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Fast Computation by Block Permanents of Cumulative Distribution Functions of Order Statistics from Several
D H Glueck1, A Karimpour-Fard, J Mandel
1Department of Preventive Medicine and Biometrics, University of Colorado Denver, Denver, Colorado, USA.
Communications in Statistics: Theory and Methods
|September 28, 2011
Summary
This study presents a new formula for the joint cumulative distribution function of order statistics from multiple populations. The method offers significant computational improvements, especially for two populations, with polynomial complexity for subsets.
Area of Science:
- Probability Theory
- Order Statistics
- Statistical Distributions
Background:
- Calculating joint cumulative distribution functions for order statistics from multiple populations is computationally intensive.
- Existing general formulas, such as Bapat and Beg's, have high worst-case computational complexity.
Purpose of the Study:
- To derive a new formula for the joint cumulative distribution function (CDF) of order statistics from several populations.
- To analyze the computational cost of the new formula and compare it with existing methods.
- To investigate the complexity for calculating the CDF of a subset of order statistics.
Main Methods:
- The study derives a formula for the joint CDF of order statistics based on the distribution functions of the individual populations.
- Computational complexity is analyzed for the general case and specifically for two populations.
- The complexity for computing the CDF of a fixed-size subset of order statistics is also examined.
Main Results:
- A novel formula for the joint CDF of order statistics from multiple populations is presented.
- For two populations, the new formula shows a dramatic improvement in computational cost compared to the general Bapat and Beg formula, though it remains exponential in the worst case.
- When only the joint CDF of a subset of order statistics is required, the complexity for two populations is reduced to polynomial.
Conclusions:
- The proposed formula provides a more computationally feasible approach for calculating joint CDFs of order statistics from multiple populations.
- The findings offer practical advantages for statistical analysis involving order statistics, particularly in scenarios with two populations or when analyzing subsets.
- Further research may explore extensions to more than two populations or different types of order statistics.
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