#### Date Thesis Awarded

6-2013

#### Document Type

Honors Thesis

#### Degree Name

Bachelors of Science (BS)

#### Department

Mathematics

#### Advisor

Tanujit Dey

#### Committee Member

Ross Iaci

#### Committee Member

M. Drew Lamar

#### Committee Member

Lawrence M. Leemis

#### Abstract

The goal of this thesis is to examine methods of statistical inference based on upper record values. This includes estimation of parameters based on samples of record values and prediction of future record values. We first define and discuss record times and record values and their distributions. Then we propose an efficient algorithm to generate random samples of record values. The algorithm, based on the conditional survivor function, has a time complexity that is linear with respect to the sample size. It is quite efficient and can be useful in simulation. Next, we discuss inference problems related to two distributions, the generalized inverted exponential distribution and the generalized Rayleigh distribution. A number of techniques are considered, including both frequentist and Bayesian techniques. The following are considered for each of these two distributions. We first consider maximum likelihood and Bayesian estimation of the unknown parameters based on upper record values. We also consider empirical Bayes estimation. Then, we derive approximate confidence intervals based on a normal approximation, bootstrap confidence intervals, and two-sided Bayes probability intervals based on record values. We derive both maximum likelihood and Bayesian methods for predicting future record values. Numerical results include both simulation and data analysis. We conduct a simulation study for each distribution. Simulation is used to compare the performance of maximum likelihood estimators, Bayesian estimators, and empirical Bayes estimators. We consider the average bias and mean squared error for these estimators. We also consider the average length and coverage probability for approximate confidence intervals. As many of the results in this paper cannot be found in closed form, we use numerical methods to find the maximum likelihood estimates and we implement the Metropolis--Hastings algorithm to compute the Bayes estimates. We use each distribution to model one real data set by estimating the parameters, computing approximate confidence intervals and two-sided Bayes probability intervals, and evaluating the fit. Finally, we illustrate Bayesian prediction methods on a simulated sample for each distribution.

#### Recommended Citation

Luckett, Daniel J., "Statistical Inference Based on Upper Record Values" (2013). *Undergraduate Honors Theses.* Paper 577.

http://publish.wm.edu/honorstheses/577

#### Creative Commons License

This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 3.0 License.

## Comments

Thesis is part of Honors ETD pilot project, 2008-2013. Migrated from Dspace in 2016.