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Math Help - Maximum Likelihood Estimator Problem

  1. #1
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    Maximum Likelihood Estimator Problem

    The lifetime, Y, of a certain type of component is known to follow the Exponential distribution. So for a sample of n such components, we know that the maximum likelihood estimator for Theta is Theta^= Y bar. The reliability of a component at a certain time t is defined to be r(t)=P(Y>t). For fixed t, find the maximum likelihood estimator for r(t).

    I'm assuming r(t) should be replaced by 1-F(Y) but I really don't understand the maximum likelihood estimator method. I can't figure out how to start.
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  2. #2
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    Ok, I'm using the notation from Maximum likelihood - Wikipedia, the free encyclopedia. So
    \mathcal{L}(\theta)=f_\theta(x_1,\ldots,x_n) is the joint PDF of the RV the n components in your sample.

    We want to find, for given values of x_1,\ldots,x_n the value of \theta that we will call \hat{\theta} that maximizes \mathcal{L}.

    That is the idea.

    What is the joint PDF? Figure out what the PDF for 1 component's lifetime as a function of r(t_0) (I add the extra 0 so we don't get confused). So Y\sim \exp(\lambda), and thus the pdf of Y is \lambda \exp(-\lambda t).

    We need to write this is a function of r(t_0). Well as you pointed out r(t_0) = 1-F(t_0)=1-(1-\exp(-\lambda t_0)) = \exp(-\lambda t_0)

    So -\log(r(t_0))/t_0 = \lambda, and so the pdf of Y is -\log(r(t_0))/t_0 \exp(\log(r(t_o))/t_0 t )

    Now at this point you could go on to write the joint PDF which is easy since the components are independent. Then maximize it (or the log of it).

    However, we just wrote down -\log(r(t_0))/t_0 = \lambda or more to the point r(t_0) = \exp(-\lambda t_0). We know MLE's have the functional invariance property (see Maximum likelihood). Since we know that the MLE for \lambda is \bar{Y}, then   \exp(-\bar{Y} t_0) is the MLE for r(t_0).

    Done. We didn't actually have to do much (since we kind of bootstrapped). But it helps to start to go through the motions of doing it outright since 1) we see how to set up the problem and 2) in doing so, we had to write down the formula that made it easy.
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  3. #3
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    Genius.
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