In josephsdavid/tstest: tstest

Practical Autocorrelation

Estimation!

$\rho_k \rightarrow 0$

For a stationary time series, if the autocorrelation approaches zero, then :

A Single realization lets us estimate mean, variance, and autovvariance!

Remember that $\rho_k = \frac{\gamma_k}{\gamma_0}$, and we use n-k pairs to calculate it (the summation)

Mean

Just calculate the mean normally for this case.

Variance

$$\mathrm{Var}\left(\bar{X}\right) = \frac{\sigma^2}{n} \sum^{n-1}_{k = -(n-1)} \left( 1 - \frac{\mid{k}\mid}{n} \right)\rho_k$$

$\sigma^2$ is calculated as normal, we will see rhok next!

remember !

Now it is time for some code!

library(glue)
xbar <- function(xs){
    mean(xs)
}

ghat_zero <- function(xs){
    summand <- (xs - xbar(xs))^2
    mean(summand)
}

ghat_one <- function(xs) {
    lhs <- xs[1:(length(xs) - 1)] - xbar(xs)
    rhs <- xs[2:length(xs)] - xbar(xs)
    summand <- lhs * rhs
    summate <- sum(summand)
    summate/length(xs)

}
rhohat_zero <- 1
rhohat_one <- function(xs) {
    ghat_one(xs) / ghat_zero(xs)
}
v <- c(76,70,66,60,70,72,76,80)
xbar(v)
ghat_zero(v)
ghat_one(v)
rhohat_one(v)