Question

A normal linear model is given as Y i =β 0 +β 1 x i1 +ε i ,i=1,…,n where Y i is the value of the response variable in the i th trial, x i1 is the value of the predictor variable in the i th trial, β 0 and β 1 are parameters, and ε i are independent N(0,σ 2 ). Find the maximum likelihood estimates (MLEs) of the parameters β 0 and β 1.

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Answer 1

The maximum likelihood estimates (MLEs) of the parameters β0 and β1 in the normal linear model can be found by maximizing the likelihood function, which is the product of the probability densities of the normal distributions for the residuals. To find the MLEs, we maximize the log-likelihood function by differentiating it with respect to β0 and β1 and setting the derivatives equal to 0.

Step-by-step explanation:

The maximum likelihood estimates (MLEs) of the parameters β0 and β1 in the normal linear model can be found by maximizing the likelihood function. In this case, the likelihood function is the product of the probability densities of the normal distributions for the residuals εi.

Since the residuals εi are assumed to be independent and normally distributed with mean 0 and variance σ2, the likelihood function can be written as:

L(β0, β1) = (1/(√(2πσ2))^n) * exp((-1/(2σ2)) * ∑(Yi - (β0 + β1xi1))2), i=1 to n

To find the MLEs of β0 and β1, we maximize the log-likelihood function:

log(L(β0, β1)) = -n/2 * log(2πσ2) - 1/(2σ2) * ∑(Yi - (β0 + β1xi1))2, i=1 to n

To find the maximum, we differentiate the log-likelihood function with respect to β0 and β1 and set the derivatives equal to 0:

d/dβ0 log(L) = 1/σ2 * ∑(Yi - (β0 + β1xi1)), i=1 to n

d/dβ1 log(L) = 1/σ2 * ∑xi1 * (Yi - (β0 + β1xi1)), i=1 to n

These equations can be solved simultaneously to find the MLEs of β0 and β1.