8.
Assessing Product Reliability
8.4. Reliability Data Analysis 8.4.1. How do you estimate life distribution parameters from censored data?


There is nothing visual about the maximum likelihood method  but it is a powerful method and, at least for large samples, very precise 
Maximum likelihood estimation begins with writing a mathematical expression
known as the Likelihood Function of the sample data. Loosely
speaking, the likelihood of a set of data is the probability of obtaining
that particular set of data, given the chosen probability distribution
model. This expression contains the unknown model parameters. The values of
these parameters that maximize the sample likelihood are known as the
Maximum Likelihood Estimates or MLEs.
Maximum likelihood estimation is a totally analytic maximization procedure. It applies to every form of censored or multicensored data, and it is even possible to use the technique across several stress cells and estimate acceleration model parameters at the same time as life distribution parameters. Moreover, MLEs and Likelihood Functions generally have very desirable large sample properties:


With small samples, MLE's may not be very precise and may even generate a line that lies above or below the data points 
There are only two drawbacks to MLEs, but they are important ones:


Likelihood equation for censored data 
Likelihood Function Examples for Reliability Data:
Let \(f(t)\) be the PDF and \(F(t)\) the CDF for the chosen life distribution model. Note that these are functions of \(t\) and the unknown parameters of the model. The likelihood function for Type I Censored data is: $$ L = C \left( \prod_{i=1}^r f(t_i) \right) [1F(T)]^{n  r} \, , $$ with \(C\) denoting a constant that plays no role when solving for the MLEs. Note that with no censoring, the likelihood reduces to just the product of the densities, each evaluated at a failure time. For Type II Censored Data, just replace \(T\) above by the random end of test time \(t_r\). The likelihood function for readout data is: $$ L = C \left( \prod_{i=1}^k \left[ F(T_i)  F(T_{i1}) \right]^{r_i} \right) [1F(T)]^{n  \sum_{i=1}^k r_i} \, , $$ with \(F(T_0)\) defined to be 0. In general, any multicensored data set likelihood will be a constant times a product of terms, one for each unit in the sample, that look like either \(f(t_i)\), \([F(T_i)  F(T_{i1})]\), or \([1F(t_i)]\), depending on whether the unit was an exact time failure at time \(t_i\), failed between two readouts \(T_{i1}\) and \(T_i\), or survived to time \(t_i\) and was not observed any longer. The general mathematical technique for solving for MLEs involves setting partial derivatives of \(\mbox{ln } L\) (the derivatives are taken with respect to the unknown parameters) equal to zero and solving the resulting (usually nonlinear) equations. The equation for the exponential model can easily be solved, however. 

MLE for the exponential model parameter \(\lambda\) turns out to be just (total # of failures) divided by (total unit test time) 
MLEs for the Exponential Model (Type I Censoring):
$$ L = C \lambda^r \mbox{ exp } \left( \lambda \sum_{i=1}^r t_i \right) \left( e^{\lambda(nr)T} \right) $$ $$ \mbox{ln } L = \mbox{ln }C + r \mbox{ ln } \lambda  \lambda \sum_{i=1}^r t_i  \lambda(nr)T $$ $$ \frac{\partial \mbox{ ln } L}{\partial \lambda} = \frac{r}{\lambda}  \sum_{i=1}^r t_i  (nr)T = 0 $$ $$ \hat{\lambda} = \frac{r}{\sum_{i=1}^r t_i + (nr)T} $$ Note: The MLE of the failure rate (or repair rate) in the exponential case turns out to be the total number of failures observed divided by the total unit test time. For the MLE of the MTBF, take the reciprocal of this or use the total unit test hours divided by the total observed failures. There are examples of Weibull and lognormal MLE analysis later in this section. 