Distribution of Rankings for Groups Exhibiting Heteroscedasticity and Correlation

By Gilbert, Scott | Journal of the American Statistical Association, March 2003 | Go to article overview

Distribution of Rankings for Groups Exhibiting Heteroscedasticity and Correlation


Gilbert, Scott, Journal of the American Statistical Association


1. INTRODUCTION

Consider a population consisting of p groups, for which [theta] is a p vector of constants with [[theta].sub.i] describing the ith group, and suppose that there is interest in the relative magnitudes of [[theta].sub.1],...,[[theta].sub.p]. This is a situation arising frequently in the analysis of two-way data. The constants are presumed to be unobservable and are estimated by some [[theta].sub.1]...,[[theta].sub.p], which have statistical rankings [r.sub.1],...,[r.sub.p], defined (in terms of ascending order) as the smallest numbers in the set {1,..., p} such that [r.sub.i]<[r.sub.j] if [[theta].sub.i]<[[theta].sub.j], [r.sub.i] = [r.sub.j] if [[theta].sub.i] = [[theta].j], for i, j = l,...,p.

The probability distribution of rankings, given by P([r.sub.i] = j), i, j, = 1,..., p, is useful in describing the success likelihood of competitors in racing events, and variations on this basic theme are common in many fields, including economics [e.g., bidding on contracts (see, e.g., Engel, Fischer, and Galetovic 2001) and global economic competition (Amato and Amato 2001)], and biology [rival contests in mating (Radcliffe and Rass 1998)]. Of particular interest is the ranking probabilities that describe the likelihood of outcomes in some future experiment for groups that may exhibit heterogeneity both in the parameters [[theta].sub.i] of interest and in other parameters (variances) and may exhibit mutual dependence.

Ranking probabilities are described for the case in which where the estimator [theta] is well approximated by a normal distribution, in a finite sample or asymptotically, with attention given to intergroup heteroscedasticity and correlation. Hence interested is not only in the group constants [[theta].sub.1],...,[[theta].sub.p], but also in the joint contribution of these and other parameters (variances and covariances) to the ranking distribution. In comparison, a great deal of previous work has focused on the problem of multiple inferences and simultaneous confidence intervals for [[theta].sub.1]..., [[theta].sub.p] (Hochberg and Tamhane 1987; Toothaker 1993; Hsu 1996). Other work on the selection problem has developed and evaluated methods for picking the best group (in population terms) using sample statistics (Bechhofer, Elmaghraby, and Morse 1959; Gibbons, Olkin, and Sobel 1977; Gupta and Panchapeakesan 1979; Lam 1989; Hoppe 1993; Mukhopadhyay and Solanky; 1994 Bechofer, Santner, and Goldsman 1995). The selection literature largely deals with the case in which group statistics [[theta].sub.1],..., [[theta].sub.p] are independent and differ in distribution only in [[theta].sub.1],..., [[theta].sub.p] with the focus on [theta] and decision procedures related to [theta]. Recently, Gupta and Miescke (1988), Nelson and Matejcik (1995), Nelson, Swann, Goldsman, and Song (2001) and Kim and Nelson (2001) have developed selection methods that allow for beteroscedastic and correlated groups.

The range of possible ranking distributions is quite diverse under normality and is analytically complex. By exploiting symmetries and the elliptical geometry of the normal distribution, the range of feasible ranking probabilities can be partially characterized under the hypothesis of equal population rankings ([[theta].sub.1] = [[theta].sub.2] = ... = [[theta].sub.p]) and under the alternative. The presence of intergroup heteroscedasticity and/or correlation greatly increases the range of feasible ranking probabilities under each hypothesis. A full characterization of the possible means and covariances consistent with a given ranking distribution is beyond the scope of this article.

Estimating the ranking distribution requires an historical sample that is suitably large relative to the planned sample. In addition to nonparametric methods based on historical event frequencies, a class of parametric estimators is proposed that includes a "plug-in" method, in which historical moment estimates are substituted for population moments in generating normal ranking probabilities, as well as methods that incorporate Bayesian uncertainty about the moment values. …

The rest of this article is only available to active members of Questia

Sign up now for a free, 1-day trial and receive full access to:

  • Questia's entire collection
  • Automatic bibliography creation
  • More helpful research tools like notes, citations, and highlights
  • A full archive of books and articles related to this one
  • Ad-free environment

Already a member? Log in now.

Notes for this article

Add a new note
If you are trying to select text to create highlights or citations, remember that you must now click or tap on the first word, and then click or tap on the last word.
One moment ...
Default project is now your active project.
Project items

Items saved from this article

This article has been saved
Highlights (0)
Some of your highlights are legacy items.

Highlights saved before July 30, 2012 will not be displayed on their respective source pages.

You can easily re-create the highlights by opening the book page or article, selecting the text, and clicking “Highlight.”

Citations (0)
Some of your citations are legacy items.

Any citation created before July 30, 2012 will labeled as a “Cited page.” New citations will be saved as cited passages, pages or articles.

We also added the ability to view new citations from your projects or the book or article where you created them.

Notes (0)
Bookmarks (0)

You have no saved items from this article

Project items include:
  • Saved book/article
  • Highlights
  • Quotes/citations
  • Notes
  • Bookmarks
Notes
Cite this article

Cited article

Style
Citations are available only to our active members.
Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

(Einhorn, 1992, p. 25)

(Einhorn 25)

1

1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

Cited article

Distribution of Rankings for Groups Exhibiting Heteroscedasticity and Correlation
Settings

Settings

Typeface
Text size Smaller Larger Reset View mode
Search within

Search within this article

Look up

Look up a word

  • Dictionary
  • Thesaurus
Please submit a word or phrase above.
Print this page

Print this page

Why can't I print more than one page at a time?

Help
Full screen

matching results for page

    Questia reader help

    How to highlight and cite specific passages

    1. Click or tap the first word you want to select.
    2. Click or tap the last word you want to select, and you’ll see everything in between get selected.
    3. You’ll then get a menu of options like creating a highlight or a citation from that passage of text.

    OK, got it!

    Cited passage

    Style
    Citations are available only to our active members.
    Sign up now to cite pages or passages in MLA, APA and Chicago citation styles.

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn, 1992, p. 25).

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences." (Einhorn 25)

    "Portraying himself as an honest, ordinary person helped Lincoln identify with his audiences."1

    1. Lois J. Einhorn, Abraham Lincoln, the Orator: Penetrating the Lincoln Legend (Westport, CT: Greenwood Press, 1992), 25, http://www.questia.com/read/27419298.

    Cited passage

    Thanks for trying Questia!

    Please continue trying out our research tools, but please note, full functionality is available only to our active members.

    Your work will be lost once you leave this Web page.

    For full access in an ad-free environment, sign up now for a FREE, 1-day trial.

    Already a member? Log in now.