Question

Let $X_1, X_2, \dots, X_n$ be a random sample of size $n$ from a gamma distribution with shape parameter $\alpha = 4$ and rate parameter $\beta = \theta$. (a) Find the Fisher Information $I(\theta)$. [8 marks] (b) Show that the maximum likelihood estimator of $\theta$ is efficient for $\theta$. [12 marks]

          Let $X_1, X_2, \dots, X_n$ be a random sample of size $n$ from a gamma distribution with shape parameter $\alpha = 4$ and rate parameter $\beta = \theta$.
(a) Find the Fisher Information $I(\theta)$.
[8 marks]
(b) Show that the maximum likelihood estimator of $\theta$ is efficient for $\theta$.
[12 marks]

Let X1, X2, …, Xn be a random sample of size n from a gamma distribution with shape parameter α = 4 and rate parameter β = θ.
(a) Find the Fisher Information I(θ).
[8 marks]
(b) Show that the maximum likelihood estimator of θ is efficient for θ.
[12 marks]

Added by Ashley L.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Madhur L

08/17/2023

Step 1

Step 1: The probability density function of a gamma distribution with shape parameter $\alpha$ and rate parameter $\beta$ is given by: $$f(x;\alpha,\beta) = \frac{\beta^\alpha}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}, x>0$$ where $\Gamma(\alpha)$ is the gamma Show more…

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Let $X_1, X_2, ..., X_n$ be a random sample of size $n$ from a gamma distribution with shape parameter $\alpha = 4$ and rate parameter $\beta = \theta$. (a) Find the Fisher Information $I(\theta)$. [8 marks] (b) Show that the maximum likelihood estimator of $\theta$ is efficient for $\theta$. [12 marks]