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Suppose that $X$ is a discrete random variable with

  • $P(X= 0) = 2θ/3$
  • $P(X=1) = 1θ/3$
  • $P(X=2) = 2(1-θ)/3$
  • $P(X=3) = (1-θ)/3$

where $0\le θ\le 1$ is a parameter. The following $n=10$ independent observations were taken from such a distribution: $( 3,0,2,1,3,2,1,0,2,1)=(x_1,\ldots,x_n)$

1)What is the maximum likehood estimate of $θ$?

2) What is the approximate standard error of the maximumlikehood estimate. (Question is from Statistics and Data Analysis by Rice, 3rd edition)

Well, we know the mle is given by $L(p)=f(x_1,\ldots,x_n|p)= \prod_{i=1}^n f(x_i|p) $ which gives the mle for $p$. (Taking the log and setting the derivative to $0$.) (I have just called $\theta$ as $p$.)

However, how should you calculate the approx. standard error in $p$? I read something about the inverse of an information matrix, but the book in question doesn't mention this so I guess there is a simpler way of doing this. Should I just calculate the standard deviation of $X$ and plug in $p$ in it?

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  • $\begingroup$ You are supposed to answer completely question 1 before tackling question 2, hence you must first reach an explicit formula for $\widehat\theta$ as a function of the sample $(x_1,x_2,\ldots,x_n)$. $\endgroup$ – Did Feb 28 '12 at 16:03
  • $\begingroup$ Not interested anymore? $\endgroup$ – Did Mar 2 '12 at 18:55

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