Estimators

alkiii321

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Mar 16, 2019
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3
We make subsequent throws of a fake cubic cube for which the probability of falling out six is 1/6 - epsilon, the probability of falling out of one is 1/6 + epsilon and the others eyes drop out with the same probability 1/6. Provide a consistent and unbalanced estimator epsilon parameter and calculate its variance. I need help i start my trip with this ad it is so hard for me :/
 

Romsek

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Nov 16, 2013
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322
\(\displaystyle \text{Let }\bar{x} \text{ be the vector of }n \text{ rolls}\)
\(\displaystyle \text{Let }o = \text{#1's in }\bar{x},~s=\text{#6's in }\bar{x}\)
\(\displaystyle p_\epsilon(\bar{x}) = \left(\dfrac 1 6\right)^{n-o-s}\left(\dfrac 1 6 - \epsilon \right)^s \left(\dfrac 1 6 + \epsilon\right)^o\)

\(\displaystyle \log(p_\epsilon(\bar{x}) = (n-o-s)\log\left(\dfrac 1 6\right) + s \log\left(\dfrac 1 6 - \epsilon\right) + o \log\left(\dfrac 1 6 + \epsilon\right)\)

The maximum likelihood estimate will result from minimizing this expression over \(\displaystyle \epsilon\)

This is done using the usual calculus minimization technique of setting the first derivative to zero and solving.

Can you finish?
 

alkiii321

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Mar 16, 2019
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3
I don't understand. What is an estimator?
 

Romsek

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Nov 16, 2013
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322
I don't understand. What is an estimator?
If you don't know what an estimator is why have you been given this problem? (Maybe your school uses a different name.)

An estimator is a function that takes a set of observations and reduces them to an estimate of some property of the underlying random variable.
 
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