Answer :
To solve this problem, we'll need to use the methods of maximum likelihood estimation, a popular technique in statistics used to estimate the parameters of a statistical model.
(a) Find the estimator for [tex]\theta[/tex] using the method of maximum likelihood.
Probability Density Function (PDF):
The given probability density function is:
[tex]f(y; \theta) = (\theta + 1)y[/tex] for [tex]0 < y < 1[/tex] and [tex]\theta > 1[/tex].Likelihood Function:
The likelihood function for a random sample [tex]Y_1, Y_2, ..., Y_n[/tex] is the product of the individual PDFs:
[tex]L(\theta) = \prod_{i=1}^{n} (\theta + 1)Y_i = (\theta + 1)^n \prod_{i=1}^{n} Y_i[/tex]Log-Likelihood Function:
The log-likelihood function is calculated as follows:
[tex]\log L(\theta) = n \log(\theta + 1) + \sum_{i=1}^{n} \log Y_i[/tex]Maximize the Log-Likelihood:
To find the estimator, differentiate the log-likelihood function with respect to [tex]\theta[/tex] and set the derivative equal to zero:
[tex]\frac{d}{d\theta}\left( n \log(\theta + 1) \right) = \frac{n}{\theta + 1} = 0[/tex]
Solving for [tex]\theta[/tex] requires setting the derivative equal to zero, but apparent complexity suggests using a software or more detailed algebraic maneuver, but conceptually, it's suggested that when estimating from observed values (specific solutions may appear from sample conditional constraints).
(b) Find the maximum likelihood estimates for [tex]\theta[/tex].
Given the sample values: 31.5, 36.9, 33.8, 30.1, 33.9, 35.2, these are out of the expected range [tex]0 < y < 1[/tex]. An implication could be exploring normalized context or computational oversight in the data transformation prior to application.
From theoretical insights, for exploration particularly in continuous domain range considerations:
- Compute through specific fitting methods or data rescaling.
The primary phase within continual updates and transformative sampling aligns these estimations more fittingly with the schema of [tex](Y_i - c) / (d - c)[/tex] with bounds in classical probability responses for utility.
Thus, applying practical constraints and statistical programming tools is recommended for precise computational handling.
