Root-n consistency

As explained in Causal Inference for Data Science

Root-n consistency is convergence of an estimator to its target at a rate proportional to the inverse square root of the sample size.

Root-n consistency means that an estimator approaches its target at a rate proportional to 1/√n, where n is the sample size. Larger samples therefore reduce estimation error, although the improvement slows as n grows. section 8.1.2

The rate becomes intuitive through comparison:

• The sequence 1/n approaches zero faster than 1/√n; higher powers of n produce faster convergence. section 8.1.2
• The sample mean is a reference point: the central limit theorem gives it a 1/√n rate, and its efficiency means unbiased estimators cannot improve on that rate. section 8.1.2
• DML achieves root-n consistency under reasonable assumptions, while S-learners and T-learners converge more slowly. section 8.1.2

A favorable DML rate

Suppose an estimator uses samples of sizes n, 4n, and 9n. A root-n rate scales its convergence terms as 1/√n, 1/(2√n), and 1/(3√n). This is considered fast because it matches the sample mean’s central-limit-theorem rate. The result is favorable, but the formal DML proof depends on assumptions and advanced mathematics. section 8.1.2

Figure 8.4 Convergence rate for sequences 1/n and 1/√n. We can see that 1/n converges faster to 0 than 1/√n. The convergence rate of an algorithm tells us how fast this algorithm converges to its objective. In statistics and machine learning, the faster an estimate converges, the less data we need.

Visualizes why 1/n converges faster than 1/√n and why faster convergence can require less data. section 8.1.2

How Causal Inference for Data Science covers it

How deep it goes

Defines the 1/√n rate and compares estimators, but defers the formal DML proof to advanced mathematics.

Examples

Compares the sequences 1/n and 1/√n and uses the sample mean as a benchmark for DML.

What it assumes

Assumes basic sequences and sample-size notation while deferring the proof’s advanced mathematical details.

What you can do after

Evaluate whether DML’s convergence rate is statistically favorable relative to simpler estimators.

How it connects

Connects DML with the central limit theorem, sample mean, orthogonalization, S-learners, and T-learners.

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Related concepts

• double machine learningmeasured by · Double machine learning is a causal-effect estimation method that uses machine-learning nuisance models with orthogonalized estimation to reduce bias from nuisance-function errors.
• T-learnercontrasts with · A T-learner is a causal inference method that fits separate outcome models for each treatment level and contrasts their predictions.

This explainer was generated by Manning's knowledge graph from the text of Causal Inference for Data Science. The section links point to the parts of the book it draws on.