The parametric case is used to study the statistical efficiency of a consistent estimator. The goal of this method is to measure the value of a parameter.

Parametric Case

The statistical efficiency of consistent estimators can vary based on the generalization error or the number of samples.

Ohio State University, Columbus

In statistics, consistency is a property of an estimator where its probabilistic limit converges to the true underlying parameter value as the sample size approaches infinity ($$	ext{plim}_{mightarrow\infty}\hat{	heta}_m=	heta$$). When an estimator is consistent, accumulating more data guarantees that empirical estimates will eventually match the true theoretical probabilities.

Consistency of an Estimator

Goodfellow, I., Bengio, Y., & Courville, A. (2016). $\mathit{Deep \ Learning.}$ MIT Press. Retrieved from [www.deeplearningbook.org](https://www.deeplearningbook.org) 

Deep Learning

Strong Consistency or "Almost Sure Consistency" is defined as follows:
$$P(lim_{m\rightarrow\infty}\hat{\theta}_m = \theta) = 1$$
So where weak consistency is the ideal limit of a probability, strong consistency is the ideal probability of a limit.

Strong Consistency of an Estimator

Statistical Efficiency

The Law of Large Numbers is a mathematical theorem guaranteeing that for repeated independent events, as the number of repetitions grows, the empirical average of the outcomes will converge to the true underlying expected probability. This principle mathematically ensures that consistent estimators become perfectly accurate given infinite data.

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