The Independence of Size and Shape before and after Scale Change

  • James E. Mosimann
  • James D. Malley
Conference paper
Part of the NATO Advanced Study Institutes Series book series (ASIC, volume 79)


Let \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{X} \) and \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{Q} \) be k-dimensional positive random vectors related by an unequal linear scale change; that is Qi = ai Xi, ai > 0, i = 1,…,k, with some ai ≠ aj. In this paper we study the independence of shape and size (or size-ratios) before and after the scale change. If, before the change, shape is independent of size, G(\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{X} \)), then the new shape after the change is also independent of G(\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{X} \)). However, shape after the change is not independent of size after, G(\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{Q}\)), unless the ratio G(\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{X} \))/G(\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{Q} \)) is degenerate. Similarly, if shape before the change is independent of a ratio of related size variables, then shape after the change cannot be independent of the same ratio applied to \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{Q} \) unless again G(\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{X} \))/G(\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{Q} \)) is degenerate. A consequence is that if proportions (shape) follow a generalized Dirichlet distribution before the change on \(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{X} \) they cannot do so after the change.

Key Words

size variables shape variables scale change proportions generalized Dirichlet distribution Dirichlet distribution isometry neutrality 


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Copyright information

© D. Reidel Publishing Company 1981

Authors and Affiliations

  • James E. Mosimann
    • 1
  • James D. Malley
    • 1
  1. 1.National Institutes of HealthBethesdaUSA

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