## Retrospective Theses and Dissertations

#### Title

Order statistics under linear trend

Dissertation

1991

#### Degree Name

Doctor of Philosophy

Statistics

H. A. David

#### Abstract

Order statistics and linear functions of order statistics are frequently used as estimators of location and scale. The moments of order statistics have been tabled for many commonly used distributions under assumptions of independence and identical distribution. Applications of estimators based on the order statistics have spread to situations in which identical distribution cannot be assumed, even though independence is still a sensible assumption. As an example, estimators based on order statistics are used in the field of communications where the median and other functions of order statistics of a moving sample are used to recover a signal contaminated with random noise. In this case, if we assume that the noise has a distribution which is independent over time, then the sample is independent. However it is not identically distributed, since the means of the observations follow the original signal. The signal could have any shape, but we will consider only the simplest case, that of (locally) linear trend. We want to describe the effect of linear trend on the moments of the order statistics and to follow up on a conjecture (David 1988) that the variances of the median increase with trend for certain types of distributions;We present computed tables of means, variances, and covariances of the order statistics of normal random variables which are assumed independent with unit variance, having means exhibiting linear trend. Then several uses of these tables are given, one in statistical process control and the other in the evaluation of estimators based on order Statistics and Probability; Estimators based on order statistics are also evaluated for a model combining linear trend with a large outlier;The theory involved with ordering in dispersion is discussed using majorization arguments and stochastic ordering. We then take a more basic case, a case in which only 2 of n observations have differing means, and look at a particular distribution, the logistic. Since the cdf of this distribution may be written in closed form, it is comparatively easy to get results about the effect of trend on the variances of order statistics, and we conclude that the variance of the median in odd samples and the variances of the extremes increase with trend.

#### DOI

https://doi.org/10.31274/rtd-180813-9265

#### Publisher

Digital Repository @ Iowa State University, http://lib.dr.iastate.edu/

James Robison-Cox

en

AAI9207253

application/pdf

91 pages

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