Estimating the Parameters of Exponential-Rayleigh Distribution under Type-I Censored Data

: This paper discusses estimating the two scale parameters of Exponential-Rayleigh distribution for singly type one censored data which is one of the most important Rights censored data, using the maximum likelihood estimation method (MLEM) which is one of the most popular and widely used classic methods, based on an iterative procedure such as the Newton-Raphson to find estimated values for these two scale parameters by using real data for COVID-19 was taken from the Iraqi Ministry of Health and Environment, AL-Karkh General Hospital. The duration of the study was in the interval 4/5/2020 until 31/8/2020 equivalent to 120 days, where the number of patients who entered the (study) hospital with sample size is (n=785). The number of patients who died during the period of study was (m=88). And the number of patients who survived during the study period was (n-m=697), then utilized one of the most important non-parametric tests which is the Chi-square test to determine if the sample (data) corresponded with the Exponential-Rayleigh distribution (ER). then, after estimating the parameters of ER distribution for singly type-I censoring data, compute the survival function, hazard function, and probability density function.


Introduction:
In many life testing and reliability studies, the experimenters and researchers may not get complete information on failure times for experimental units, the Data obtained from experiments are called censoring data 1 . The censored sample is divided into three types: Right-censored sample, Leftcensored sample, and Interval-censored sample. The right-censored sample is also divided into three branches, singly type one censoring sample, singly type two censoring sample, and progressively censoring sample. This paper depends on the most important Rights censored data which type one censored data.
In 2013, Hussain I, et al, estimated the unknown parameters of the generalized Rayleigh distribution for singly type one censored sample by using one of the most important classical estimation methods which maximum likelihood estimation method 2 .
In 2016, Makhdoom I, et al, estimated the parameters of an Exponential distribution using approximation forms of Lindley, and approximation of Tierney and Kadane under type two censoring samples. And to compare the effectiveness of the various methods, a Monte Carlo simulation is used 1 .
In 2020, Abadi QS, et al, used the (MLE) method to estimate three parameters of a new mixture distribution consisting of one shape parameter and two scale parameters for type-I censored data and progressively censored data 3,4 .
In 2021, Heidari KF, et al, generalized the estimated parameters of Rayleigh distribution by using the Bayesian method for type two censoring data under the squared error loss function 5 .
In 2021, Mazaal AR, et al, based on a singly type II censored sample for Weibull Stress-Strength with considering the stress-strength Reliability estimation. And considered the Bayesian analysis using the loss function under (Extension of Jeffery and Gamma) which are prior functions 6 .

The
properties of Exponential-Rayleigh distribution This paper depends on the formula that was proven by Hussein LK, Hussein IH, and Rasheed HA in 2021, by using the mix between the cumulative of each distribution as follows 7 : Let T = max (v, w), where vandw are two independent random variables, then:

Parameters Estimation
This section shows the derivative and estimates the unknown parameters of (ER) distribution, using the maximum likelihood estimation method for type-one censoring sample. Type-One Censoring Data 3 It is one of the most common rightcensored types, the time of experiments denoted by t, which is fixed, while the number of failures time denoted by m observed is random. However, n represents the individuals or items that are placed in the study, and the units or individuals that did not fail or survive are denoted by (n-m).

Maximum Likelihood Estimation Method for Type-One Censoring Data
The maximum likelihood estimation method is one of the most popular and widely used classic methods. This method was initially used in 1912 by R.A. Fisher in his first statistical papers. Maximum likelihood describes a technique for estimating the unknown parameters of any distribution 8 .
Taking the natural logarithm for both sides of the equation: Deriving Eq.7 partially with respect to and respectively, and setting it equal to zero: Now, putting θ as a function f(θ) and put as a Note that Eq.10 and Eq.11 are so difficult to solve, then employing them by an iterative method such as Newton Raphson procedure 9 to find the value of θand β as: The error term denoted by , which is very small and assumed value, is the value for the difference between the new values of and in the new iterative with the previous values of and in the past iterative. The error term is formulated as: +1 (θ) = θ k+1 − θ k +1 (β) = β k+1 − β k Where θ k and β k are initial values that are assumed.

Results and Discussion:
This study is based on a sample and real data taken from the Iraqi Ministry of Health and Environment, AL-Karkh General Hospital about COVID-19. The duration of the study was the The formula of this test is given by: Where: O i refers to the observed in class i. E i refers to the expected frequency in class i. k refers to the number of class.
The calculated value (15.32008 ) is less than the tabulated value (21.67) with the degree of freedom (9) and level of significance (0.01). That means accepting the null hypothesis H 0 and the data is distributed as (ER) distribution.
After that, employing MATLAB programming (version 2021) to estimate values of the following parameters:θ = 0.0100, β = 0.0183. When the initial values are θ 0 = 0.0120, β 0 = 0.0001. Substitute the estimated values of parameters and the values of the lifetime in Eq.1 , Eq.2 , Eq.3. And Eq.4 Then get the values of f̂(t), F(t), Ŝ(t), and ĥ(t) in the following