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Maximum likelihood estimation

Definition​

Definition

Suppose that for any observation (x1,…,xn)(x_1, \dots, x_n) of the random variables (X1,…,Xn)(X_1, \dots, X_n) there exists a single value θ\theta, denoted θ^nMV\hat\theta_n^{MV} such that the likelihood is maximised.

L(θ^nMV)(x1,…,xn;x1,…,xn)=max⁡θL(θ;x1,…,xn)L(\hat\theta_n^{MV})(x_1, \dots, x_n; x_1, \dots, x_n) = \max_\theta L(\theta; x_1, \dots, x_n)

Then we say that θ^nMV(x1,...,xn)\hat{\theta}_n^{MV}(x_1, ..., x_n) is a maximum likelihood estimate of θ\theta.

The corresponding random variable θ^nMV(X1,...,Xn)\hat\theta_n^{MV}(X_1, ..., X_n), denoted θ^nMV\hat\theta_n^{MV} or θ^MV\hat\theta^{MV} is called the maximum likelihood estimator of θ\theta.

Results​

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