I1 mean-change testing

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.

set.seed(12)
y <- rnorm(100) + rep(c(0, 3), each=50)
fit <- i1_test(y)
fit
#>   pvalue      i1 etahat    cpmag
#> 1      0 47.1701     51 3.799967
diagnostics <- attr(fit, "diagnostics")
diagnostics
#> $sigma
#> [1] 0.8434712
#> 
#> $rho
#> [1] 0
#> 
#> $calibration
#> [1] "reference"
#> 
#> $calibration_key
#> [1] "0"
#> 
#> $w1
#> [1] 22
#> 
#> $statistic
#> [1] 47.17013
#> 
#> $calibration_n
#> [1] 100
#> 
#> $calibration_length_matches
#> [1] TRUE
#> 
#> $calibration_provenance
#> [1] "supplied Gaussian reference density; n=100, w1=22"
#> 
#> $inference_scope
#> [1] "iid Gaussian, n=100, w1=22"
fit$cpmag * diagnostics$sigma
#> [1] 3.205162

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.

i1_test(y, archeck=1, calibration="none")
#>   pvalue      i1 etahat    cpmag
#> 1     NA 50.5476     51 4.072048
i1_test(y, w1=16, calibration="none")
#>   pvalue      i1 etahat    cpmag
#> 1     NA 33.1022     51 3.566918
set.seed(13)
long_series <- rnorm(300)
i1_stt(long_series)
#> [1] 13.41701
i1_test(long_series, calibration="none")
#>   pvalue     i1 etahat     cpmag
#> 1     NA 13.417    160 0.6909279

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.