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)
Just calculate the mean normally for this case.
$$\mathrm{Var}\left(\bar{X}\right) = \frac{\sigma^2}{n} \sum^{n-1}_{k = -(n-1)} \left( 1 - \frac{\mid{k}\mid}{n} \right)\rho_k$$
$\sigma^2$ 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)
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