Question

*Please show work and explain steps*

Assume Y1, ... , Yn are IID continuous variables with PDF f(yi; θ), where f is dependent on a parameter θ.

Complete the following:

a) Derive the likelihood, L(θ), and the log-likelihood, l(θ), in terms of the function f.

b) Find dl/d(theta) in terms of f(yi; θ) and df/d(theta). Note that dl/d(θ) is usually referred to as the score function.

c) Show that E[dl/d(θ)]= 0. Hint: you can use without proof the following: ∫ (∂g(y,θ)/∂θ) dy= ∂/∂θ ∫g(y,θ)dy, where g(y,θ) is a function that depends on y and θ. This trick is called “differentiating under the integral sign”.

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Answer 1

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