Derive the maximum likelihood estimator for b = (b0, b1)T and σ2 under the model Yi = b0 + b1Xi1 + εi, where ε1, ..., εn are independent and εi ∼ N (0, σ2X2 )
Question
Derive the maximum likelihood estimator for b = (b0, b1)T and σ2 under the model Yi = b0 + b1Xi1 + εi, where ε1, ..., εn are independent and εi ∼ N (0, σ2X2 )
Solution
To derive the maximum likelihood estimator for b = (b0, b1)T and σ2 under the given model Yi = b0 + b1Xi1 + εi, where ε1, ..., εn are independent and εi ∼ N (0, σ2X2), we can follow these steps:
Step 1: Write the likelihood function The likelihood function is the joint probability density function (PDF) of the observed data. In this case, the observed data consists of the pairs (Yi, Xi1) for i = 1 to n. Since εi ∼ N (0, σ2X2), the likelihood function can be written as:
L(b0, b1, σ2) = f(Y1, X1; b0, b1, σ2) * f(Y2, X2; b0, b1, σ2) * ... * f(Yn, Xn; b0, b1, σ2)
where f(Yi, Xi1; b0, b1, σ2) is the PDF of the normal distribution with mean b0 + b1Xi1 and variance σ2X2.
Step 2: Take the logarithm of the likelihood function Taking the logarithm of the likelihood function simplifies the calculations and does not affect the location of the maximum. Therefore, we have:
log L(b0, b1, σ2) = log f(Y1, X1; b0, b1, σ2) + log f(Y2, X2; b0, b1, σ2) + ... + log f(Yn, Xn; b0, b1, σ2)
Step 3: Simplify the logarithm of the likelihood function Since the PDF of the normal distribution is given by:
f(Yi, Xi1; b0, b1, σ2) = (1 / √(2πσ2X2)) * exp(-(Yi - b0 - b1Xi1)2 / (2σ2X2))
we can substitute this expression into the logarithm of the likelihood function.
Step 4: Maximize the logarithm of the likelihood function To find the maximum likelihood estimators, we need to maximize the logarithm of the likelihood function with respect to b0, b1, and σ2. This can be done by taking the partial derivatives of the logarithm of the likelihood function with respect to each parameter and setting them equal to zero.
Step 5: Solve the equations Solving the equations obtained from the partial derivatives will give us the maximum likelihood estimators for b0, b1, and σ2.
Note: The detailed calculations for each step are not provided here, but by following these steps, you can derive the maximum likelihood estimators for b = (b0, b1)T and σ2 under the given model.
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