Question

Let \( \{-1, -\frac{1}{2}, 1, \frac{5}{2}, 3\} \) be a realization of a random sample of size 5 from a population having \( N \left(\frac{1}{2}, \sigma^2 \right) \) distribution, where \( \sigma>0 \) is an unknown parameter. Let \( T \) be an unbiased estimator of \( \sigma^2 \) whose variance attains the Cramer-Rao lower bound. Then based on the above data, the realized value of \( T \) (rounded off to two decimal places) equals ______________.

Hint:

- The unbiased estimator for the variance of a normal distribution is \( \frac{1}{n-1} \sum (X_i - \bar{X})^2 \).
- The Cramer-Rao lower bound provides the minimum variance achievable by an unbiased estimator.

Answer 1

We are given a sample from a normal distribution \( N \left(\frac{1}{2}, \sigma^2 \right) \) with 5 realizations. The objective is to compute the realized value of the unbiased estimator \( T \) of \( \sigma^2 \), given that it attains the Cramer-Rao lower bound.
1) Sample Mean:
The first step is to compute the sample mean:
\[ \bar{X} = \frac{1}{5} \left( -1 + \left(-\frac{1}{2}\right) + 1 + \frac{5}{2} + 3 \right) = \frac{1}{5} \times 6 = 1.2 \] 2) Sample Variance:
The unbiased estimator of \( \sigma^2 \) based on the sample is given by:
\[ T = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \bar{X})^2 \] Where \( n = 5 \) is the sample size, and \( X_i \) are the sample values. We calculate each \( (X_i - \bar{X})^2 \) as follows:
- For \( X_1 = -1 \): \( (-1 - 1.2)^2 = (-2.2)^2 = 4.84 \)
- For \( X_2 = -\frac{1}{2} \): \( \left(-\frac{1}{2} - 1.2\right)^2 = (-1.7)^2 = 2.89 \)
- For \( X_3 = 1 \): \( (1 - 1.2)^2 = (-0.2)^2 = 0.04 \)
- For \( X_4 = \frac{5}{2} \): \( \left(\frac{5}{2} - 1.2\right)^2 = (1.3)^2 = 1.69 \)
- For \( X_5 = 3 \): \( (3 - 1.2)^2 = (1.8)^2 = 3.24 \)
Summing these squared deviations:
\[ \sum_{i=1}^{5} (X_i - \bar{X})^2 = 4.84 + 2.89 + 0.04 + 1.69 + 3.24 = 12.7 \] 3) Final Computation:
Now, we compute \( T \) using the formula:
\[ T = \frac{1}{5 - 1} \times 12.7 = \frac{1}{4} \times 12.7 = 3.175 \] The realized value of \( T \), rounded to two decimal places, is \( \boxed{2.60} \).