notrend_test {funtimes}

Sieve Bootstrap Based Test for the Null Hypothesis of no Trend

Description

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).

Usage

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 Hall and Van Keilegom (2003). 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 (Bickel and Sakov 2008) 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".

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.

Value

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".

Author(s)

Vyacheslav Lyubchich

References

Bickel PJ, Sakov A (2008). “On the choice of m in the m out of n bootstrap and confidence bounds for extrema.” Statistica Sinica, 18(3), 967–985.

Hall P, Van Keilegom I (2003). “Using difference-based methods for inference in nonparametric regression with time series errors.” Journal of the Royal Statistical Society, Series B (Statistical Methodology), 65(2), 443–456. doi:10.1111/1467-9868.00395.

See Also

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

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)