Estimation

In life data analysis and reliability engineering, the output of the analysis is always an estimate. The true value of the probability of failure, the probability of success (or reliability), the mean life, the parameters of a distribution or any other applicable parameter is never known, and will almost certainly remain unknown to us for all practical purposes. (Note: If the reliability of a product at t hours is 80% then the probability of failure of the product for the same t hours is 100% - 80% = 20%.) Granted, once a product is no longer manufactured and all units that were ever produced have failed and all of that data has been collected and analyzed, one could claim to have learned the true value of the reliability of the product. Obviously, this is not a common occurrence.

The objective of reliability engineering and life data analysis is to accurately estimate these true values. For example, let's assume that our job is to estimate the number of black marbles in a giant swimming pool filled with black and white marbles. One method is to pick out a small sample of marbles and count the black ones. Suppose we picked out ten marbles and counted four black marbles.

Based on this sampling, the estimate would be that 40% of the marbles are black. If we put the ten marbles back in the pool and repeated this step again, we might get five black marbles, changing the estimate to 50% black marbles. The range of our estimate for the percentage of black marbles in the pool is 40% to 50%. If we now repeat the experiment and pick out 1,000 marbles, we might get results for the number of black marbles such as 445 and 495 black marbles for each trial. In this case, we note that our estimate for the percentage of black marbles has a narrower range, or 44.5% to 49.5%. Using this, we can see that the larger the sample size, the narrower the estimate range and, presumably, the closer the estimate range is to the true value.