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

8.4.1.2.

Maximum likelihood estimation

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 MLE's.  

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, MLE's and Likelihood Functions generally have very desirable large sample properties: 

  • they become unbiased minimum variance estimators as the sample size increases
  • they have approximate normal distributions and approximate sample variances that can be calculated and used to generate confidence bounds
  • likelihood functions can be used to test hypotheses about models and parameters 
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 MLE's, but they are important ones: 
  • With small numbers of failures (less than 5, and sometimes less than 10 is small), MLE's can be heavily biased and the large sample optimality properties do not apply
  • Calculating MLE's often requires specialized software for solving complex non-linear equations. This is less of a problem as time goes by, as more statistical packages are upgrading to contain MLE analysis capability every year.
Additional information about maximum likelihood estimatation can be found in Chapter 1.
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:  $$ L = C \left( \prod_{i=1}^r f(t_i) \right) [1-F(T)]^{n - r} \, , $$ with \(C\) denoting a constant that plays no role when solving for the MLE's. 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: $$ L = C \left( \prod_{i=1}^k \left[ F(T_i) - F(T_{i-1}) \right]^{r_i} \right) [1-F(T)]^{n - \sum_{i=1}^k r_i} \, , $$ 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 \(f(t_i)\), \([F(T_i) - F(T_{i-1})]\), or \([1-F(t_i)]\), depending on whether the unit was an exact time failure at time \(t_i\), failed between two readouts \(T_{i-1}\) and \(T_i\), or survived to time \(t_i\) and was not observed any longer. 

The general mathematical technique for solving for MLE's 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 non-linear) 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) MLE's for the Exponential Model (Type I Censoring):

$$ L = C \lambda^r \mbox{ exp } \left( -\lambda \sum_{i=1}^r t_i \right) \left( e^{-\lambda(n-r)T} \right) $$ $$ \mbox{ln } L = \mbox{ln }C + r \mbox{ ln } \lambda - \lambda \sum_{i=1}^r t_i - \lambda(n-r)T $$ $$ \frac{\partial \mbox{ ln } L}{\partial \lambda} = \frac{r}{\lambda} - \sum_{i=1}^r t_i - (n-r)T = 0 $$ $$ \hat{\lambda} = \frac{r}{\sum_{i=1}^r t_i + (n-r)T} $$

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.

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