statistics

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

𝑓𝛼,𝛽(π‘₯)=𝛽𝛼Γ(𝛼)π‘₯π›Όβˆ’1π‘’βˆ’π›½π‘₯,for allπ‘₯β‰₯0.
The Ξ“ function is defined by

Ξ“(𝑠)=∫∞0π‘₯π‘ βˆ’1π‘’βˆ’π‘₯𝑑π‘₯.
As usual, the constant 𝛽𝛼Γ(𝛼) is a normalization constant that gives ∫∞0𝑓𝛼,𝛽(π‘₯)𝑑π‘₯=1.

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

𝛽=1𝛼for some 𝛼>0.
That is, 𝑋1,…,π‘‹π‘›βˆΌGamma(𝛼,1𝛼) random variables for some 𝛼>0 . The pdf for 𝑋𝑖 is therefore

𝑓𝛼(π‘₯)=1Ξ“(𝛼)𝛼𝛼π‘₯π›Όβˆ’1π‘’βˆ’π‘₯/𝛼,for all π‘₯β‰₯0.

What is the limit, in probability, of the sample average π‘‹βŽ―βŽ―βŽ―βŽ―βŽ―π‘› of the sample in terms of 𝛼 ?

π‘‹βŽ―βŽ―βŽ―βŽ―βŽ―π‘›βˆ’β†’βˆ’βˆ’π‘›β†’βˆžπ = ?

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Use the result from the previous problem to give a consistent estimator 𝛼̂ of 𝛼 in terms of π‘‹βŽ―βŽ―βŽ―βŽ―βŽ―π‘› .

(Enter barX_n for π‘‹βŽ―βŽ―βŽ―βŽ―βŽ―π‘› )

𝛼̂ =?

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

π‘₯=?
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What is its asymptotic variance of 𝛼̂ ?

𝑉(𝛼̂ )=

  1. 👍 4
  2. 👎 0
  3. 👁 529
  1. Using the previous part, find confidence intervals for 𝛼 with asymptotic level 90% using both the β€œsolving" and the β€œplug-in" methods. Use 𝑛=25 , and π‘‹βŽ―βŽ―βŽ―βŽ―βŽ―π‘›=4.5 .
    (Enter your answers accurate to 2 decimal places. Use the Gaussian estimate π‘ž0.05β‰ˆ1.6448 for best results.)

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

    1. 👍 3
    2. 👎 0
  2. (a) alpha^2
    (b) barX_n^(1/2)
    (c1) alpha^2
    (c2) Normal Distribution
    (c3) ?
    (d)?

    1. 👍 6
    2. 👎 0
  3. c1 answer above is wrong

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    2. 👎 4
  4. I got green tick with alpha^2 for c1

    1. 👍 3
    2. 👎 0
  5. c3 and d?

    1. 👍 1
    2. 👎 0

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