Question

Point estimates often need to be nested in layers of analysis, and it is the Invariance Principle that provides a pathway for doing so. An example would be estimating Mu after having to estimate Alpha and Beta for some distributions. In simpler statistics class exercises (like those we've seen up until now), this is typically avoided by providing the lower level parameters within exercises or problems (e.g. asking you for Mu by giving you the Alpha and Beta). The only real options we've had prior to this unit for estimating lower level parameters has been trial-and-error: collecting enough data to form a curve that we then use probability plots against chosen parameter values until we find a combination of parameters that "fits" the data we've collected. That approach works in the simplest cases, but fails as our problem grows larger and more complex. Even for a single distribution (e.g., Weibull) there are an infinite number of possible Alpha-Beta combinations. We can't manually test them all.

Point estimation gets us around all of that by providing the rules needed to actually calculate lower level parameters from data. We sometimes need to be able to collect a lot more data to use this approach, but it's worth it. We'll be able to calculate more than one possible value for many parameters, so it's important that we have rules for selecting from among a list of candidates.

Discuss what some of those rules are, and how they get applied in your analysis. If an engineering challenge includes "more than one reasonable estimator," (Devore, p. 249, Example 6.1 in Section 6.1) how do engineers know which to pick, and what issues arise statistically and in engineering management when making those choices?

Answer 1

Alpha and beta are probabilities of type 1 and type 2 errors each. According to the situation we decide which type of error is more precious to us(i.e which type of error will affect us the most. eg in medical testing, no diagnosis of a disease when disease is there can prove to be more fatal that actually diagnosing a disease that is not present.) hence, once we have decided on the important error, we set it to a value that minimises the other error as well. this can be done by plotting a graph for your distribution and using computers make things less complex.