## Electronic Theses and Dissertations

#### Degree Name

MS (Master of Science)

#### Program

Mathematical Sciences

5-2000

Edith Seier

#### Committee Members

Robert M. Price Jr.

#### Abstract

L-moments are defined as linear combinations of expected values of order statistics of a variable.(Hosking 1990) L-moments are estimated from samples using functions of weighted means of order statistics. The advantages of L-moments over classical moments are: able to characterize a wider range of distributions; L-moments are more robust to the presence of outliers in the data when estimated from a sample; and L-moments are less subject to bias in estimation and approximate their asymptotic normal distribution more closely.

Hosking (1990) obtained an asymptotic result specifying the sample L-moments have a multivariate normal distribution as n approaches infinity. The standard deviations of the estimators depend however on the distribution of the variable. So in order to be able to build confidence intervals we would need to know the distribution of the variable.

Bootstrapping is a resampling method that takes samples of size n with replacement from a sample of size n. The idea is to use the empirical distribution obtained with the subsamples as a substitute of the true distribution of the statistic, which we ignore. The most common application of bootstrapping is building confidence intervals without knowing the distribution of the statistic.

The research question dealt with in this work was: How well do bootstrapping confidence intervals behave in terms of coverage and average width for estimating L-moments and ratios of L-moments? Since Hosking's results about the normality of the estimators of L-moments are asymptotic, we are particularly interested in knowing how well bootstrap confidence intervals behave for small samples.

There are several ways of building confidence intervals using bootstrapping. The most simple are the standard and percentile confidence intervals. The standard confidence interval assumes normality for the statistic and only uses bootstrapping to estimate the standard error of the statistic. The percentile methods work with the (α/2)th and (1-α/2)th percentiles of the empirical sampling distribution. Comparing the performance of the three methods was of interest in this work.

The research question was answered by doing simulations in Gauss. The true coverage of the nominal 95% confidence interval for the L-moments and ratios of L-moments were found by simulations.

#### Document Type

Thesis - unrestricted