on patients, as shown in the application section. To establish the validity of the test, we calculated the power of the proposed test and efficiency on both complete and censored data, compared the results to those of existing tests, and then applied the test to a range of real-world data.


Introduction
The estimation and testing of hypotheses were the two basic divisions of statistical inference. Generally, we have no idea what the true value of population parameters is. Thus, we have to estimate them. We do, however, have some theories concerning the true values of population parameters. The null hypothesis is frequently denoted by the sign H 0 , and the alternative hypothesis known as H 1 . The null and alternative hypotheses must be defined prior to conducting any significant statistical test.
Various life distribution classes have been proposed during the last century to model various elements of aging. See, The survivor function is defined as A fundamental property of surviving dissemination is the ability to age. The manuscript introduces a number of classes to categorize distributions.
The first aging is NBU. Here, FðxÞ has NBU property iff The class NBUE distribution is the second aging class, and a life distribution has the NBUE property, represented by Definition 1. A survival function FðxÞ considered NBU in the increasing concave order (X ∈ NBUð2Þ) iff The previous inequality has a physical meaning: in terms of rising concave ordering, the lifetime of a worn object with age t > 0 is stochastically shorter than that of a new one. Barlow and Proschan [7], Abu-Youssef and Bakr [8], Abu-Youssef et al. [9], Franco et al. [10], Deshpande et al. [11], Hu and Xie [12], Qureshi and Yusuf [13], Hassan and Said [14], Sania Qureshi and Rashid Jan [15], Abu-Youssef et al. [16,17], and Lia and Xie [18], and others proposed the probabilistic features of the aging distributions in the literature. The following are the relationships between the previous classes: We have some genuine data in this research investigation, and we are trying to figure out whether H 0 : data is exponential or H 1 : data is NBU (2).
In this research study, we have got some real data, and we are trying to figure out whether H 0 : data is exponential or H 1 : data is NBU (2). To decide which H 0 or H 1 is true or to reach to a conclusion, we must first establish test statistics. The test statistic is a random variable that determines how closely a sample result matches one of these hypotheses under consideration.
Al-Gashgari et al. [19] and Gadallah [20] followed Atallah et al. [21] in developing a new approach of exponential testing that is more generic and flexible than goodness of fit.
This is how the research is structured: in Section 2, we produce a test statistic for the complete data using the Laplace transform approach. In Section 3, Monte Carlo null distribution critical points for data sets 5(5)100 are illustrated using Mathematica 8. For popular alternatives, Section 4 tabulates Pitman asymptotic efficiency, and Section 5 estimates the power of the test. In Section 6, a recommended test for appropriately filtered data is proposed. Finally, we discuss sets of medical real data in Section 7 to highlight the importance of our test.
Lemma 2 provides a measure of deviation. As a result, it might be utilized to create a testing technique.
Proof. Let us call the deviation from H 0 the measure of departure as where Hence, the result follows.
We may derive the empirical estimator from (7) for δ as So, we can say that b δ is where The U-statistic theory can be used to obtain the limiting distribution of b δðsÞ.

Computational and Mathematical Methods in Medicine
Set Then, Let The asymptotic normality of the statistic (7) was demonstrated in Theorem 1.

Theorem 3.
According to U-statistics theory, the statistic has the following characteristics: (i) As n ⟶ ∞, with mean of 0 and variance σ 2 , ð b δ n − δÞ is asymptotically normal, such that (ii) Under H 0 , the variance reduces to Proof.
(i) We obtain the following results using U-statistic theory (Lee [22]): (ii) In H 0 , it is obvious that μ 0 = E½ξðXÞ = 0, and the variance is

Critical Points
Based on 10,000 simulated sample sizes n = 5 (5) 100 from the exponential distribution, we calculate the upper percentage points of the test statistic for our test in this section using Mathematica 8 Programming. Table 1 shows a statistically significant percentile points at s = 0:4, 0:6.
Asymptotic normality improves, as the critical values decreases and their sample size increases, as in Table 1.

Pittman Efficiency
We can compare our test to the other courses to determine the quality of this procedure. Here, we choose the test U n presented by Kayid et al. [23] and δ ð2Þ F n presented by Mahmoud and Abdul Alim [24].
In this section, the PAE of our test is evaluated.
We investigate the PAE of our test δ, where  Table 2.
The PARE of δ with respect to U n and δ ð2Þ F n is shown in Table 3. Tables 2 and 3 show the good performance of the selected families. Table 4 illustrates the efficacy of the suggested test. For the LFR, Makeham, and Weibull distributions, these powers were estimated using 10,000 simulated samples with n = 10 , 20, and 30 at a significance level of 0.05 (programming using Mathematica 8).

Estimates of Power
The suggested test has good validity for the Makeham and Weibull families and acceptable validity for the LFR family, as shown in Table 4.

Test Statistic for Censored Data
One of the most significant improvements in the life sciences has come from a property of survival data: if some of the participants did not observe the event of interest at the end of this research or at the time of analysis, the data is skewed. Some patients may still be alive or disease-free at the end of the experiment because the timing of survival or the end of disease is unknown. Individuals are referred to as censored observations or censored times when they cannot be followed up with after a research session.
In this section, a statistical test is developed to assess H 0 and H 1 with data that has been randomly right censored.
In the case of censored data, use Definition 1 and the Kaplan-Meier estimator; we can write the measure of departure as follows: Table 5 shows a statistically significant percentile points of δ c in (25), at s = 0:4, 0:6.
As shown in Table 5, the asymptotic normality improves, as the critical values decrease and their sample size increase.

Applications
We apply the conclusions in this work to certain real data sets to demonstrate their utility.  Table 6), which represents the ages of the 40 patients with leukemia, taken in years. In the two situations of b δð0:4Þ and b δð0:6Þ as n = 40, we calculate the statistics in (7): b δð0:4Þ = 1:5 and b δð0:6Þ = 0:43, both of which are above the critical value of Table 1. Consequently, the data set has the property of NBU (2).

Example 2.
Consider the data in Bakr et al. [3] (see Table 7), which represent 39 liver cancer patients taken from El-Minia Cancer Center, Ministry of Health, Egypt, which entered in 1999.      Table 1. Consequently, the data set has no property of NBU (2).

Application 2: A Case of Censored Data
Example 3. We use data from Mahmoud et al. [25] (see Tables 8 and 9 Table 5. Consequently, the data set has the property of NBU (2). Example 4. Consider the survival intervals in weeks reported by Lee and Wolfe [26] (see Tables 10 and 11) for 61 patients with unresectable lung cancer who were treated with cyclophosphamide. There are 33 uncensored observations and 28 censored observations for patients whose therapy was terminated owing to an evolving condition.
In  Table 5. Consequently, the data set has the property of NBU (2) Table 12 show the notations and abbreviations.

Conclusion
We devised a statistical testing approach (Laplace transform technique) for the largest class of life distribution NBU (2) in both complete and censored data to aid in the quality assessment of prospective cancer therapies. Our testing revealed whether the suggested therapies had a good or negative influence on patient survival as shown in the four discussed examples. The effectiveness of the suggested statistical test was computed and compared to existing tests to confirm that it produced good findings. Regardless of the kind of treatment technique, the suggested test can be used to evaluate the efficacy of any treatment strategy in any field of medical study. This test, however, should not be used to compare two distinct treatment options. On the other hand, the efficiency of our proposed tests is compared to the tests of Kayid et al. [14] and δ ð2Þ F n presented by Qureshi and Jan [15], which are based on Pitman's asymptotic relative efficiency and employ several wellknown life distributions, namely, the LFR and the Weibull family. Finally, the paper's findings are tested using actual real-world data.

Data Availability
The data are mentioned along the paper.

Conflicts of Interest
The authors declare that they have no conflicts of interest.