The Gamma distribution Gamma(𝛼,𝛽) with paramters 𝛼>0 , and 𝛽>0 is defined by the density

Question

𝑓𝛼,𝛽(𝑥)=𝛽𝛼Γ(𝛼)𝑥𝛼−1𝑒−𝛽𝑥,for all𝑥≥0.
The Γ function is defined by

Γ(𝑠)=∫∞0𝑥𝑠−1𝑒−𝑥𝑑𝑥.
As usual, the constant 𝛽𝛼Γ(𝛼) is a normalization constant that gives ∫∞0𝑓𝛼,𝛽(𝑥)𝑑𝑥=1.

In this problem, let 𝑋1,…,𝑋𝑛 be i.i.d. Gamma variables with

𝛽=1𝛼for some 𝛼>0.
That is, 𝑋1,…,𝑋𝑛∼Gamma(𝛼,1𝛼) random variables for some 𝛼>0 . The pdf for 𝑋𝑖 is therefore

𝑓𝛼(𝑥)=1Γ(𝛼)𝛼𝛼𝑥𝛼−1𝑒−𝑥/𝛼,for all 𝑥≥0.

What is the limit, in probability, of the sample average 𝑋⎯⎯⎯⎯⎯𝑛 of the sample in terms of 𝛼 ?

𝑋⎯⎯⎯⎯⎯𝑛−→−−𝑛→∞𝐏 = ?

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Use the result from the previous problem to give a consistent estimator 𝛼̂ of 𝛼 in terms of 𝑋⎯⎯⎯⎯⎯𝑛 .

(Enter barX_n for 𝑋⎯⎯⎯⎯⎯𝑛 )

𝛼̂ =?

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For the Delta method to apply, at what value of 𝑥 does 𝑔 need to be continuously differentiable? (Your answer should be in terms of 𝛼 .)

𝑥=?
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What is its asymptotic variance of 𝛼̂ ?

𝑉(𝛼̂ )=

Answer 1

Using the previous part, find confidence intervals for 𝛼 with asymptotic level 90% using both the “solving" and the “plug-in" methods. Use 𝑛=25 , and 𝑋⎯⎯⎯⎯⎯𝑛=4.5 .
(Enter your answers accurate to 2 decimal places. Use the Gaussian estimate 𝑞0.05≈1.6448 for best results.)

Isolve=[ , ]
Iplug-in=[ , ]

Answer 2

(a) alpha^2
(b) barX_n^(1/2)
(c1) alpha^2
(c2) Normal Distribution
(c3) ?
(d)?

Answer 3

c1 answer above is wrong

Answer 4

I got green tick with alpha^2 for c1

Answer 5

c3 and d?

Answer 6

Isolve=[ 1.89, 2.37]
Iplug-in=[ 1.88, 2.36]