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

• A. \(\frac{7}{5}\) and 2
• B. \(\frac{47}{60}\) and \(\frac{32}{15}\)
• C. \(\frac{7}{5}\) and \(\frac{32}{15}\)
• D. \(\frac{7}{5}\) and \(\frac{47}{60}\)

Answer 1

The Correct Option is C

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)

1. We begin by reviewing the given sample: \(0.2, 1.2, 1.4, 0.3, 0.9, 0.7\).
2. 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 \(1.4\).
3. The MLE of \(\theta\) is found by taking the maximum value from the observed sample, which gives us \(\theta_{MLE} = 1.4\).
4. Thus, the MLE of \(\theta\) is \(\frac{7}{5}\) (since \(\frac{7}{5} = 1.4\)).

Step 2: Method of Moments Estimate (MME)

1. Calculate the sample mean, \(\bar{x} = \frac{0.2 + 1.2 + 1.4 + 0.3 + 0.9 + 0.7}{6} = \frac{4.7}{6}\).
2. 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:
3. Solving the integral, we find:
4. 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.