---
title: "I1 mean-change testing"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{I1 mean-change testing}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>")
library(DCIi1)
```

## Reference test

The null is a constant mean with independent Gaussian noise. This release
uses the supplied empirical calibration only for 100 observations and w1=22.
The finite-sample probabilities are approximate; they are not Gumbel p-values.

```{r}
set.seed(12)
y <- rnorm(100) + rep(c(0, 3), each=50)
fit <- i1_test(y)
fit
diagnostics <- attr(fit, "diagnostics")
diagnostics
fit$cpmag * diagnostics$sigma
```

etahat is the last observation before a candidate boundary, whether or not
the test rejects. The signed cpmag is in standardized units. Multiplication
by sigma expresses that local contrast in data units.

## Formula and scope

Adaptive future-minus-past contrasts are summed symmetrically. The maximum
absolute aggregate is taken over w,...,n-w, following the research script.
The standard deviation estimate is IQR(diff(y))/[-2 sqrt(2) Phi_inverse(.25)].
WD mode divides that scale by (1-rho), exactly as in Appendix B.1, and estimates
rho using 1+2 cor(diff(y)[-last],diff(y)[-first]). The radical covers 2 only.

```{r}
i1_test(y, archeck=1, calibration="none")
i1_test(y, w1=16, calibration="none")
set.seed(13)
long_series <- rnorm(300)
i1_stt(long_series)
i1_test(long_series, calibration="none")
```

Statistic-only results have pvalue=NA. Reference mode rejects other lengths,
windows and WD mode. Input checks do not establish Gaussian independence.
Zero IQR and inadmissible dependence estimates error explicitly.

## Finite-sample evidence

In 10000 independent Gaussian null series (seed 26102403), rejection rates
were .0086, .0415 and .0857 at nominal .01, .05 and .10. The same fixed density
was used throughout. These conservative rates do not prove exact size or
validity under other distributions. A reported zero is a finite-grid tail
approximation. Rounded p-values limit significance resolution.

There is no adjustment for repeated testing. The package targets mean shifts;
a linear trend is not the constant-mean null. For multiple descriptive
candidates, see the DCI vignette.

## Reference

Waychal, N., Laha, A. K. and Sinha, A. (2026).
Closely spaced change-point detection with an application to geopolitical
shocks in oil and gas markets. <https://doi.org/10.48550/arXiv.2610.23476>.