Math Problem Statement

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?

Solution

The question states:

If T1T_1 and T2T_2 are two unbiased estimators of T(θ)T(\theta) with variances σ12\sigma_1^2 and σ22\sigma_2^2, and correlation ρ\rho, what is the best unbiased linear combination of T1T_1 and T2T_2, and what is the variance of such a combination?


Solution:

1. Best Linear Combination:

The best linear unbiased estimator (BLUE) is given as a linear combination: T=aT1+bT2T = a T_1 + b T_2 where aa and bb 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 TT is: Var(T)=a2σ12+b2σ22+2abρσ1σ2\text{Var}(T) = a^2 \sigma_1^2 + b^2 \sigma_2^2 + 2ab \rho \sigma_1 \sigma_2

Using the condition a+b=1a + b = 1, substitute b=1ab = 1 - a: Var(T)=a2σ12+(1a)2σ22+2a(1a)ρσ1σ2\text{Var}(T) = a^2 \sigma_1^2 + (1-a)^2 \sigma_2^2 + 2a(1-a) \rho \sigma_1 \sigma_2

Simplify: Var(T)=a2σ12+(σ222ρσ1σ2)a+σ22\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 aa and set to zero: dda[Var(T)]=2aσ122(1a)ρσ1σ22(1a)σ22=0\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: 2aσ122ρσ1σ2+2aρσ1σ22σ22+2aσ22=02a \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(σ12+σ222ρσ1σ2)=σ22ρσ1σ2a (\sigma_1^2 + \sigma_2^2 - 2\rho \sigma_1 \sigma_2) = \sigma_2^2 - \rho \sigma_1 \sigma_2

Thus: a=σ22ρσ1σ2σ12+σ222ρσ1σ2a = \frac{\sigma_2^2 - \rho \sigma_1 \sigma_2}{\sigma_1^2 + \sigma_2^2 - 2 \rho \sigma_1 \sigma_2}

From a+b=1a + b = 1: b=σ12ρσ1σ2σ12+σ222ρσ1σ2b = \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 aa and bb back into the variance expression: Var(T)=σ12σ22(1ρ2)σ12+σ222ρσ1σ2\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=aT1+bT2,a=σ22ρσ1σ2σ12+σ222ρσ1σ2,b=σ12ρσ1σ2σ12+σ222ρσ1σ2T = 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: Var(T)=σ12σ22(1ρ2)σ12+σ222ρσ1σ2\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}


Would you like further clarification or assistance with similar problems?

Relative Questions:

  1. What does the correlation ρ\rho imply about the relationship between T1T_1 and T2T_2?
  2. How does setting ρ=0\rho = 0 simplify the above expressions?
  3. What happens when σ12=σ22\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.

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Math Problem Analysis

Mathematical Concepts

Unbiased Estimation
Linear Combinations
Variance Minimization
Correlation

Formulas

T = aT₁ + bT₂
a + b = 1
Var(T) = a²σ₁² + b²σ₂² + 2abρσ₁σ₂
a = (σ₂² - ρσ₁σ₂) / (σ₁² + σ₂² - 2ρσ₁σ₂)
b = (σ₁² - ρσ₁σ₂) / (σ₁² + σ₂² - 2ρσ₁σ₂)
Var(T) = (σ₁²σ₂²(1-ρ²)) / (σ₁² + σ₂² - 2ρσ₁σ₂)

Theorems

Best Linear Unbiased Estimator (BLUE)

Suitable Grade Level

Undergraduate Level (Statistics/Probability)