The law of large numbers overview
I have already several posts about the law of large numbers:
- start with the intuition, which is illustrated using Excel;
- simulations in Excel show that convergence is not as fast as some textbooks claim;
- to distinguish the law of large numbers from the central limit theorem read this;
- the ultimate purpose is the application to simple regression with a stochastic regressor.
Here we busy ourselves with the proof.
Measuring deviation of a random variable from a constant
Let be a random variable and some constant. We want a measure of differing from the constant by a given number or more. The set where differs from by or more is the outside of the segment , that is, .
Now suppose has a density . It is natural to measure the set by the probability . This is illustrated in Figure 1.
Convergence to a spike formalized
Once again, check out the idea. Consider a sequence of random variables and a parameter . Fix some and consider a corridor of width around . For to converge to a spike at we want the area to go to zero as we move along the sequence to infinity. This is illustrated in Figure 2, where, say, has a flat density and the density of is chisel-shaped. In the latter case the area is much smaller than in the former. The math of this phenomenon is such that should go to zero for any (the narrower the corridor, the further to infinity we should move along the sequence).
Definition. Let be some parameter and let be a sequence of its estimators. We say that converges to in probability or, alternatively, consistently estimates if as for any .
The law of large numbers in its simplest form
Let be an i.i.d. sample from a population with mean and variance . This is the situation from the standard Stats course. We need two facts about the sample mean : it is unbiased,
and its variance tends to zero
(2) as .
(by the Chebyshev inequality, see Extension 3))
Since this is true for any , the sample mean is a consistent estimator of the population mean. This proves Example 1.
The above proof applies in the next more general situation.
Theorem. Let be some parameter and let be a sequence of its estimators such that: a) for any and b) . Then converges in probability to .
This statement is often used on the Econometrics exams of the University of London.
In the unbiasedness definition the sample size is fixed. In the consistency definition it tends to infinity. The above theorem says that unbiasedness for all plus are sufficient for consistency.