ETC3400 Chap.4 Convergence, Consistency and Delta Methods
Convergence, Consistency and Delta Methods
Convergence, Consistency and Delta Methods distinguishes pathwise, probabilistic and distributional limits before using them in likelihood theory. Almost sure convergence implies convergence in probability, which implies convergence in distribution, but reverse implications need additional conditions. Consistency means convergence in probability to the true parameter and is logically separate from unbiasedness.
A Taylor expansion of the likelihood score connects the random score at the truth with the estimation error, while a central limit theorem, Hessian law of large numbers and Slutsky step produce asymptotic normality. Continuous mapping and the delta method then carry these results through transformations.
What this chapter covers
- 01
Almost sure convergence and sample path interpretation
- 02
Convergence in probability and consistency
- 03
Convergence in distribution through cumulative distribution functions
- 04
Hierarchy among convergence modes and the constant limit special case
- 05
Continuous mapping for probability and distributional limits
- 06
Taylor expansion of the likelihood score around the truth
- 07
Slutsky combination of score and Hessian limits
- 08
Scalar and vector delta methods for transformed estimators
Worked example · free
Transform an asymptotic variance with the delta method
- +1Use g(x)=square root x and evaluate the derivative at sigma squared.
- +2The derivative is 1 divided by 2 sigma because g prime at sigma squared equals 1/(2 sigma).
- +2Multiply the original variance by the squared derivative: [1/(4 sigma squared)] times 2 sigma to the fourth.
- +2The result is square root n times (square root S n minus sigma) converging to N(0, sigma squared divided by 2).
Key terms
- Almost sure convergence
- Convergence along sample paths outside an event of probability zero, a stronger mode than convergence in probability.
- Convergence in probability
- A limit in which the probability of a deviation larger than any fixed positive tolerance tends to zero.
- Convergence in distribution
- Convergence of distribution functions at continuity points of the limiting distribution, often used for approximate inference.
- Consistent estimator
- An estimator that converges in probability to the true parameter as sample size grows.
- Continuous mapping theorem
- A theorem that carries a convergence result through a function continuous at the limiting value.
- Slutsky theorem
- A rule for combining a distributional limit with quantities converging in probability to constants.
- Delta method
- A first order Taylor transformation of an asymptotic distribution using a derivative or Jacobian evaluated at the limit.
Convergence, Consistency and Delta Methods FAQ
Does consistency mean an estimator is unbiased?
No. Consistency describes behaviour as sample size grows, while unbiasedness is an exact finite sample expectation statement. A biased estimator can become consistent when its bias vanishes with sample size.
When does convergence in distribution imply convergence in probability?
The implication holds when the limiting random variable is a constant. For a nonconstant limit, distributional convergence alone does not generally control the distance between variables on the same probability space.
Why does asymptotic normality use a Taylor expansion?
The score is zero at the MLE but random at the truth. Expanding between those points writes estimation error as inverse curvature times the true score, whose limit can be analysed.
What if the delta method derivative is zero?
The first order approximation degenerates and reports no informative variation. A higher order expansion or another transformation method is needed rather than declaring the estimator variance to be zero.
Exam move
Build a theorem table with columns for notation, definition, strength and typical use. On every limit problem, write the convergence mode above the arrow before manipulating it. For MLE asymptotic normality, memorise the logic rather than the final formula: zero score at the estimator, Taylor expansion around the truth, central limit theorem for the score, probability limit for the average Hessian, then Slutsky.
Practise continuous mapping and delta method on the same transformation so their different outputs are clear. For vector transformations, annotate Jacobian dimensions before forming GVG prime. Finally, state the assumptions that make the information matrix finite and nonsingular. Create a one-page implication map and annotate every arrow that is not reversible.
For each exercise, circle whether the limit is constant or random before invoking a theorem. Re-derive the score expansion with every evaluation point labelled, because confusing the Hessian at the truth, an intermediate value and the estimate is a common source of unsupported steps.
In delta-method drills, calculate the derivative before substituting the limit and pause when it is zero; that signals a higher-order problem rather than zero uncertainty. Use a short simulation only after the proof to compare the finite-sample distribution with its proposed limit at several sample sizes.
Record which feature improves—centering, spread or shape—and do not present numerical convergence as a replacement for theorem conditions.
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