Maximum Likelihood Estimate

**dagmary** · July 17th 2008, 03:55 AM

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?

**mr fantastic** · July 17th 2008, 04:58 AM

Originally Posted by dagmary

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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