# R/hypothesis_quantiles.R In wwntests: Hypothesis Tests for Functional Time Series

#### Documented in Q_WS_hyp_test

```# V_WS_hyp_test comuputes the 1-alpha qunatile of the beta * chi-squared distribution with nu
#   degrees of freedom, where beta and nu are obtained from a Welch-Satterthwaite approximation
#   of the test statistic V_K. This quantile is used to conduct an approximate size alpha test
#   of the hypothesis H'_0_K.
# Input: f_data = the functional data matrix with functions in columns
#        K = specifies the range of lags 1:K for the test statistic V_K
#        alpha = the significance level to be used in the hypothesis test
#        M = optional argument specifying the sampling size in the related Monte Carlo method
#        low_disc = boolean value specifiying whether or not to use low-discrepancy sampling
#                   for the Monte-Carlo method (only Sobol Sampling is currently supported)
# Output: scalar value of the 1-alpha quantile of the beta * chi-square distribution with nu
#         degrees of freedom (which approximates V_K)
V_WS_quantile <- function(f_data, K, alpha=0.05, M=NULL, low_disc=FALSE) {
mean_V_K <- mean_hat_V_K(f_data, K)
var_V_K <- variance_hat_V_K(f_data, K, M=M, low_disc=low_disc)
beta <- var_V_K / (2 * mean_V_K)
nu <- 2 * (mean_V_K^2) / var_V_K
quantile <- beta * qchisq(1 - alpha, nu)
statistic <- t_statistic_V(f_data, K)
p_val <- pchisq(statistic / beta, nu, lower.tail = FALSE)
list(statistic = statistic, quantile = quantile, p_value = p_val)
}

V_WS_quantile_iid <- function(f_data, K, alpha=0.05) {
mean_V_K <- mean_hat_V_K_iid(f_data, K)
var_V_K <- variance_hat_V_K_iid(f_data, K)
beta <- var_V_K / (2 * mean_V_K)
nu <- 2 * (mean_V_K^2) / var_V_K
quantile <- beta * qchisq(1 - alpha, nu)
statistic <- t_statistic_V(f_data, K)
p_val <- pchisq(statistic / beta, nu, lower.tail = FALSE)
list(statistic = statistic, quantile = quantile, p_value = p_val)
}

# Q_WS_hyp_test comuputes the 1-alpha qunatile of the beta * chi-squared distribution with nu
#   degrees of freedom, where beta and nu are obtained from a Welch-Satterthwaite approximation
#   of the test statistic Q_h. This quantile is used to conduct an approximate size alpha test
#   of the hypothesis H_0_h.
# Input: f_data = the functional data matrix with functions in columns
#        lag = specifies the lag used for the test statistic Q_h
#        alpha = the significance level to be used in the hypothesis test
#        M = optional argument specifying the sampling size in the related Monte Carlo method
#        low_disc = boolean value specifiying whether or not to use low-discrepancy sampling
#                   for the Monte-Carlo method (only Sobol Sampling is currently supported)
# Output: scalar value of the 1-alpha quantile of the beta * chi-square distribution with nu
#         degrees of freedom (which approximates Q_h).
Q_WS_quantile <- function(f_data, lag, alpha=0.05, M=NULL, low_disc=FALSE) {
mean_Q_h <- mean_hat_Q_h(f_data, lag)
var_Q_h <- variance_hat_Q_h(f_data, lag, M=M, low_disc=low_disc)
beta <- var_Q_h / (2 * mean_Q_h)
nu <- 2 * (mean_Q_h^2) / var_Q_h
quantile <- beta * qchisq(1 - alpha, nu)
statistic <- t_statistic_Q(f_data, lag)
p_val <- pchisq(statistic / beta, nu, lower.tail = FALSE)
list(statistic = statistic, quantile = quantile, p_value = p_val)
}

# Q_WS_quantile_iid computes the size alpha test of the hypothesis H_0_h using the WS
#   Approximation under the assumption that the data follows a strong white noise.
# Input: f_data = the functional data matrix with functions in columns
#        alpha = the significance level to be used in the hypothesis test
# Output: scalar value of the 1-alpha quantile of the beta * chi-square distribution with nu
#         degrees of freedom (which approximates Q_h) (computed under a strong white noise
#         assumption).
Q_WS_quantile_iid <- function(f_data, alpha=0.05) {
mean_Q_h <- mean_hat_Q_h_iid(f_data)
var_Q_h <- variance_hat_Q_h_iid(f_data)
beta <- var_Q_h / (2 * mean_Q_h)
nu <- 2 * (mean_Q_h^2) / var_Q_h
quantile <- beta * qchisq(1 - alpha, nu)
statistic <- t_statistic_Q(f_data, lag = 1)
p_val <- pchisq(statistic / beta, nu, lower.tail = FALSE)
list(statistic = statistic, quantile = quantile, p_value = p_val)
}

#' Compute size alpha single-lag hypothesis test under weak or strong white noise assumption
#'
#' \code{Q_WS_hyp_test} Computes the size alpha test of a single lag hypothesis under a weak white noise
#' or strong white noise assumption using a Welch-Satterthwaite Approximation.
#'
#' @param f_data the functional data matrix with observed functions in the columns
#' @param lag the lag to use to compute the single lag test statistic
#' @param alpha the significance level to be used in the hypothesis test
#' @param iid boolean value, if given TRUE, the hypothesis test will use a strong-white noise assumption.
#' By default is FALSE, in which the hypothesis test will use a weak-white noise assumption.
#' @param M Number of samples to take when applying a Monte-Carlo approximation
#' @param low_disc Boolean value indicating whether or not to use low-discrepancy sampling in the Monte
#' Carlo method. Note, low-discrepancy sampling will yield deterministic results.
#' @param bootstrap boolean value, if given TRUE, the hypothesis test is done by approximating the
#' limiting distribution of the test statistic via a block bootstrap algorithm. FALSE by default
#' @param block_size the block size to be used in the block bootstrap method (in each bootstrap sample).
#' 10 by default.
#' @param straps the number of bootstrap samples to take; 300 by default
#' @param moving boolean value; determines whether or not the block bootstrap should be moving
#' @return A list containing the p-value, the quantile, and a boolean value indicating whether or not the
#' hypothesis is rejected.
#'
#' @import stats
Q_WS_hyp_test <- function(f_data, lag, alpha=0.05, iid=FALSE,
M=NULL, low_disc=FALSE, bootstrap=FALSE,
block_size='adaptive', straps=300, moving = FALSE) {
statistic <- t_statistic_Q(f_data, lag)
if (bootstrap == TRUE) {
block_size <- ceiling(adaptive_bandwidth(f_data, kernel = 'Bartlett'))
}
bootsraps <- list()
bootstrap_samples <- block_bootsrap(f_data, block_size, B = straps, moving = moving)
stats_distr <- lapply(bootstrap_samples, t_statistic_Q, lag=lag)
statistic <- t_statistic_Q(f_data, lag=lag)
quantile <- quantile(as.numeric(stats_distr), 1 - alpha)
p_value <- sum(statistic > stats_distr) / length(stats_distr)
list(statistic = as.numeric(statistic), quantile = as.numeric(quantile),
p_value = as.numeric(p_value), block_size = block_size)
} else if (iid == FALSE) {
results <- Q_WS_quantile(f_data, lag, alpha=alpha, M=M, low_disc=low_disc)
statistic <- results\$statistic
quantile <- results\$quantile
p_val <- results\$p_val
reject <- statistic > quantile
list(statistic = statistic, quantile = quantile, p_value = p_val)
} else {
results <- Q_WS_quantile_iid(f_data, alpha=alpha)
statistic <- results\$statistic
quantile <- results\$quantile
p_val <- results\$p_val
reject <- statistic > quantile
list(statistic= statistic, quantile = quantile, p_value = p_val)
}
}
```

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wwntests documentation built on July 2, 2020, 2:57 a.m.