Question

Suppose that X1,…,Xn be a random sample form a random sample from a distribution for which the pdf f(x;θ) is as follow: f(x;θ)={θxθ−1,0,for 0 0). Find the MLE of θ.

Answer 1

The question involves finding the MLE of θ for a given pdf. The process includes constructing a likelihood function and maximizing it by taking the derivative of its logarithm and solving for θ.

Step-by-step explanation:

The question involves finding the maximum likelihood estimate (MLE) of a parameter θ from a given probability density function (pdf) for a random sample. The pdf is defined as f(x; θ) = θxθ-1, for x between 0 and 1. To find the MLE of θ, we need to construct the likelihood function from the product of the individual pdfs for all observations in the sample and then proceed to find the θ that maximizes this likelihood.

Steps to find the MLE of θ:

1. Write down the likelihood function L(θ) as the product of f(xi; θ) across all sample points x1, ..., xn.

2. Take the natural logarithm of the likelihood function to get the log-likelihood function, which is often easier to maximize.

3. Differentiate the log-likelihood function with respect to θ to find the critical points.

4. Find the value of θ that maximizes the likelihood function, given the critical points and boundary conditions.

This is a standard procedure in statistics for estimating parameters of a distribution.