Math Problem Statement
Solution
The question states:
If and are two unbiased estimators of with variances and , and correlation , what is the best unbiased linear combination of and , 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: where and 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 is:
Using the condition , substitute :
Simplify:
To minimize variance, differentiate with respect to and set to zero:
Simplify:
Thus:
From :
4. Variance of the Best Linear Combination:
Substitute and back into the variance expression:
Final Answers:
-
Best Linear Combination:
-
Variance of the Combination:
Would you like further clarification or assistance with similar problems?
Relative Questions:
- What does the correlation imply about the relationship between and ?
- How does setting simplify the above expressions?
- What happens when ?
- How does this approach generalize to more than two estimators?
- 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)
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