notrend_test function

Sieve Bootstrap Based Test for the Null Hypothesis of no Trend

A combination of time series trend tests for testing the null hypothesis of no trend, versus the alternative hypothesis of a linear trend (Student's t-test), or monotonic trend (Mann--Kendall test), or more general, possibly non-monotonic trend (WAVK test).

notrend_test(
  x,
  B = 1000,
  test = c("t", "MK", "WAVK"),
  ar.method = "HVK",
  ar.order = NULL,
  ic = "BIC",
  factor.length = c("user.defined", "adaptive.selection"),
  Window = NULL,
  q = 3/4,
  j = c(8:11)
)

Arguments

• x: a vector containing a univariate time series. Missing values are not allowed.

• B: number of bootstrap simulations to obtain empirical critical values. Default is 1000.

• test: trend test to implement: Student's t-test ("t", default), Mann--Kendall test ("MK"), or WAVK test ("WAVK", see WAVK).

• ar.method: method of estimating autoregression coefficients. Default "HVK" delivers robust difference-based estimates by if(!exists(".Rdpack.currefs")) .Rdpack.currefs <-new.env();Rdpack::insert_citeOnly(keys="Hall_VanKeilegom_2003;textual",package="funtimes",cached_env=.Rdpack.currefs) . Alternatively, options of ar function can be used, such as "burg", "ols", "mle", and "yw".

• ar.order: order of the autoregressive model when ic = "none", or the maximal order for IC-based filtering. Default is round(10*log10(length(x))), where x is the time series.

• ic: information criterion used to select the order of autoregressive filter (AIC of BIC), considering models of orders $p=$ 0,1,...,ar.order. If ic = "none", the AR($p$) model with $p=$ ar.order is used, without order selection.

• factor.length: method to define the length of local windows (factors). Used only if test = "WAVK". Default option "user.defined" allows to set only one value of the argument Window. The option "adaptive.selection" sets method = "boot" and employs heuristic $m$-out-of-$n$ subsampling algorithm if(!exists(".Rdpack.currefs")) .Rdpack.currefs <-new.env();Rdpack::insert_citeOnly(keys="Bickel_Sakov_2008",package="funtimes",cached_env=.Rdpack.currefs) to select an optimal window from the set of possible windows length(x)*q^j whose values are mapped to the largest previous integer and greater than 2. Vector x is the time series tested.

• Window: length of the local window (factor), default is round(0.1*length(x)). Used only if test = "WAVK". This argument is ignored if

  factor.length = "adaptive.selection".

• q: scalar from 0 to 1 to define the set of possible windows when factor.length = "adaptive.selection". Used only if test = "WAVK". Default is $3/4$. This argument is ignored if factor.length = "user.defined".

• j: numeric vector to define the set of possible windows when factor.length = "adaptive.selection". Used only if test = "WAVK". Default is c(8:11). This argument is ignored if factor.length = "user.defined".

Returns

A list with class "htest" containing the following components: - method: name of the method.

• data.name: name of the data.

• statistic: value of the test statistic.

• p.value: $p$-value of the test.

• alternative: alternative hypothesis.

• estimate: list with the following elements: employed AR order and estimated AR coefficients.

• parameter: window that was used in WAVK test, included in the output only if test = "WAVK".

Details

This function tests the null hypothesis of no trend versus different alternatives. To set some other shape of trend as the null hypothesis, use wavk_test. Note that wavk_test employs hybrid bootstrap, which is an alternative to the sieve bootstrap employed by the current function.

Examples

## Not run:

# Fix seed for reproducible simulations:
set.seed(1)

#Simulate autoregressive time series of length n with smooth linear trend:
n <- 200
tsTrend <- 1 + 2*(1:n/n)
tsNoise <- arima.sim(n = n, list(order = c(2, 0, 0), ar = c(0.5, -0.1)))
U <- tsTrend + tsNoise
plot.ts(U)

#Use t-test
notrend_test(U)

#Use Mann--Kendall test and Yule-Walker estimates of the AR parameters
notrend_test(U, test = "MK", ar.method = "yw")

#Use WAVK test for the H0 of no trend, with m-out-of-n selection of the local window:
notrend_test(U, test = "WAVK", factor.length = "adaptive.selection")
# Sample output:
##      Sieve-bootstrap WAVK trend test
##
##data:  U
##WAVK test statistic = 21.654, moving window = 15, p-value < 2.2e-16
##alternative hypothesis: (non-)monotonic trend.
##sample estimates:
##$AR_order
##[1] 1
##
##$AR_coefficients
##    phi_1
##0.4041848
## End(Not run)

References

if(!exists(".Rdpack.currefs")) .Rdpack.currefs <-new.env();Rdpack::insert_all_ref(.Rdpack.currefs)

See Also

ar, HVK, WAVK, wavk_test, vignette("trendtests", package = "funtimes")

Author(s)

Vyacheslav Lyubchich