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MLE and method of moments

Why this is asked: MLE is invariant and asymptotically efficient but can be biased and need not be unique; moment estimators are easy but usually inefficient.

Maximum likelihood and the method of moments

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When the MLE is not found by differentiatinginteractive

The uniform case, where calculus finds nothing and the answer sits on the boundary.

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The trap here

“The MLE is unbiased” — false

σ\sigmâ2=(1/n)∑(Xi−Xˉ)2^{2} = (1/n)\sum(X_{i} - {\bar{X}})^{2} for N(μ,σ2)N(\mu, \sigma^{2}), or X(n)X_{(n)} for Uniform(0,θ)(0,\theta)

Both underestimate systematically; E[X(n)]=nθ/(n+1)E[X_{(n)}] = n\theta/(n+1).

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Check yourself

The probability of a head in tossing a coin is p ∈ (0, 1). The coin is independently tossed 25 times and heads appear 10 times. The Bayes estimate of p with respect to the prior Beta(5, 5) and squared error loss is

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Open this in the full syllabus view · Unit 4