ECE565: ESTIMATION AND DETECTION PROJECT Polya’s Distribution Parameters Estimation Yousef Qassim, Arash Abbasi 1. INTRODUCTION This project focuses on estimating the parameters of beta binomial distribution, with the parameters α 1 and α 2. The distribution is given by where n(x) is the length of x, ,and n(x i ) is the length of observation vector x i . This distribution is one dimensional version of the multivariate Polya distribution. The Beta binomial distribution is a family of discrete probability distribution on finite support arising when the probability of success p of a known number of Bernoulli trials is random. The probability of success is drawn randomly from the Beta distribution with the parameters α 1 and α 2 , and the observation vectors x i s are drawn from the Binomial distribution with probability vector p. An example of this distribution is a model known as Polya urn, where two colored balls green and blue placed with a probability p. Then a ball is drawn from this urn randomly, i.e. binomial distribution. The goal of this project is to calculate the CRLB, and estimate the parameters α 1 and α 2 using maximum likelihood and method of moments. Finally, compare the results of MSE ML and MOM with the CRLB. 2. FIM and CRLB 2.1 Theoretical Background To compute the FIM and CRLB, it is essential to find the log likelihood function of In order to compute FIM, it is enough to compute FIM for one of the observation vector x i . m k i k k i k 1 1 m i=1 k k i i k k k k Γ( α) n(x )! Γ(n (x )+α )) p(x ,x ,...,x )= ( ), n (x )! Γ(n(x )+ α )) Γ(α )
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Yousef Qassim, Arash Abbasi 1. INTRODUCTIONweb.engr.oregonstate.edu/~qassimy/Documents/ECE565_Est_Project_… · Yousef Qassim, Arash Abbasi 1. INTRODUCTION ... The goal of this project
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ECE565: ESTIMATION AND DETECTION PROJECT
Polya’s Distribution Parameters Estimation
Yousef Qassim, Arash Abbasi
1. INTRODUCTION
This project focuses on estimating the parameters of beta binomial distribution, with the
parameters α1 and α2. The distribution is given by
where n(x) is the length of x, ,and n(xi) is the length of observation vector
xi. This distribution is one dimensional version of the multivariate Polya distribution. The Beta
binomial distribution is a family of discrete probability distribution on finite support arising when the
probability of success p of a known number of Bernoulli trials is random. The probability of success
is drawn randomly from the Beta distribution with the parameters α1 and α2, and the observation
vectors xis are drawn from the Binomial distribution with probability vector p. An example of this
distribution is a model known as Polya urn, where two colored balls green and blue placed with a
probability p. Then a ball is drawn from this urn randomly, i.e. binomial distribution.
The goal of this project is to calculate the CRLB, and estimate the parameters α1 and α2 using
maximum likelihood and method of moments. Finally, compare the results of MSE ML and MOM
with the CRLB.
2. FIM and CRLB
2.1 Theoretical Background
To compute the FIM and CRLB, it is essential to find the log likelihood function of
In order to compute FIM, it is enough to compute FIM for one of the observation vector xi.