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Statistics is one of the branches of mathematics related to data collection, processing, analyzing and conclusion. One of the analyzes used in the branch of statistics is the analysis of life time data. Life time data analysis is one of the most useful statistical techniques for analyzing testing on the survival or reliability of a component or individual called survival analysis. The survival analysis includes various statistical techniques that are useful for analyzing random variables. Random variables in survival analysis are either a survival or failure time (Collett, 1997). In conducting survival analysis, survival data is needed which includes the survival time and the status of the survival time of the object under study.

The survival data is a non-negative random variable because it consists of survival time of a set of observation objects. Survival data can be either uncensored or censored data. But in fact, to observe the survival time of all observation objects takes much time and cost so that the survival data is usually censored data. Censored data is data that can’t be observed as a whole because the object of observation is lost/out so it’s not known the actual time of the incident, the object of observation has an occurrence outside which the researcher noticed, or until the end of the research object has not experienced an event that is noticed. The censored data consisted of right, left and interval censored data. The right censored data consists of two types, namely type I and type II censored data. The right censored type II data is the time-sequence data where the observation is stopped after the first object r obtained by the occurrence of n the observed object with 1 < r < n. (Klein & Moeschberger, 1997). By observing various survival data on the right censored type II, a survival graph was obtained that could represent survival function. Figure 1 is a survival graph obtained from the data.

 

Figure 1. Survival graph of right-censored survival data type II.From the survival graph, the survival function is exp(-x^2/theta). After the survival function is obtained, then the probability density function (PDF) can be searched. The probability density function (PDF) of the data is f(x)=2x/theta exp(-x^2/theta). It turns out that the function is a probability density function of Rayleigh distribution.  Things that can not be separated from the assessment of distribution is about parameter estimates. Estimated parameters can be point estimates and interval estimates. One of the methods for finding point estimates is Bayes method. The Bayes method is a method that uses or incorporates subjective (past) knowledge of the parameters to be assessed with information obtained from the sample data. The preceding information is the prior information obtained from the distribution of parameters called the prior distribution. The prior distribution used in this final project is Invers Gamma Prior. The information from the data is summarized in the likelihood function. Combine prior information and information from the data will produce posterior information (Walpole, 2012). The conclusion of the estimated parameters is based on posterior information following a particular distribution called the posterior distribution. The posterior distribution of  can be written as follows with  is the prior distribution and  is a likelihood function. If  and  are not simple mathematical functions, then to solve the integral on the denominator will be difficult. One way to solve this difficulty is to limit ourselves to the prior distribution which is then known as the prior conjugate distribution. A prior is said to be conjugate prior to the given model if and only if the resulting posterior distribution comes from the same distribution as that prior distribution but the resulting parameters are different (Willmot, 2004). After obtaining the posterior distribution, then searched the estimated survival data of right censored type II that distributed Rayleigh with Square Error Loss Function. After obtaining the estimated point then also see the properties of the estimated parameters obtained. The estimated parameters obtained turn out to be biased estimates so it needs to be checked that the resulting variance should be minimum. As an illustration we will use examples of right censored survival data type II that distributed Rayleigh to see the estimated points obtained by the Bayes method and the properties of the parameter estimates.

Bayes method, conjugate prior, rayleigh distribution, right censored survival data type II, survival analysis