Estimate the variance of the estimators and construct normal and non-normal confidence intervals.

A point estimate without uncertainty bounds is useless. The exam tests three CI machines: Wald (normal), log-transform (non-normal but closed-form), and likelihood ratio (drop the asymptotic shortcut entirely). For exponential data, a fourth, exact chi-square interval, sidesteps asymptotics altogether.

Constructing the MLE: the score equation. A point estimate has to come from somewhere. The likelihood ranks every possible parameter value by how plausibly it generated the observed data. The MLE is the value where that plausibility peaks. For smooth likelihoods the peak sits where the derivative of $\ln L$, the score function, hits zero.

Four-step recipe:

• Write the likelihood: $L(\theta )={\prod }_{i=1}^{n}f(x_i;\theta )$. The data is fixed; $\theta$ varies.
• Take the log: $\ell (\theta )={\sum }_{i=1}^n\ln f(x_i;\theta )$. Sums are easier than products and the maximizer is unchanged.
• Differentiate and set to zero: $U(\theta )={\ell }^{\prime }(\theta )=0$. This is the score equation.
• Solve for $\hat{\theta }$, then verify ${\ell }^{\prime \prime }(\hat{\theta })<0$ so it is a maximum, not a minimum or saddle.

Common mistakes

• Using $1/I_{ii}$ instead of $[I^{-1}{]}_{ii}$ when parameters are correlated. The correct SE is the square root of the i-th diagonal of the inverse.
• Forgetting the squared derivative in the delta method. The factor is $[g^{\prime }(\hat{\theta }){]}^2$, not $g^{\prime }(\hat{\theta })$.
• Reporting a Wald CI that crosses zero for a positive parameter and silently rounding to zero. Switch to log-transform or LR.

Bottom line

• Score equation: the MLE solves ${\ell }^{\prime }(\hat{\theta })=0$. Write $L$, log, differentiate, set to zero, solve, then confirm a maximum with ${\ell }^{\prime \prime }(\hat{\theta })<0$.
• Variance from Fisher information: $\widehat{\text{Var}}(\hat{\theta })=I(\hat{\theta }{)}^{-1}$; for multiple parameters use the square root of the i-th diagonal of the inverse Hessian, not $1/I_{ii}$.
• Delta method for transformed estimators: $\text{Var}(g(\hat{\theta }))\approx [g^{\prime }(\hat{\theta }){]}^2\text{Var}(\hat{\theta })$.
• Wald CI (normal): $\hat{\theta }\pm z_{\alpha /2}\widehat{\text{SE}}$. Symmetric, uses $z$ because asymptotic, and can cross zero for positive parameters.

Exam shortcut

Exponential MLE: $\hat{\theta }=\bar{X}$, $\widehat{\text{Var}}(\hat{\theta })={\bar{X}}^2/n$. No calculus needed. Exponential rate MLE: $\hat{\lambda }=1/\bar{X}=n/\sum x_i$. Score equation recipe: $L\to \ln L\to {\ell }^{\prime }\to$ set to zero $\to$ solve $\to$ check ${\ell }^{\prime \prime }<0$. For 95% LR CI, drop the log-likelihood by 1.92 from its max; the crossing points are the endpoints. For 90%, drop by 1.35. For a tail probability or percentile CI, compute the delta-method SE; switch to log-transform the moment the Wald endpoint leaves $[0,1]$ or turns negative.

The full lesson (about 3,035 words, 20 min read) adds 3 worked examples, all 10 common mistakes, a self-check, free in the app.

Learning objectives

• 4b

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