Distributions of Sums of Nonidentical Random Variables

Authored by: Wim J. van der Linden

Handbook of Item Response Theory Volume Two Statistical Tools

Print publication date:  February  2016
Online publication date:  March  2017

Print ISBN: 9781466514324
eBook ISBN: 9781315373645
Adobe ISBN:

10.1201/b19166-8

 Download Chapter

 

Abstract

An obvious example of the subject of this chapter is the probability distribution of the number-correct score of a test taker on a test with dichotomous or polytomous items. However, similar types of distributions are met in statistical tests of cheating on tests, for instance, as null distributions of the number of matching responses between two test takers suspected of answer copying or the number of answer changes by a single test taker reviewing his answers in the detection of fraudulent erasures on answer sheets. In each of these cases, the desired distribution is found as the convolution of more elementary distributions. Although they may seem to lead to complicated combinatorics, a simple recursive algorithm exists that allows us to evaluate the convolutions efficiently and accurately. The same algorithm can be used to evaluate the posterior odds of certain events given sum scores or the combinatorial expressions that arise in conditional maximum-likelihood (CML) estimation of the parameters in the Rasch model. In addition to these discrete cases, the chapter addresses the distribution of the total time spent by a test taker on a test, which is found as the convolution of the distributions of his response times (RTs) on the items.

 Cite
Search for more...
Back to top

Use of cookies on this website

We are using cookies to provide statistics that help us give you the best experience of our site. You can find out more in our Privacy Policy. By continuing to use the site you are agreeing to our use of cookies.