## The Delta Method: Applications

This week I am going to present three applications of the Delta method theorem. The first is a direct one and it is about the behavior in distribution of the sample variance. The second one is an hypothesis test in the variance when the sample is normal. Finally, the third is an interesting application in […]

## The Delta method: Main Result

Let $latex {T_{n}}&fg=000000$ an estimator of $latex {\theta}&fg=000000$, we want to estimate the parameter $latex {\phi(\theta)}&fg=000000$ where $latex {\phi}&fg=000000$ is a known function. It is natural to estimate $latex {\phi(\theta)}&fg=000000$ by $latex {\phi(T_{n})}&fg=000000$. Now, we can then ask: How the asymptotic properties of $latex {T_{n}}&fg=000000$ could be transfer to $latex {\phi(T_{n})}&fg=000000$?

## Weak Law of Large Numbers and Central Limit Theorem via the Levy’s continuity theorem

The Levy’s continuity theorem is a very important tool in the statistical machinery. For example, it will give us two simple proofs to two classical statistical problems: The Law of Large Numbers and the Central Limit Theorem.

## Revista de Matemática: Teoría y Aplicaciones. Vol 19, No 2 (2012)

The newest edition of the mathematical journal of Costa Rica “Revista de Matemática: Teoría y Aplicaciones. Vol 19, No 2 (2012).” is available here http://bit.ly/PDfXDK

## Characteristic functions and the Lévy’s continuity theorem

Photo of Paul Lévy. Source: MacTutor and Ra-bird. Using $latex {(ii)}&fg=000000$ of the Pormanteau lemma, it is possible to show convergence in distribution for a random vectors sequence via one “transformation”. The most important transform is the characteristic function

## The probability versions for the Big-O and little-o notations

We introduce here some notation very useful in probability and statistics. Definition 1 For a given sequence of random variables $latex {R_{n}}&fg=000000$, $latex {(i)}&fg=000000$ $latex {X_{n}=o_{P}(R_{n})}&fg=000000$ means $latex {X_{n}=Y_{n}R_{n}}&fg=000000$ with $latex {Y_{n}}&fg=000000$ converging to $latex 0&fg=000000$ in probability. $latex {(ii)}&fg=000000$ $latex {X_{n}=O_{P}(R_{n})}&fg=000000$ means $latex {X_{n}=Y_{n}R_{n}}&fg=000000$ with the family $latex {(Y_{n})_{n}}&fg=000000$ uniformly thigh.