Nile application

Data

Annual minimum Nile water levels

The worked application uses 663 yearly minimum water-level observations from the Roda gauge near Cairo. The series is often studied in the long-memory literature, although structural breaks can generate similar persistent behaviour.

Observations	663
Period	622 to 1284
Frequency	Annual
Response	`NileMin`, minimum Nile water level
Source	`longmemo::NileMin`; originally compiled from Tousson (1925) and Beran (1994)

R / load the Nile data

library(SKN)
data("NileMin", package = "longmemo")
nile_levels <- as.numeric(NileMin)

First six observations

Year	Minimum water level
622	1157
623	1088
624	1169
625	1169
626	984
627	1322

Specification

A reproducible single-bootstrap application

The reported result is generated from the current package source with a fixed random seed. The non-studentized route keeps the website build computationally light while exercising the public SKNtest() interface on real time-series data.

Setting	Value
Route	Non-studentized single bootstrap
`B1`	1000 replications
Criterion	AIC
Maximum lag	\(\lceil (\log n)^2/2 \rceil\)
Random seed	20260813
Selected lag	13

Results

None of the four null hypotheses is rejected

R / Nile application

set.seed(20260813)
nile_result <- SKNtest(
  NileMin,
  B1 = 1000,
  crit = "AIC",
  studentized = FALSE
)
nile_result

Output

Selected lag: 13
pS = 0.195
pR = 0.177
pK = 0.684
pN = 0.352

Computed result

Bootstrap decisions at the 5% level

Test	p-value	Decision
Sample skewness (pS)	0.195	Do not reject symmetry at 5%
Robust skewness (pR)	0.177	Do not reject symmetry at 5%
Kurtosis (pK)	0.684	Do not reject mesokurtosis at 5%
Combined normality (pN)	0.352	Do not reject the joint skewness-kurtosis conditions at 5%

At the 5% level, neither symmetry test rejects symmetry, the kurtosis test does not reject mesokurtosis, and the combined test does not reject the joint skewness-kurtosis conditions associated with normality.

These non-rejections describe this application; they do not prove that the marginal distribution satisfies the corresponding conditions. Likewise, persistence in the series is not by itself evidence of long-range dependence because structural breaks may produce similar behaviour.

Default route

Run the studentized double bootstrap

The package default uses the studentized procedure. When B2 is omitted, the package sets B2 = 100 and prints a message. This route is substantially more computationally demanding and is therefore shown without being run during website generation.

R / studentized alternative

set.seed(20260813)
studentized_result <- SKNtest(
  NileMin,
  B1 = 1000,
  crit = "AIC"
)