Question:
Let 0.2, 1.2, 1.4, 0.3, 0.9, 0.7 be the observed values of a random sample of size 6 from a continuous distribution with the probability density function
\(f(x) = \begin{cases} 1, & 0< x \le \frac{1}{2} \\ \frac{1}{2\theta-1}, & \frac{1}{2} \lt x \le \theta \\ 0, & \text{otherwise,}\end{cases}\)
where θ > \(\frac{1}{2}\) is unknown. Then the maximum likelihood estimate and the method of moments estimate of θ, respectively, are
Let 0.2, 1.2, 1.4, 0.3, 0.9, 0.7 be the observed values of a random sample of size 6 from a continuous distribution with the probability density function
\(f(x) = \begin{cases} 1, & 0< x \le \frac{1}{2} \\ \frac{1}{2\theta-1}, & \frac{1}{2} \lt x \le \theta \\ 0, & \text{otherwise,}\end{cases}\)
where θ > \(\frac{1}{2}\) is unknown. Then the maximum likelihood estimate and the method of moments estimate of θ, respectively, are
\(f(x) = \begin{cases} 1, & 0< x \le \frac{1}{2} \\ \frac{1}{2\theta-1}, & \frac{1}{2} \lt x \le \theta \\ 0, & \text{otherwise,}\end{cases}\)
where θ > \(\frac{1}{2}\) is unknown. Then the maximum likelihood estimate and the method of moments estimate of θ, respectively, are
Updated On: Nov 25, 2025
- \(\frac{7}{5}\) and 2
- \(\frac{47}{60}\) and \(\frac{32}{15}\)
- \(\frac{7}{5}\) and \(\frac{32}{15}\)
- \(\frac{7}{5}\) and \(\frac{47}{60}\)
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The Correct Option is C
Solution and Explanation
To solve this problem, we need to find the Maximum Likelihood Estimate (MLE) and the Method of Moments Estimate (MME) for the parameter \(\theta\) given a sample and the probability density function (PDF) of the form:
\(f(x) = \begin{cases} 1, & 0< x \le \frac{1}{2} \\ \frac{1}{2\theta-1}, & \frac{1}{2} \lt x \le \theta \\ 0, & \text{otherwise} \end{cases}\)
Step 1: Maximum Likelihood Estimate (MLE)
- We begin by reviewing the given sample: \(0.2, 1.2, 1.4, 0.3, 0.9, 0.7\).
- For MLE, we need to ensure that all sample values fall within the interval \((0, \theta]\). The largest observed value in the sample is \(1.4\). Therefore, \(\theta\) must be greater than or equal to \(1.4\).
- The MLE of \(\theta\) is found by taking the maximum value from the observed sample, which gives us \(\theta_{MLE} = 1.4\).
- Thus, the MLE of \(\theta\) is \(\frac{7}{5}\) (since \(\frac{7}{5} = 1.4\)).
Step 2: Method of Moments Estimate (MME)
- Calculate the sample mean, \(\bar{x} = \frac{0.2 + 1.2 + 1.4 + 0.3 + 0.9 + 0.7}{6} = \frac{4.7}{6}\).
- To find the MME, set the sample mean equal to the expected value of the distribution. The expected value \(E(X)\) for \(x > \frac{1}{2}\) is:
- Solving the integral, we find:
- Solve for \(\theta\) using algebra, leading to the MME as \(\theta_{MME} = \frac{32}{15}\).
Thus, the Maximum Likelihood Estimate and Method of Moments Estimate for \(\theta\) are \( \frac{7}{5} \) and \(\frac{32}{15}\), respectively.
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