Parameter Estimation of Inverse Rayleigh Distribution under Competing Risk Model for Masked data

This paper tries to derive maximum likelihood estimators (MLEs) for the parameters of the inverse Rayleigh distribution (IRD) when the observed data is masked. MLEs, asymptotic confidence intervals (ACIs) and boot-p confidence intervals (boot-p CIs) for the lifetime parameters have been discussed. The simulation illustrations provided that as the sample size increases the estimated value approaches to the true value, and the mean square error decreases with the increase in sample size, and mean square error increases with increase in level of masking, the ACIs are always symmetric and the boot-p CIs approaches to symmetry as the sample size increases whereas the mean life time due to the local spread of the disease is less than that due to the metastasis spread in case of real data set.


INTRODUCTION
A random variable (rv) X is said to follow inverse Rayleigh distribution (IRD) if its probability density function (pdf) is given by .
The reliability function of IRD with pdf (1) is given as follows Trayer (1964) introduced the IRD and discussed its application in reliability theory.Voda (1972) discussed some properties of the maximum likelihood estimator (MLE) of θ and the property that the distribution of lifetimes of several types of experimental units can be approximated by the IRD.Mukherjee and Saran (1984) noticed that the hazard rate of IRD is increasing till , then it decreases and then becomes stable after some time.Various methods of estimation for IRD are developed by Gharraph (1993) and Mukherjee and Maiti (1996).A number of researchers have been discussed the IRD (Soliman et al. 2010, Dey 2012).
The theory of competing risk studies the subjects that are exposed to more than one cause of failure, and failure due to one cause excludes the chance of failure due to other causes.An investigator may use this theory to study the reliability characteristics of the components of series system.The failure of one component causes the breakdown of the system, the data can be collected in a pair of the lifetime of the system and the component which causes the failure.Crowder (2001) has presented appropriate competing risk methodology for analysis of such data.
In reliability analysis/competing risk analysis, with multiple causes of failures, there may be cases for which the exact cause of failure may be unknown.It makes the data incomplete.The incompleteness may be due to deficiency of knowledge about the exact cause of failure, identifiability of cause due to time consuming process etc.Such a data is termed as "masked" in literature.Sarhan (2003) discussed the estimations of parameters in Pareto reliability model in the presence of masked data.For some more citations, one may refer to Basu et al. (2003) and Lin et al. (1993).
In this context, this paper is to consider the competing risk analysis of masked data.Assuming the life times of components to be IRD, we obtain MLEs, asymptotic confidence intervals (ACIs) and boot-p confidence intervals (boot-p CIs) for the lifetime parameters of individual components using system lifetime data.We derive expressions to estimate the MLEs of parameter and to compute ACIs and boot-p CIs for parameters.This paper also attempts the simulation and application of proposed methods using real data sets.

MODELS
Some of the used notations and assumptions are given below.
 n #systems under observation  k #components in each system  1 n #systems failed due to component 1 x and a set i S of system's components that may cause it to fail. Masking is s-independent of the cause of failure.

Maximum Likelihood Estimation
Suppose an experiment is conducted with n identical 2component systems.


With these notations we write the likelihood function of the data as follows.

  
(3) Using ( 1) and (2), we get the from (3) that Taking logarithm of (4) and differentiating it partially with respect to 1  and 2  , respectively, we get From ( 5) and ( 6), we observe that the likelihood equations of 1  and 2  cannot be solved analytically.Therefore, we use iterative methods to evaluate these MLEs.Thus, using ( 5) and ( 6), we write the following expression for 1  and 2  in order to obtain their MLEs through numerical procedure.