# Estimation of a common location parameter

- Schätzungsprobleme bezüglich des gemeinsamen Lageparameters

Wang, Xiaofang; Kamps, Udo (Thesis advisor); Cramer, Erhard (Thesis advisor)

*Aachen (2014, 2016)*

Dissertation / PhD Thesis

Dissertation, RWTH Aachen, 2014

Abstract

In this thesis we focused on the estimation problem relating the common location parameter of several type II censored sequential order statistics (SOSs). In Chapter 1, the concept, density function and practical applications of SOSs were introduced. The density function of SOSs depends on a baseline function F which can be an arbitrary continuous function. Here, we restricted ourselves to some particular F chosen from a location-scale family. In Chapter 2, the unknown scale parameter(s) in one (or m) n-r+1-out-of system(s) was estimated; given that the system parameters, model parameters, the g-functions and the common location parameter were known. In this case, the family of probability measures of SOSs formed a one-parametric (or m-parametric) exponential family. The maximum likelihood estimator (MLE) and the uniformly minimum variance unbiased estimator (UMVUE) based on a sample of one or m different n-r+1-out-of-n system(s) were respectively given in Section 2.1 and 2.2. The (asymptotic) properties of the estimators were obtained by applying the properties of the one-dimensional or multidimensional exponential family. In Chapter 3, we estimated the unknown common location parameter and scale parameters in m different n-r+1-out-of-n systems, given that the system parameters, model parameters and the related g-functions were known. In this case, the family of probability measures of SOSs loses the structure of exponential family and we had to calculate the MLE and UMVUE and their properties with basic methods. These works were done in Section 3.1 and 3.2. In section 3.3, a modified maximum likelihood estimator (MMLE) of the common location parameter was introduced. To obtain the conditions such that the MMLE dominates the UMUVE in terms of the mean squared error, we presented some important identities. In Section 3.4, we proposed a class of estimators of the common location parameter, each of which dominates the MLE in terms of risk under a group of convex loss functions. Then, we found out that the estimators in this class are also Pitman closer than the MLE. In Chapter 4, we studied the interval estimation of the common location parameter with the model introduced in Chapter 3. In Section 4.1, two general ideas for constructing confidence sets were introduced. In Section 4.2, we gave some confidence intervals of the common location parameter by using some famous methods in meta-analysis (e.g. Fairweather's method, Cohen and Sackrowitz's method, Jordan and Krishnammoorthy's method, Tippet's Method, Wilkinson's method, Fisher's method, inverse normal method and logit method). In Chapter 5, theoretical results in Chapter 3 and Chapter 4 were illustrated by means of simulation study. In section 5.1, the simulations were carried out with randomly selected system parameters and model parameters. We computed the simulation results of some confidence levels and compared them with the real values. Moreover, we obtained general patterns about the size of estimators and lower bounds. In Section 5.2, chosen values were assigned to the system parameters and model parameters. In this case, the estimators given in Chapter 3 were compared in terms of relative risk improvements, whereas the mean lengths of the confidence intervals presented in Chapter 4 were listed for different confidence coefficients.

### Identifier

- URN: urn:nbn:de:hbz:82-rwth-2016-008849
- RWTH PUBLICATIONS: RWTH-2016-00884