Event Type:

Probability Seminar

Date/Time:

Tuesday, November 21, 2017 - 16:00 to 17:00

Location:

BEXL 207

Guest Speaker:

Institution:

Oregon State University

Abstract:

This talk concerns a somewhat innocent question motivated by an observation concerning the use of Chebyshev bounds on sample estimates of *p* in the binomial distribution with parameters *n*, *p*. Namely, what moment order produces the best Chebyshev estimate of *p*? If S_{n}(*p*) has a binomial distribution with parameters *n*, *p*, then it is readily observed that argmax_{0 ≤ p ≤ 1}**E**S_{n}^{2}(*p*) = argmax_{0 ≤ p ≤ 1}*np*(1-*p*) = 1/2, and **E**S_{n}^{2} = *n*/4. Rabi Bhattacharya observed (personal communication) that while the second moment Chebyshev sample size for a *95%* confidence estimate within

±5 percentage points is *n* = 2000, the fourth moment yields the substantially reduced polling requirement of *n* = 775. Why stop at fourth moment? Is the argmax achieved at *p* = 1/2 for higher order moments and, if so, does it help, and can one easily compute the moment bound **E**S_{n}^{2m}(1/2)? As captured by the title of this talk, answers to these questions lead to a simple rule of thumb for best choice of moments in terms of an effective sample size for Chebyshev concentration inequalities. This talk is based on joint work with Chris Jennings-Shafer and Dane Skinner.