genwwn.thpower: Compute (approximation) to the theoretical power of the `genwwn.test` test for ARMA processes (including, of course, white noise itself).

Description

Compute (approximation) to the theoretical power of the genwwn.test test. Note: this function does no simulation, it merely computes an approximation to the likely statistical power (or size) of the genwwn.test function. It can be useful to establish the reverse question: what sample size do I require to achieve a certain power for a given ARMA process?

Usage

genwwn.thpower(N = 128, ar = NULL, ma = NULL, plot.it = FALSE,
	sigsq = 1, alpha = 0.05, away.from = "standard",
	filter.number = 10, family = "DaubExPhase", verbose = FALSE)

Value

A list containing the following components.

C.alpha.c: The critical value for the test, which is the nominal size critical value after correction for multiple hypothesis tests (correction using Bonferroni).
th.power: The computed theoretical power
norspecwd: The wavelet coefficients of the true spectrum
norspecvarwd: The squared wavelet transform of the squared normalized spectrum
all.hwc: All of the wavelet coefficients from the normalized true specturm as a single vector
all.sdwc: The `true' standard deviations of the wavelet coefficients

Arguments

N: The length of the series you want to get a theoretical power result for.
ar: Autoregressive parameters. A vector with p entries for AR(p) with the first entry being the value for lag-one term (alpha_1), the second entry being the value for the lag-two term (alpha_2) etc. If this argument is NULL then there are no AR terms.
ma: Similar to the ar argument except for MA terms. A vector of length q for MA(q) parameters, with first entry being beta_1, the second being beta_2, etc. If this argument is NULL then there are no MA terms.
plot.it: If TRUE then two plots are produced. The first is of the time series spectrum you are considering (controlled by the N, ar and ma arguments.) The second is a plot of the wavelet coefficients of the normalized spectrum.
sigsq: The theoretical innovation variance (also the variance of white noise if ar=ma=NULL.
alpha: The nominal size of the test for this theoretical power calculation.
away.from: Describes how many fine scales to exclude, the same as in genwwn.test. This can be an integer up to the number of scales. However, mostly you can leave this at "standard" where the scales calculation is automatically determined.
filter.number: The number of vanishing moments in the Daubechies series of wavelets.
family: The wavelet family.
verbose: If TRUE then informative messages are printed during the progress of the function.

Author

Delyan Savchev and Guy Nason

Details

Function calculates the value of the power function at the specified arguments. It does this by: (i) specifying the functional spectrum of the ARMA process (which can be flat, ie white noise); (ii) calculating the variance of the ARMA process by numerical integration of the spectrum; (iii) calculating the spectrum values at the Fourier frequencies; (iv) calculating the wavelet coefficients at the exact spectrum values; (v) computing the exact variance of the wavelet coefficients of the squared normalized spectrum; (vi) computing the approximate power of the whole lot.

References

Nason, G.P. and Savchev, D. (2014) White noise testing using wavelets. Stat, 3, 351-362. tools:::Rd_expr_doi("10.1002/sta4.69")

Examples

Run this code

#
# Calculate what the theoretical actual size is likely to be for the
# genwwn.test for a white noise sequence of T=64, nominal size=0.05
#
genwwn.thpower(N=64)$th.power
#[1] 0.04894124
#
# This is pretty close to the nominal size of 5%. Good.
#
# What is the power of detection for the AR(1) process with alpha=0.3?
# Let's say with sample size of T=32
#
genwwn.thpower(N=32, ar=0.3)$th.power 
#[1] 0.2294128
#
# That's pretty poor, we'll only detect about 23% of cases. Can we achieve
# a power of 90%? Actually, it turns out that by repeating these above
# functions with N=128 gives a power of 61%, and for N=256 we get a power of
# 90%.

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