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The question states:

If \( T_1 \) and \( T_2 \) are two unbiased estimators of \( T(\theta) \) with variances \( \sigma_1^2 \) and \( \sigma_2^2 \), and correlation \( \rho \), what is the best unbiased linear combination of \( T_1 \) and \( T_2 \), and what is the variance of such a combination?

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### **Solution:**

#### 1. **Best Linear Combination**:
The best linear unbiased estimator (BLUE) is given as a linear combination:
\[
T = a T_1 + b T_2
\]
where \( a \) and \( b \) are constants to be determined.

#### 2. **Unbiasedness Condition**:
For the estimator to remain unbiased:
\[
E[T] = a E[T_1] + b E[T_2] = \theta \implies a + b = 1
\]

#### 3. **Minimizing the Variance**:
The variance of \( T \) is:
\[
\text{Var}(T) = a^2 \sigma_1^2 + b^2 \sigma_2^2 + 2ab \rho \sigma_1 \sigma_2
\]

Using the condition \( a + b = 1 \), substitute \( b = 1 - a \):
\[
\text{Var}(T) = a^2 \sigma_1^2 + (1-a)^2 \sigma_2^2 + 2a(1-a) \rho \sigma_1 \sigma_2
\]

Simplify:
\[
\text{Var}(T) = a^2 \sigma_1^2 + (\sigma_2^2 - 2\rho \sigma_1 \sigma_2)a + \sigma_2^2
\]

To minimize variance, differentiate with respect to \( a \) and set to zero:
\[
\frac{d}{da}[\text{Var}(T)] = 2a \sigma_1^2 - 2(1-a) \rho \sigma_1 \sigma_2 - 2(1-a) \sigma_2^2 = 0
\]

Simplify:
\[
2a \sigma_1^2 - 2 \rho \sigma_1 \sigma_2 + 2a \rho \sigma_1 \sigma_2 - 2 \sigma_2^2 + 2a \sigma_2^2 = 0
\]
\[
a (\sigma_1^2 + \sigma_2^2 - 2\rho \sigma_1 \sigma_2) = \sigma_2^2 - \rho \sigma_1 \sigma_2
\]

Thus:
\[
a = \frac{\sigma_2^2 - \rho \sigma_1 \sigma_2}{\sigma_1^2 + \sigma_2^2 - 2 \rho \sigma_1 \sigma_2}
\]

From \( a + b = 1 \):
\[
b = \frac{\sigma_1^2 - \rho \sigma_1 \sigma_2}{\sigma_1^2 + \sigma_2^2 - 2 \rho \sigma_1 \sigma_2}
\]

#### 4. **Variance of the Best Linear Combination**:
Substitute \( a \) and \( b \) back into the variance expression:
\[
\text{Var}(T) = \frac{\sigma_1^2 \sigma_2^2 (1-\rho^2)}{\sigma_1^2 + \sigma_2^2 - 2 \rho \sigma_1 \sigma_2}
\]

---

### **Final Answers**:
1. **Best Linear Combination**:
\[
T = a T_1 + b T_2, \quad a = \frac{\sigma_2^2 - \rho \sigma_1 \sigma_2}{\sigma_1^2 + \sigma_2^2 - 2 \rho \sigma_1 \sigma_2}, \quad b = \frac{\sigma_1^2 - \rho \sigma_1 \sigma_2}{\sigma_1^2 + \sigma_2^2 - 2 \rho \sigma_1 \sigma_2}
\]

2. **Variance of the Combination**:
\[
\text{Var}(T) = \frac{\sigma_1^2 \sigma_2^2 (1-\rho^2)}{\sigma_1^2 + \sigma_2^2 - 2 \rho \sigma_1 \sigma_2}
\]

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Would you like further clarification or assistance with similar problems? 

### Relative Questions:
1. What does the correlation \( \rho \) imply about the relationship between \( T_1 \) and \( T_2 \)?
2. How does setting \( \rho = 0 \) simplify the above expressions?
3. What happens when \( \sigma_1^2 = \sigma_2^2 \)?
4. How does this approach generalize to more than two estimators?
5. Can we apply this method to biased estimators? Why or why not?

**Tip**: Always confirm whether the estimators are unbiased when applying these formulas.

If T₁ and T₂ are two unbiased estimators of T(θ) with variances σ₁² and σ₂², and correlation ρ, what is the best unbiased linear combination of T₁ and T₂, and what is the variance of such a combination?

Learn how to determine the best unbiased linear combination of two estimators with given variances and correlation. This solution includes step-by-step derivations for the coefficients and variance, using key statistical concepts such as unbiased estimation and variance minimization. Suitable for undergraduate students studying statistics.

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