Sample Question #133 (statistics)

Given a sample of *N *observations: *X*_{1}, *X*_{2}, …, *X*_{N}, please write out its likelihood function.

(Comment: a really good question that tests your understanding of one of the basic concepts in statistical estimations)

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I really like this question. Many students only memorize how to form an MLE, but neglect to understand what this "likelihood function" is to begin with.

A likelihood function simply relates the parameters of the distribution, given the observations, to the theoretical density function of the population from which the observations were supposedly drawn.

For the data sample in the question, the likelihood function, in its general form, is:

L(a_1, …, a_k | X_1, …, X_N) proportional to f(X_1, …, X_N | a_1, …, a_k),

where the a_i’s are the k parameters of the model. Usually we can set the two sides to equal to each other. If we add the additional assumptions that the X’s are independent observations, the right-hand side becomes a product of the individual cdf’s evaulated at each X value. MLE then proceeds by maximizing this product.