Question

Suppose X₁, X₂,....., Xₙ are i.i.d. random variables with finite mean μ and variance σ² ∈(0,[infinity]). We want to estimate σ² by an estimator σ² Consider σ² = 1/n ∑ⁿᵢ₌₁ (Xᵢ −

Xₙ )² , where Xₙ = 1/n ∑ⁿᵢ₌₁ Xᵢ Prove that σ² → σ² Hint: You can use 1/n ∑ⁿᵢ₌₁ (Xᵢ - μ)² → σ² . Xₙ → X and Yₙ → Y iff (Xₙ,, Yₙ ) → (X,Y), where (Xₙ )ₙ∈ₙ and (Yₙ )ₙ∈ₙ denote sequences of random variables.

Answer 1

To prove that σ² → σ², we show that the estimator σ² converges in probability to the true value σ² as the sample size increases.

Step-by-step explanation:

To prove that σ² → σ², we need to show that the estimator σ² converges in probability to the true value σ² as the sample size increases.

Let's start by rewriting the estimator σ² = 1/n ∑ⁿᵢ₌₁ (Xᵢ − Xₙ )² in terms of the sample mean Xₙ, which is Xₙ = 1/n ∑ⁿᵢ₌₁ Xᵢ. We can substitute Xₙ in the estimator formula to get σ² = 1/n ∑ⁿᵢ₌₁ (Xᵢ − (1/n ∑ⁿᵢ₌₁ Xᵢ))².

Next, we need to use the property (Xₙ,, Yₙ ) → (X,Y) to show that the estimator converges to the true value. This property states that if sequences of random variables (Xₙ), (Yₙ) converge to X and Y respectively, then the joint sequence (Xₙ, Yₙ) converges to (X,Y).