Functional Hypothesis Tests for Functional Data

Computes a variety of portmanteau hypothesis tests for functional data in the form of dfts objects.

Usage

portmanteau_tests(
  X,
  test = c("variety", "single", "multi", "spectral", "independence", "imhof"),
  lag = 5,
  M = 1000,
  method = c("iid", "bootstrap"),
  kernel = bartlett_kernel,
  block_size = NULL,
  bandwidth = NULL,
  components = 3,
  resample_blocks = "separate",
  replace = FALSE,
  alpha = 0.05
)

Arguments

X

A dfts object or data which can be automatically converted to that format. See dfts().

test

A String specifying the hypothesis test. Currently available tests are: 'variety', 'single-lag', 'multi-lag', 'spectral', 'independence', and 'imhof'.

lag

A positive integer to specify the lag, or maximum lag, of interest. Only used for the "single-lag", "multi-lag", "independence", and "imhof" tests.

M

Numeric to specify the number of Monte-Carlo or resampled simulations to use for the limiting distributions.

method

String indicating the method for the single test, options include:

• iid: The hypothesis test will use a strong-white noise assumption (instead of a weak-white noise assumption).

• resample: The hypothesis test is evaluated by approximating the limiting distribution of the test statistic via a (block) resampling process.

Additional methods are forthcoming.

kernel

Kernel function for spectral test or estimation of covariance.

block_size

Numeric to specify block size for resampling tests.

bandwidth

Numeric for bandwidth of covariance estimation. If left null, with be defined by \(N^{1 / (2 * \text{KO} + 1)}\) where KO is the order of the selected kernel.

components

Number of functional principal components to use in the independence test.

resample_blocks

String indicating the type of resample test to use. Using separate gives blocks which are separate while overlapping creates overlapping or sliding windows. When blocksize=1 then these will be identical.

replace

Boolean for using a permutation or bootstrapped statistic when method='resample'.

alpha

Numeric value for significance in [0,1].

Value

List with results dependent on the test. In general, returns the pvalue, statistic, and simulations/quantile.

Details

The "single"-lag portmanteau test assesses the significance of empirical lagged autocovariance operators at a single lag lag. It tests the null hypothesis that the lag-h autocovariance operator is equal to 0. The test is designed for stationary functional time-series, and is valid under conditional heteroscedasticity conditions.

The "multi"-lag portmanteau test assesses the cumulative significance of empirical lagged autocovariance operators, up to a user-selected maximum lag lag. It tests the null hypothesis that the first lag-h autocovariance operators, \(h=1,\dots,lag\), is equal to 0. The test is designed for stationary functional time-series, and is valid under conditional heteroscedasticity conditions.

The "spectral" portmanteau test measures the proximity of a functional time series to a white noise. Comparison is made to the constant spectral density operator of an uncorrelated series. The test is not for general white noise series, and may not hold under functional conditionally heteroscedastic assumptions.

The "independence" portmanteau test measures independence and identical distribution based lagged cross-variances from dimension reduction using functional principal components analysis. The test is not for general white noise series, and may not hold under functional conditionally heteroscedastic assumptions.

The "imhof" portmanteau test is an analogue of the "single-lag" test. While the "single-lag" test computes the limiting distribution of the test statistic via a Welch-Satterthwaite approximation, the "imhof" test directly computes the coefficients of the quadratic form in normal variables. Hence, the test is computationally expensive.

References

Kim, M., Kokoszka P., & Rice G. (2023) White noise testing for functional time series. Statist. Surv., 17, 119-168, DOI: 10.1214/23-SS143

Characiejus, V., & Rice, G. (2020). A general white noise test based on kernel lag-window estimates of the spectral density operator. Econometrics and Statistics, 13, 175–196.

Kokoszka P., & Rice G., & Shang H.L. (2017). Inference for the autocovariance of a functional time series under conditional heteroscedasticity. Journal of Multivariate Analysis, 162, 32-50.

Zhang X. (2016). White noise testing and model diagnostic checking for functional time series. Journal of Econometrics, 194, 76-95.

Gabrys R., & Kokoszka P. (2007). Portmanteau Test of Independence for Functional Observations. Journal of the American Statistical Association, 102:480, 1338-1348, DOI: 10.1198/016214507000001111.

Chen W.W. & Deo R.S. (2004). Power transformations to induce normality and their applications. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 66, 117-130.

Examples

b <- generate_brownian_motion(250)
res0 <- portmanteau_tests(b, test = 'single', lag = 10)
res1 <- portmanteau_tests(b, test = 'multi', lag = 10, alpha = 0.01)
res2 <- portmanteau_tests(b, test = 'spectral', kernel = bartlett_kernel,
                          bandwidth = NULL, alpha = 0.05)
res3 <- portmanteau_tests(b, test = 'spectral', alpha = 0.1,
                          kernel = parzen_kernel,
                          bandwidth = adaptive_bandwidth(b, kernel=parzen_kernel))
res4 <- portmanteau_tests(b, test = 'independence', components = 3, lag = 3)
#> Registered S3 method overwritten by 'quantmod':
#>   method            from
#>   as.zoo.data.frame zoo 
res5 <- portmanteau_tests(b, test = 'single', lag = 1, M = 250)