Question

Let \( \{0.90, 0.50, 0.01, 0.95\} \) be a realization of a random sample of size 4 from the probability density function

\[ f(x) = \begin{cases} \frac{\theta}{(1-\theta)} x^{(2\theta-1)/(1-\theta)}, & 0<x<1 \\ 0, & \text{otherwise} \end{cases} \]

where \( 0.5 \leq \theta<1 \). Then the maximum likelihood estimate of \( \theta \) based on the observed sample equals

\[ \underline{\hspace{2cm}} \]

(round off to 2 decimal places).

Hint:

To estimate parameters using the maximum likelihood method, construct the likelihood function and maximize it with respect to the parameter of interest.

Answer 1

Correct Answer: 0.5

The likelihood function is constructed using the given probability density function. Taking the log-likelihood and maximizing it, we find the maximum likelihood estimate of \( \theta \): \[ \hat{\theta} \approx 0.50. \] Thus, the value is \( 0.50 \).