Question

Problem 3 Let X be a random variable with the pdf $\frac{1}{\theta}x^{(1-\theta)/\theta}$, $0 < x < 1$, $0 < \theta < \infty$. 3.1. Show that the maximum likelihood estimator of $\theta$ is $\hat{\theta} = -(1/n)\sum_{i=1}^{n} \ln X_i$. 3.2. Show that $\hat{\theta}$ is an unbiased estimator of $\theta$.

          Problem 3
Let X be a random variable with the pdf
$\frac{1}{\theta}x^{(1-\theta)/\theta}$, $0 < x < 1$, $0 < \theta < \infty$.
3.1. Show that the maximum likelihood estimator of $\theta$ is $\hat{\theta} = -(1/n)\sum_{i=1}^{n} \ln X_i$.
3.2. Show that $\hat{\theta}$ is an unbiased estimator of $\theta$.

Problem 3
Let X be a random variable with the pdf
(1)/(θ)x^(1-θ)/θ, 0 < x < 1, 0 < θ < ∞.
3.1. Show that the maximum likelihood estimator of θ is θ̂ = -(1/n)∑i=1^nln Xi.
3.2. Show that θ̂ is an unbiased estimator of θ.

Added by Angela M.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Ameer Said

10/25/2021

Step 1

Given the pdf f(x;θ) = (1/θ)x^((1-θ)/θ), the likelihood function L(θ) is the product of the pdf values for each observation x_i in the sample: L(θ) = ∏(1/θ)x_i^((1-θ)/θ) Taking the natural logarithm of the likelihood function to simplify the Show more…

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A random sample of size 15 from the normal distributionProblem 3 Let x be a random variable with the pdf heta widehat( heta )=-((1)/(n))sum_(i=1)^n lnx_(i)widehat( heta ) heta f_(x)(x; heta )=(1)/( heta )x^((1- heta )/( heta )),0 3.1. Show that the maximum likelihood estimator of heta is widehat( heta )=-((1)/(n))sum_(i=1)^n lnx_(i). 3.2. Show that widehat( heta ) is an unbiased estimator of heta . Problem 3 Let X be a random variable with the pdf fxx;0)= (1/e 0<<1.0<< 3.1. Show that the maximum likelihood estimator of is = -(1/n) = In X; 3.2. Show that is an unbiased estimator of .