Asymptotic Distribution
======================
Definition
An Asymptotic Distribution is a concept used in statistics and probability theory to describe the limiting behavior of a sequence of random variables, as the Sample Size increases without bound. It provides a way to estimate the underlying distribution of a variable when the number of observations grows large.
Types of Asymptotic Distributions
There are several types of asymptotic distributions, including:
- Limiting Distribution: The Limiting Distribution is a probability distribution that is obtained by taking the limit of a sequence of random variables as the Sample Size increases.
- Asymptotic Independence: A sequence of random variables is said to be asymptotically independent if the conditional distribution of one variable given the values of other variables does not depend on those variables.
- Asymptotic Equivalence: Two sequences of random variables are said to be asymptotically equivalent if they have the same Limiting Distribution.
Properties of Asymptotic Distributions
Asymptotic distributions have several key properties:
- Convergence in Probability: An Asymptotic Distribution converges in probability to its Limiting Distribution, meaning that the probability of the random variable deviating from the Limiting Distribution decreases as the Sample Size increases.
- Consistency: An Asymptotic Distribution is said to be consistent if it is consistent with the true distribution of a population Parameter, meaning that as the Sample Size increases without bound, the sample estimates of the Parameter converge to its True Value at the same rate.
- Convergence in Law: A sequence of random variables converges in law to an Asymptotic Distribution, meaning that the Limiting Distribution is the same as that described above.
Examples
Example 1: Limiting Distribution
Consider a sequence of binomial random variables, where each observation has a probability p of success and a probability q = 1 - p of failure. Let X_n be the number of successes in n observations.
As n increases without bound, X_n / n converges to a Limiting Distribution D(p) that is normal with mean np and variance p(1-p).
Example 2: Asymptotic Independence
Consider two independent random variables, X_1 and X_2, which are also binomial random variables. Let Y = X_1 + X_2.
As the Sample Size increases without bound, Y / (n(X_1) + n(X_2)) converges to a Limiting Distribution D(p) that is normal with mean 0 and variance p(1-p)/p.
Example 3: Asymptotic Equivalence
Consider two sequences of binomial random variables, X_n and Y_n, where each observation has the same probability p. Let Z = X_1 + Y_1.
As the Sample Size increases without bound, Z / n(X_1) + Z / n(Y_1) converges to a Limiting Distribution that is normal with mean 0 and variance 2p(1-p)/p, regardless of the value of p.
Conclusion
Asymptotic distributions provide a powerful tool for analyzing the behavior of random variables as the Sample Size increases without bound. They can be used to estimate the underlying distribution of a variable, analyze the Consistency of estimators, and compare different models.