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Normal distribution X_i ~ N(θ,θ^2). X_1,....,X_n. iid. Determine MLE θ(hat)_n for θ.

Normal distribution X_i ~ N(θ,θ^2). X_1,....,X_n. iid.
Determine MLE θ(hat)_n for θ.

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Answer #1

The pdf of normal distribution is

Let X1, X2, ...Xn is a random sample from the normal distribution. So we have

.

.

.

The likelihood function is

Taken log of both sides gives

Differentiating above with respect to theta gives

Equating it to zero gives

The root of above equation is

Ignoring negative values of theta gives

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