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Thread: Maximum likelihood function

  1. #1
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    Maximum likelihood function

    I'm doing it for a binomial random variable X with one observation X=x. All I want to know is whether I should write my estimator in terms of X or x.

    I believe it to be X. If that is so, should my working be in terms of x then at the final stage change x to X.

    Thanks
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  2. #2
    Super Member ILikeSerena's Avatar
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    Re: Maximum likelihood function

    Quote Originally Posted by Duke View Post
    I'm doing it for a binomial random variable X with one observation X=x. All I want to know is whether I should write my estimator in terms of X or x.

    I believe it to be X. If that is so, should my working be in terms of x then at the final stage change x to X.

    Thanks
    Hi Duke!

    An estimator is a function that estimates a parameter of your model, which in this case is a binomial distribution that has parameters n and p.
    Typically you would use a sample to do the estimation.

    What you would have is:

    X the random variable
    x a specific (possible) observation
    $\displaystyle \bar x$ the sample mean
    $\displaystyle \hat p = {\bar x \over n}$ the estimator for the parameter p of the binomial distribution.
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  3. #3
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    Re: Maximum likelihood function

    Ok but I mean in general would I want the estimator in terms of the random variable or the specific observations. For example you say (correctly) that p= $\displaystyle {\bar x \over n}$. But if I wanted E(p), I would do $\displaystyle E({\bar X \over n})$. So should it really be $\displaystyle \hat p =({\bar X \over n})$?
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  4. #4
    Super Member ILikeSerena's Avatar
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    Re: Maximum likelihood function

    The notation $\displaystyle \bar X$ represents the mean of n random variables, but I do not think that is what you intend.


    Typically x would represent the sample, and you would write $\displaystyle \hat p(x) = {\bar x \over n}$.

    Your expectation would be written as: $\displaystyle E(\hat p) = E[\hat p(X)] = E[X / n]$, which is an expectation based on a random variable, of which samples are taken to estimate $\displaystyle \hat p$.
    Btw, this expection is equal to the parameter p.


    In other words, the choice for the symbol depends on what you're talking about.
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