part (a)

First, I want to say that I am using the following deﬁntion of the Gamma function (using $β$ instead of $λ$ ) in the deﬁntion. Since The data given has units of time and are not rate (i.e. 1/time). So I am using this deﬁnition of Gamma PDF

|-----------------------| |f (t) = 1--1-- tα −1e− tβ| | βΓ (α ) | ------------------------

Now to answer part (A).

Yes. The following shows the histogram of the data, and a plot of a Gamma distribution with the shape parameter $α = 1$ and scale parameter $β$ set to the average of the data.

Part(b)

Using method of moments. We need 2 equations since we have to estimate 2 parameters $α,λ.$ For Gamma

Now from the data itself, calculate the First and Second moments and equate to the above and solve for $α, λ$ and these will be our estimate. This little code does the above

Now using the MLE method. For $α$

Hence we obtain the 2 equations

From the second equation, set it to zero we obtain

ˆ -n-ˆα-- αˆ- λ = ∑n = X¯ Xi i

Substitute the above in the ﬁrst equation and set to zero we obtain

And solve for $αˆ$ . Once we ﬁnd $ˆα$ we then ﬁnd also $ˆλ = ˆα¯X-$

This code below does the above.

Part(C)

Now Fit this model again, and compare the MLE ﬁtting to the method of moments ﬁtting

This plot shows more closely the ﬁtting on top of each others. They are very close so hard to see the diﬀerence other than near the high frequency part.

The ﬁts above both look reasonable.

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Part(d)

Use bootstrap method.

For the method of moments.

Try for $n = 500$ be the same size $.$ Use the method of moments parameters to generate an $n$ random variables from Gamma distribution. First time use the parameters estimated from the data as shown above.

Now, use the sample generated above to estimate the parameters from it again using also the method of moments. Use these parameters to generate another $n$ random variables. repeat this process for say $N = 5000$ and ﬁnd the variances of the parameters $α,λ$ , and hence we ﬁnd the standard error which is the square root of these variances.

Here is the code to do the above and the result

(Last minute update), I am getting large result for standard error from the bootstrap method. I think I have something wrong. Here is the result I get and the code

For Method of moments, I get standard error for alpha=918 and for lambda=18

For MLE I get

Standard error for alpha=1.68697*10^8

Standard error for lambda=60.2585

Part (e) and (f)

sorry, run out of time.