Provide an intuitive motivation for the definition of MLEs in the continuous case (see Sec...

Provide an intuitive motivation for the definition of MLEs in the continuous case (see Sec. 6.5) by going through steps (a) through (c) below. As before, the observed data are X₁, X₂, …, X_n, and are IID realizations of a random variable X with density f_θ. Bear in mind that the X_i’s have already been observed, so are to be regarded as fixed numbers rather than variables.

(a) Let ε be a small (but strictly positive) real number, and define the phrase “getting a value of X near X_i” to be the event {X_i – ε < X < X_i + ε}. Use the mean-value theorem from calculus to argue that P(getting a value of X near X_i) ≈ 2ε f_θ(X_i), for any i = 1, 2, …, n.

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(b) Define the phrase “getting a sample of n IID values of X near the observed data” to be the event (getting a value of X near X₁, getting a value of X near X₂, …, getting a value of X near X_n). Show that P(getting a sample of n IID values of X near the observed data) ≈ (2ε)ⁿf_θ(X₁)f_θ(X₂) …f_θ(X_n), and note that this is proportional to the likelihood function L(θ).

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(c) Argue that the MLE is the value of θ that maximizes the approximate probability of getting a sample of n IID values of X near the observed data, and in this sense “best explains” the data that were actually observed.