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Math Help - Maximum Likelihood Estimate

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    Maximum Likelihood Estimate

    I need help with this question :


    Suppose a town has bicycles with license numbers 1, . . . , N . You observe
    5 bicycles and the highest number you observe is 60. What is the MAximum Likelihood Estimate(MLE) for the number of bicycles in the town? Does the MLE provide a reasonable estimate?
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    Quote Originally Posted by dagmary View Post
    I need help with this question :


    Suppose a town has bicycles with license numbers 1, . . . , N . You observe
    5 bicycles and the highest number you observe is 60. What is the MAximum Likelihood Estimate(MLE) for the number of bicycles in the town? Does the MLE provide a reasonable estimate?
    Let x_1, \, x_2, \, x_3, \, x_4, \, x_5 be a random sample of five observations from a discrete uniform distribution with pdf f(x_i) = \frac{1}{N}. The largest observation is 60 (that is, x_{(5)} = 60).

    Then L = f(x_1, \, x_2, \, x_3, \, x_4, \, x_5) = f(x_1) \, f(x_2) \, f(x_3) \, f(x_4) \, f(x_5) = \frac{1}{N^5}.

    Note that L is a monotonically decreasing function of N and so nowhere in the interval 0 < N < \infty is \frac{dL}{dN} equal to zero. However, note that:

    1. L increases as N decreases, and

    2. N must be equal to or greater than 60.

    Therefore the value of N that maximises L is N = 60 (that is, N = x_{(5)}).

    Note: This estimator is not an unbiased estimator of N.
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