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Consider the family of probability mass functions given by

f(x;k) = 3(4^(k-x)) x = k + 1, k + 2,....

and indexed by parameter k E Z. For a random sample of size n, derive with justification:

a) the method of moments estimator for k.

b) the maximum likelihood estimator for k.

The MME isn't too bad, after a bit of algebra I get the sample mean - 4/3. The MLE is a bit more difficult. I don't think any calculus is useful here? In the likelihood function I need to maximise (nk -(sumXi)) which means that the biggest value of k will do this. But what is confusing me is that the parameter, k, is in the support of X. I dont know if this affects the summation or what. So pretty stuck at this point if anybody could help? Thanks

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1 Answer 1

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Assume that you have a sample $\bar x=x_1,\ldots,x_n$ from the probability distribution given by $f(x;k)$. The likelihood function is

$L(k;\bar x)=\prod_{i=1}^n 3\cdot 4^{k-x_i}=(3\cdot4^k)^n4^{-\sum_{i=1}^n x_i},$

if $x_i>k\ \forall i,i=1,\ldots,n$ and 0 otherwise. $L(k;\bar x)$ is maximized by the largest possible value of $k$. Because the support of the distribution is given by $k+1,k+2,\ldots$, we have that $x_i>k \ \forall i,i=1,\ldots,n$ (otherwise $f(x_i;k)=0$). This is equivalent to $\min_i x_i>k$. Hence, we should maximize $L(k;\bar x)$ with the condition that $\min_i x_i>k$, so the MLE is $k=\min_i x_i-1$.

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  • $\begingroup$ Thanks so much man, really appreciate it. Makes a lot of sense $\endgroup$
    – Joshdn10
    Nov 7, 2014 at 16:23

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