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SOME NEW APPLICATIONS OF P–P PLOTS

Published online by Cambridge University Press:  28 March 2013

Isha Dewan
Affiliation:
Indian Statistical Institute, New Delhi, India E-mail: isha@isid.ac.in
Subhash Kochar
Affiliation:
Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, Portland, OR E-mail: kochar@pdx.edu

Abstract

The P–P plot is a powerful graphical tool to compare stochastically the magnitudes of two random variables. In this note, we introduce a new partial order, called P–P order based on P–P plots. For a pair of random variables (X1, Y1) and (X2, Y2) one can see the relative precedence of Y2 over X2 versus that of Y1 over X1 using P–P order. We show that several seemingly very technical and difficult concepts like convex transform order and super-additive ordering can be easily explained with the help of this new partial order. Several concepts of positive dependence can also be expressed in terms of P–P orders of the conditional distributions.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2013 

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References

1.Avérous, J. & Dortet-Bernadet, J.L. (2000). LTD and RTI dependence orderings. Canadian Journal of Statististics 28: 151157.CrossRefGoogle Scholar
2.Avérous, J., Genest, C., & Kochar, S.C. (2005). On the dependence structure of order statistics. Journal of Multivariate Analysis 94: 159171.CrossRefGoogle Scholar
3.Barlow, R.E. & Proschan, F. (1981). Statistical theory of reliability and life testing. Silver Spring, Maryland: To Begin with.Google Scholar
4.Capera, P. & Genest, C. (1990). Concepts de dependance et orders stochastiques pour des lois bidimensionnelles. Canadian Journal of Statistics 18: 315326.CrossRefGoogle Scholar
5.Dolati, A., Genest, C., & Kochar, S.C. (2008). On the dependence between the extreme order statistics in the proportional hazards model. Journal of Multivariate Analysis 99: 777786.CrossRefGoogle Scholar
6.Fang, Z. & Joe, H. (1992). Further developments on some dependence orderings for continuous bivariate distributions. Annals of Institute of Statistical Mathematics 44: 501517.CrossRefGoogle Scholar
7.Holmgren, E.B. (1989). Using P–P plots in Meta-analysis as general measures of treatment effects. Technical report - Stanford University.Google Scholar
8.Holmgren, E.B. (1995). The P–P plot as a method for comparing treatment effects. Journal of American Statistical Association 90: 360365.CrossRefGoogle Scholar
9.Khaledi, B. & Kochar, S.C. (2000). Some new results on stochastic comparisons of parallel systems. Journal of Applied Probability 37: 11231128.CrossRefGoogle Scholar
10.Kochar, S.C., Li, X., & Shaked, M. (2002). The total time on test transform and the excess wealth stochastic orders of distributions. Advances in Applied Probability 34(2002), 826845.CrossRefGoogle Scholar
11.Kochar, S.C. & Weins, D. (1987). Partial orderings of life distributions with respect to their aging properties. Naval Research Logistic 34, 823829.3.0.CO;2-R>CrossRefGoogle Scholar
12.Kochar, S.C. & Xu, M. (2009). Comparisons of parallel systems according to convex transform order. Journal of Applied Probability 46: 342352.CrossRefGoogle Scholar
13.Lehmann, E.L. (1966). Some concepts of dependence. Annals of Mathematical Statistiscs 37: 11371153.CrossRefGoogle Scholar
14.Marshall, A.W. & Olkin, I. (2007). Life Distributions, structure of nonparametric, semiparametric, and parametric families. New York: Springer.Google Scholar
15.Schriever, B.F. (1987). An ordering for positive dependence. Annals of Statistics 15: 12081214.CrossRefGoogle Scholar
16.van Zwet, W.R. (1970). Convex transformations of random variables. Mathematical Centre Tracts No. 7, 2nd ed.Amsterdam: Mathematical Centre.Google Scholar
17.Yanagimoto, T. & Okamoto, M. (1969). Partial orderings of permutations and monotonicity of a rank correlation statistic. Annals of Institute of Statistical Mathematics 21: 489506.CrossRefGoogle Scholar