#### Date

2014

#### Document Type

Dissertation

#### Degree

Doctor of Philosophy

#### Department

Mathematics

#### First Adviser

Eisenberg, Bennett

#### Other advisers/committee members

Huang, Wei-Min; Wu, Ping-Shi; Harlow, Gary

#### Abstract

In this thesis, the model being considered is a left Type-II censoring scheme with the underlying density being of the formf (x) = 1/&mu&xexp(-(x&minus&theta)/&mu), x&ge&thetaThis is what is known as an exponential distribution with scale parameter &mu and location parameter &theta. Our main focus is on the advantage of knowing one of the parameters in point estimation. For example, we ask to what advantage does an observer have in knowing &theta when estimating &mu to one who does not know either of the parameters. Our criteria between comparisons is Mean Square Error. (MSE) One of the most interesting results is that the relative advantage in knowing &theta when estimating &mu is the same as the relative advantage in knowing &mu when estimating &theta. Essentially all of our work revolves around considering a given proportion of the data 1 &minus p that is left censored and determining the asymptotic MSE of our estimators. In the work particularly done by Balakrishnan and Cohen, (See [4].) they derive formulas for the MSEs of the estimators with a much more general doubly Type-II censoring scheme, but they do not fix the proportion being censored and ask questions relating to the asymptotic nature of the MSEs for the estimators. The beauty of the ratio identity is that the limiting ratio turns out to be very close to the function y = p, where 1 &minus p is the proportion of the data that is censored. This is a consequence oftwo of our estimators nearly attaining the Cramer-Rao Lower Bound (CRLB) based on all the data for fairly small values of p. Another interesting result is how close the MSEs of these two estimators are as a function of p asymptotically.

#### Recommended Citation

Haines, Christopher, "The Relative Advantage of Knowing an Initiation Time or a Mean Time When Estimating a Parameter from an Exponential Distribution Using Type-II Left Censored Data" (2014). *Theses and Dissertations*. 1498.

https://preserve.lehigh.edu/etd/1498