Statistics 8054 (Geyer, Spring 2024) AR(1) Example

Theory

An AR(1) time series satisfies

X_{n + 1} = ρ X_n + Y_n

where the Y_n are IID Normal(0, τ²).

The lag k autocovariance is

γ_k = ρ^k τ² ⁄ (1 - ρ²)

The asymptotic variance in the central limit theorem is

σ² = γ₀ (1 + ρ) ⁄ (1 - ρ)

The approximate, large k, variance of the empirical estimates of the γ_k is approximately 1 ⁄ n times

γ₀² + 2 ∑_{j = 1}^∞ γ_j²

Simulation

Batch Means

Overlapping Batch Means

Behavior of Empirical Autocovariances

Estimators Using Reversibility

Γ_k = γ_{2 k} + γ_{2 k + 1}

is strictly positive, strictly decreasing, strictly convex function of k.

Truncate sequence at first negative estimate of Γ_k. Gives initial positive sequence estimator.

Take maximimal decreasing function that fits under above. Gives initial decreasing sequence estimator.

Take maximimal decreasing convex function that fits under above. Gives initial convex sequence estimator.

The Old Way with Lots of Code

The New Way with New Function

R statements n <- 1e5 rho <- 0.99 x <- arima.sim(model = list(ar = rho), n = n) library(mcmc) out <- initseq(x) ##### initial positive sequence black ##### initial decreasing sequence red ##### initial convex sequence blue plot(seq(along = out$Gamma.pos) - 1, out$Gamma.pos, xlab = "k", ylab = expression(Gamma[k]), type = "l", lwd = 2) lines(seq(along = out$Gamma.dec) - 1, out$Gamma.dec, col = "red", lwd = 2) lines(seq(along = out$Gamma.con) - 1, out$Gamma.con, col = "blue", lwd = 2) # asymptotic 95% confidence interval for mean of x mean(x) + c(-1, 1) * qnorm(0.975) * sqrt(out$var.con / length(x)) # estimated asymptotic variance out$var.con # theoretical asymptotic variance (1 + rho) / (1 - rho) * 1 / (1 - rho^2) # illustrating use with batch means blen <- 5 stopifnot(blen * floor(n / blen) == n) bm <- apply(matrix(x, nrow = 5), 2, mean) initseq(bm)$var.con * blen