Question

If V1, V2...Vn1 and W1,W2...WN2 are independent random samples of size n1 and n2 from normal populations with the means μ1= α+β and μ2=α-β and the common variance σ2 = 1, find maximum likelihood estimators for α and β

Answer 1

To find the maximum likelihood estimators for α and β, set up the likelihood function and maximize it.

Step-by-step explanation:

To find the maximum likelihood estimators for α and β, we need to set up the likelihood function and maximize it.

1. The likelihood function is given by L(α, β) = f₁(x₁) × f₂(x₂) × ... × fₙ₁(xₙ₁) × g₁(w₁) × g₂(w₂) × ... × gₙ₂(wₙ₂), where fᵢ(xi) is the probability density function of the sample Vᵢ, and gⱼ(wj) is the probability density function of the sample Wⱼ.
2. Taking the logarithm of the likelihood function, we get l(α, β) = ln(L(α, β)).
3. To find the maximum likelihood estimators, we differentiate l(α, β) with respect to α and β, and set the derivatives equal to zero. Solving these equations will give us the estimators for α and β.