Using Cauchy Distribution To Estimate Survival Function

: This paper intends to estimate the unlabeled two parameters for Cauchy distribution model depend on employing the maximum likelihood estimator method to obtain the derivation of the point estimators for all unlabeled parameters depending on iterative techniques , as Newton – Raphson method , then to derive “Lindley approximation estimator method and then to derive Ordinary least squares estimator method. Applying all these methods to estimate related probability functions; death density function, cumulative distribution function, survival function and hazard function (rate function)”. “When examining the numerical results for probability survival function by employing mean squares error measure and mean absolute percentage measure, this may lead to work on the best method in modeling a set of real data”

Where  : "is scale parameter"  : "is location parameter" "The cumulative distribution function for this distribution is": There is no chance to find the estimators for the parameters ) , (   , and it is kind of difficulty to process the nonlinear equations thus, it is better to make use of iterative methods in numerical analysis as Newton-Raphson method which is the best way to get the estimate values and number of iteration".
"The Newton-Raphson method requires an initial value of each unknown parameters" ) , (   . This method follows : We assumed that   , have the following Gamma conjugate prior distribution such that : ~) , ( a n "We make used of Lindley's approximate R which approximate the ratio of the two integrals to obtain Bayes estimators approximation that can be resulted as follows" : Using equation (6) we get the following:  (49)

Ordinary Least Squares Estimator Method (OLSEM):
"The OLSEM is the most used way to estimate parameters in linear or nonlinear model. Researchers make use of this method to lessen the sum squares differences concerning observed sample values and expected estimated values by linear approximation".
There is no chance to find the estimators for the parameters ) , (

 
, and it is kind of difficulty to process the nonlinear equations thus, it is better to make use of iterative methods in numerical analysis as Newton-Raphson method which is the best way to get the estimate values and number of iteration.
The Newton-Raphson method requires an initial value of each unknown parameters ) ,

Results and Discussion:
The Educational Hospital in Diwaniyah province was the place from which the data was gathered.
Keeping in mind that this work relies on data taken from real life, it is reached to select this kind of cancer (Breast cancer) because it is remarkable widespread and deadly in Iraq; this disease has failure time (death time) which is phenomenon in this paper.
The study of this paper covers a period of six months; it begins from Jun 2019 until December 2019; it is an experiment that includes (14) patients.(12) patients were dead and (2) patients remain alive .
When applying the test statistic (Kolmogorov-Smirnov) depending upon statistical programming (EasyFit 5.5 Professional) in order to fit Cauchy distribution data , it is discovered that the calculated value is (0. 11034) , this means data is distributed according to Cauchy distribution . The null and alternative hypotheses are as follows : When applying MATLAB (R2014a) , the estimated parameters results are as follows : The assumed initial values for two-parameters are as follows:    1-We notice in both methods that the estimated values of the probability survival function decrease with increasing failure times (an inverse relationship between them). 2-We notice in both methods that the estimated values of the potential risk function increase with increasing times of failure (a direct relationship between them). 3-It is recommended to use (LAEM) of Cauchy distribution of Breast cancer by employing MSE criterion.