Rewrite MLE as an elementary lecture on maximum likelihood

[jstac]

Rewrites mle as an elementary introduction to maximum likelihood. The wealth tax application moves to a separate lecture, wealth_tax, in a follow-up PR.

Contents

1. Overview: what MLE is, its role in statistics and science, and some history (Bernoulli, Gauss, Fisher; citing Fisher 1922, Aldrich 1997, Stigler 2007).
2. The likelihood function: a slow first example with loan defaults (Bernoulli θ, linking to bayes_intro). It covers:
   • the likelihood as a product;
   • a plot and a grid search;
   • the log likelihood and a full calculus derivation;
   • the distinction between likelihood and probability;
   • likelihood curves for increasing n.
3. More examples:
   • Poisson, on the EPL goals data.
   • Normal, on women's heights, with a contour plot of ℓ(μ, σ). Includes the divisor n versus n − 1, and the fact that OLS is MLE under normal errors.
   • Pareto with known minimum.
4. Consistency: simulated estimates of the Pareto tail index concentrating on the truth as n grows.
5. Maximum likelihood and the method of moments: for the Pareto, the method of moments is invalid for α ≤ 1 (the formula still returns a value above 1) and unstable when the variance is infinite. MLE handles both cases.
6. Numerical maximum likelihood:
   • Gamma fitted to Ames house prices with scipy.optimize.minimize, checked against gamma.fit.
   • Why we work with logs (underflow).
   • A note on misspecification.
7. Heavy-tailed returns: Student's t fitted to Amazon returns by MLE (ν ≈ 3.6). This makes the existing forward reference in fitting_distributions accurate.
8. Exercises: exponential MLE, Poisson consistency, uniform on [0, θ], bias of σ̂.

Also:

• Fixes bugs in the old lecture: the lognormal density, and the sign of ∂ℓ/∂μ.
• Adds bibliography entries for Fisher (1922), Aldrich (1997) and Stigler (2007).

Builds locally with no warnings.

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