(iv) Suppose we are given the sample data : 0.95 0.73 0.87 0.72 0.95 0.68 0.88 0.85 0.99 0.79 0.8 0.94 0.94 0.91 0.91 0.87 Plot l(x; m) as a function of m for the given sample data and compute the value of m* (as defined in the previous part). (v) What is the positive integer m that maximizes the function for this sample? This is the MLE estimate of m for this sample¹. (Hint: you could eyeball the plot to get the answer. A more mathematical approach is to test which of the two nearest integers to m* would give a higher value of l.)
(iv) Suppose we are given the sample data : 0.95 0.73 0.87 0.72 0.95 0.68 0.88 0.85 0.99 0.79 0.8 0.94 0.94 0.91 0.91 0.87 Plot l(x; m) as a function of m for the given sample data and compute the value of m* (as defined in the previous part). (v) What is the positive integer m that maximizes the function for this sample? This is the MLE estimate of m for this sample¹. (Hint: you could eyeball the plot to get the answer. A more mathematical approach is to test which of the two nearest integers to m* would give a higher value of l.)
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Can you show me the answer to question 4 and 5? Thank you so much for your help
![This exercise gives a less standard example of MLE. Let m be a positive integer, and
U₁, U₂,,Um are i.i.d. Uniform([0, 1]) RV's. Let X = max(U₁, U₂,...,Um), i.e. X is the
highest outcome among m random numbers independently and uniformly chosen between 0
and 1.
(i) What is the p.d.f. of the random variable X?
(ii) Suppose we don't know the value of m and wish to determine it from an i.i.d. sample
x = (x₁, x2,,xn) of the random variable X. Verify that the log-likelihood function
equals
n
l(x; m) = n log m - (m − 1) Σlog(1/x;).
=
i=1
(iii) Given £₁, X2,
‚¤n, find a formula for the real number m* such that the function l
above is maximized at m m*. (However, m* is not the MLE estimator for m because
it is not necessarily an integer.)
(iv) Suppose we are given the sample data :
0.95 0.73 0.87 0.72 0.95 0.68 0.88 0.85
0.99 0.79 0.8 0.94 0.94 0.91 0.91 0.87
Plot l(x; m) as a function of m for the given sample data ☎ and compute the value
of m* (as defined in the previous part).
(v) What is the positive integer m that maximizes the function for this sample? This is
the MLE estimate of m for this sample¹. (Hint: you could eyeball the plot to get the
answer. A more mathematical approach is to test which of the two nearest integers to
m* would give a higher value of l.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F33efa0ee-e3c4-4640-bf2a-d63d72536f00%2Ffa88f8e0-9f7c-4a2b-9d66-d7b978fc40ec%2Ftx0vzh8_processed.png&w=3840&q=75)
Transcribed Image Text:This exercise gives a less standard example of MLE. Let m be a positive integer, and
U₁, U₂,,Um are i.i.d. Uniform([0, 1]) RV's. Let X = max(U₁, U₂,...,Um), i.e. X is the
highest outcome among m random numbers independently and uniformly chosen between 0
and 1.
(i) What is the p.d.f. of the random variable X?
(ii) Suppose we don't know the value of m and wish to determine it from an i.i.d. sample
x = (x₁, x2,,xn) of the random variable X. Verify that the log-likelihood function
equals
n
l(x; m) = n log m - (m − 1) Σlog(1/x;).
=
i=1
(iii) Given £₁, X2,
‚¤n, find a formula for the real number m* such that the function l
above is maximized at m m*. (However, m* is not the MLE estimator for m because
it is not necessarily an integer.)
(iv) Suppose we are given the sample data :
0.95 0.73 0.87 0.72 0.95 0.68 0.88 0.85
0.99 0.79 0.8 0.94 0.94 0.91 0.91 0.87
Plot l(x; m) as a function of m for the given sample data ☎ and compute the value
of m* (as defined in the previous part).
(v) What is the positive integer m that maximizes the function for this sample? This is
the MLE estimate of m for this sample¹. (Hint: you could eyeball the plot to get the
answer. A more mathematical approach is to test which of the two nearest integers to
m* would give a higher value of l.)
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