This is the program for the 2010 Joint Statistical Meetings in Vancouver, British Columbia.
Abstract Details
Activity Number:
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210
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Type:
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Invited
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Date/Time:
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Monday, August 2, 2010 : 2:00 PM to 3:50 PM
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Sponsor:
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Section on Nonparametric Statistics
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Abstract - #305947 |
Title:
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Diversity in a Sample and Large Number of Small Probabilities: The Case of Questionnaires
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Author(s):
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Estate V. Khmaladze*+
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Companies:
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Victoria University of Wellington
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Address:
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Kelburn Parade, Wellington, 600, New Zealand
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Keywords:
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Karlin-Rouault law ;
probabilities of unseen events ;
Good-Turing indices ;
Zipf's law
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Abstract:
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Consider a questionnaire with $q$ binary questions, which is filled by $N$ individuals, thus providing $N$ ``opinions" $\overrightarrow x = \{x_i\}_{i=1}^q$, with each $x_i$ being $0$ or $1$. For $q$ large, most probabilities $p(\overrightarrow x)$ are small and many of them are very small. However, how many how small probabilities do we have typically is not quite arbitrary and under general assumptions follows the asymptotics $$ \sum_{\overrightarrow x} {\mathbb I}_{\{2^q p(\overrightarrow x)>z\}} \sim c_q z^{-u}, \; {\rm as} \: q\to\infty $$ with specified $c_q$. If $\mu_q$ and $\mu_q(k)$ denote the number of different opinions and the number of opinions occurring $k$ times in the sample, we show that \eqref{inv} is equivalent to the convergence $$ \frac{\mu_q(k)}{\mu_q}\to \frac {u\Gamma(k-u)}{\Gamma(k+1)\Gamma(1-u)} $$
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