Scalar mean inference

This vignette uses a synthetic series. It shows the scalar case of affine-equivariant adjusted-range self-normalization, where the construction reduces exactly to adjusted-range self-normalization: the scaled estimation error is divided by the adjusted range (maximum minus minimum) of the centered partial-sum path.

Setup

The series is a stationary first-order autoregression with mean 0.3 and autoregressive coefficient 0.5. The parameter of interest is the mean.

library(aersn)
set.seed(2026)
n <- 300
y <- 0.3 + as.numeric(arima.sim(list(ar = 0.5), n))
fit <- aersn_mean(y)
fit
#> Affine-equivariant adjusted-range self-normalization (increment hull)
#>   model: sample mean (psi_t = Y_t - Ybar) 
#>   n = 300 observations; q = 1 parameter
#>   centering: calendar time, tau(r) = r 
#>   estimate:
#>  theta 
#> 0.4159 
#>   hull diagnostics: numerical rank 1 of 1  ; condition 1.000 ; min projected range 2.083 ; spread 1.000 
#> Use aersn_test(), confint(), aersn_contrast(), aersn_region(), plot().

The influence contribution of observation t for the sample mean is \(Y_t - \bar Y_n\). The centered path \(\hat G_n(k/n) = n^{-1/2}\sum_{t \le k}(Y_t - \bar Y_n)\) starts and ends at zero:

plot(fit$path)

Test of a point null

For \(H_0: \theta_0 = 0\) the statistic is \(T_n(0) = |\sqrt n(\bar Y_n - 0)| / \{\max_k \hat G_n(k/n) - \min_k \hat G_n(k/n)\}\). Its limit is \(|M| = |Z| / R\), where \(Z\) is standard normal and \(R\) is the range of an independent standard Brownian bridge. Two reference laws are available:

the closed-form continuous-path law (reference = "continuous"), whose five-percent critical value is 1.7058; and
the matched-grid law, simulated for a Brownian bridge on the same grid of n intervals as the sample path. Matched-grid quantiles are larger because a path observed on a grid has a smaller range than the continuous path. The manuscript uses matched-grid quantiles as critical values.

aersn_test(fit, null = 0, reference = "continuous")
#> 
#>  Adjusted-range increment hull test, q = 1
#> 
#> data:  fit
#> T = 3.459, q = 1, n = 300
#> alternative hypothesis: true parameter is not equal to the null value
#> null value: theta = 0 
#> estimate:   theta = 0.4159 
#> critical value at level 0.95: 1.706  (reject)
#> p-value = 0.0005637 (closed-form law)
#> reference: closed-form continuous-path law of |M| (q = 1)
aersn_test(fit, null = 0, draws = 20000, seed = 1)
#> 
#>  Adjusted-range increment hull test, q = 1
#> 
#> data:  fit
#> T = 3.459, q = 1, n = 300
#> alternative hypothesis: true parameter is not equal to the null value
#> null value: theta = 0 
#> estimate:   theta = 0.4159 
#> critical value at level 0.95: 1.837  (reject)
#> p-value = 0.00130 (Monte Carlo s.e. 0.00025, resolution 0.000050)
#> reference: matched-grid Monte Carlo law for increment-hull gauge: q = 1, uniform grid with n = 300 intervals, 20000 draws, seed 1

The matched-grid p-value is a Monte Carlo estimate; its standard error and resolution (one over the number of draws) are reported. A one-sided alternative uses the signed ratio:

aersn_test(fit, null = 0, alternative = "greater", reference = "continuous")
#> 
#>  Adjusted-range increment hull test, q = 1
#> 
#> data:  fit
#> T = 3.459, q = 1, n = 300
#> alternative hypothesis: true parameter is greater than the null value
#> null value: theta = 0 
#> estimate:   theta = 0.4159 
#> critical value at level 0.95: 1.397  (reject)
#> p-value = 0.0002818 (closed-form law)
#> reference: closed-form continuous-path law of |M| (q = 1)

Confidence interval

The confidence interval is \(\bar Y_n \pm (c/\sqrt n)\,\mathcal R(\hat G_n)\), where \(c\) is the reference quantile.

confint(fit, level = 0.95, draws = 20000, seed = 1)
#> Simultaneous (joint-region projection) Adjusted-range increment hull confidence intervals, level 0.95
#>       lower  upper
#> theta 0.195 0.6368
#> critical value 1.837 (reference dimension 1); matched-grid Monte Carlo law for increment-hull gauge: q = 1, uniform grid with n = 300 intervals, 20000 draws, seed 1
confint(fit, level = 0.95, reference = "continuous")
#> Simultaneous (joint-region projection) Adjusted-range increment hull confidence intervals, level 0.95
#>        lower upper
#> theta 0.2108 0.621
#> critical value 1.706 (reference dimension 1); closed-form continuous-path law of |M| (q = 1)

The interval endpoints are exactly the boundary of the set of null values that the test does not reject:

ci <- confint(fit, draws = 20000, seed = 1)
aersn_gauge(fit, ci[1, "upper"])      # equals the critical value
#> [1] 1.83715
attr(ci, "critical.value")
#> [1] 1.83715

What is and is not guaranteed

The test is asymptotically valid when the partial sums of \(Y_t - \theta_0\) satisfy a functional central limit theorem with positive long-run variance. No bandwidth, kernel, or block length is chosen. Under strong persistence the finite-sample null rejection rate exceeds the nominal level; the manuscript’s simulations report, for example, 14.4 percent at autoregressive coefficient 0.9 with n = 500 and q = 2.