Please see attached mathematical statistics question below.part(a)How to show or prove that ˆθ = −n/ Σ(ln(Xi)) is the mle of θ? PLEASE DO NOT REJECT QUESTION IF YOU DO NOT KNOW HOW TO SOLVE THIS PROBLEM! Let $ X_1, \ldots, X_n $ be a random sample from a Beta($\theta, 1$) distribution, where $\theta > 0$.(a) Show that $ \hat{\theta} = \frac{n}{\sum_{i=1}^n \ln X_i} $ is the MLE (maximum likelihood estimator) of $\theta$.(b) Find the distribution of $ Y = - \ln X $ (using section 1.7).(c) Use Part (b) to show that $ W = -\sum_{i=1}^n \ln X_i $ has a gamma distribution $\Gamma(n, 1/\theta)$.(d) Show that $ 2\theta W $ has a $\chi^2(2n)$ distribution.(e) Using Part (d), find $ c_1, c_2 $ so that \[ P \left( c_1 < \frac{2\theta n}{\hat{\theta}} < c_2 \right) = 1 - \alpha, \]for $ \alpha \in (0, 1) $. Next obtain a $ (1-\alpha)100\% $ confidence interval for $\theta$.(f) For $ \alpha = 0.02 $ and $ n = 15 $, compare the length of this interval with the length of the interval found in Example 6.2.6.

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Description: Please see attached mathematical statistics question below.part(a)How to show or prove that ˆθ = −n/ Σ(ln(Xi)) is the mle of θ? PLEASE DO NOT REJECT QUESTION IF YOU DO NOT KNOW HOW TO SOLVE THIS PROBLEM! Let $ X_1, \ldots, X_n $ be a random sample

Answered: Please see attached mathematical…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

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Please see attached mathematical statistics question below.

part(a)

How to show or prove that ˆθ = −n/ Σ(ln(Xi)) is the mle of θ?

PLEASE DO NOT REJECT QUESTION IF YOU DO NOT KNOW HOW TO SOLVE THIS PROBLEM!

Let $ X_1, \ldots, X_n $ be a random sample from a Beta($\theta, 1$) distribution, where $\theta > 0$.

(a) Show that $ \hat{\theta} = \frac{n}{\sum_{i=1}^n \ln X_i} $ is the MLE (maximum likelihood estimator) of $\theta$.

(b) Find the distribution of $ Y = - \ln X $ (using section 1.7).

(c) Use Part (b) to show that $ W = -\sum_{i=1}^n \ln X_i $ has a gamma distribution $\Gamma(n, 1/\theta)$.

(d) Show that $ 2\theta W $ has a $\chi^2(2n)$ distribution.

(e) Using Part (d), find $ c_1, c_2 $ so that
\[ P \left( c_1 < \frac{2\theta n}{\hat{\theta}} < c_2 \right) = 1 - \alpha, \]
for $ \alpha \in (0, 1) $. Next obtain a $ (1-\alpha)100\% $ confidence interval for $\theta$.

(f) For $ \alpha = 0.02 $ and $ n = 15 $, compare the length of this interval with the length of the interval found in Example 6.2.6.

Expert Solution

Step 1

$f (x) = \frac{x^{θ - 1} {(1 - x)}^{1 - 1}}{B (θ, 1)}, 0 < x < 1$

Consider the likelihood function,

$L (θ) = \prod_{i = 1}^{n} f (x) = \prod_{i = 1}^{n} \frac{x^{θ - 1}}{B (θ, 1)} = {[\frac{1}{B (θ, 1)}]}^{n} \prod_{i = 1}^{n} x^{θ - 1}$

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