stat_Vn: Compute the CUSUM Statistic
In CPAT: Change Point Analysis Tests

Description Usage Arguments Details Value References Examples

This function computes the CUSUM statistic (and can compute weighted/trimmed variants, depending on the values of kn and tau).

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stat_Vn(dat, kn = function(n) {     1 }, tau = 0, estimate = FALSE,
  use_kernel_var = FALSE, custom_var = NULL, kernel = "ba",
  bandwidth = "and", get_all_vals = FALSE)

`dat`	The data vector
`kn`	A function corresponding to the trimming parameter t_T in the trimmed CUSUM variant; by default, is a function returning 1 (for no trimming)
`tau`	The weighting parameter τ for the weighted CUSUM statistic; by default, is 0 (for no weighting)
`estimate`	Set to `TRUE` to return the estimated location of the change point
`use_kernel_var`	Set to `TRUE` to use kernel methods for long-run variance estimation (typically used when the data is believed to be correlated); if `FALSE`, then the long-run variance is estimated using \hat{σ}^2_{T,t} = T^{-1}≤ft( ∑_{s = 1}^t ≤ft(X_s - \bar{X}_t\right)^2 + ∑_{s = t + 1}^{T}≤ft(X_s - \tilde{X}_{T - t}\right)^2\right), where \bar{X}_t = t^{-1}∑_{s = 1}^t X_s and \tilde{X}_{T - t} = (T - t)^{-1} ∑_{s = t + 1}^{T} X_s
`custom_var`	Can be a vector the same length as `dat` consisting of variance-like numbers at each potential change point (so each entry of the vector would be the "best estimate" of the long-run variance if that location were where the change point occured) or a function taking two parameters `x` and `k` that can be used to generate this vector, with `x` representing the data vector and `k` the position of a potential change point; if `NULL`, this argument is ignored
`kernel`	If character, the identifier of the kernel function as used in cointReg (see `getLongRunVar`); if function, the kernel function to be used for long-run variance estimation (default is the Bartlett kernel in cointReg)
`bandwidth`	If character, the identifier for how to compute the bandwidth as defined in cointReg (see `getBandwidth`); if function, a function to use for computing the bandwidth; if numeric, the bandwidth value to use (the default is to use Andrews' method, as used in cointReg)
`get_all_vals`	If `TRUE`, return all values for the statistic at every tested point in the data set

The definition of the statistic is

T^{-1/2} \max_{1 ≤q t ≤q T} \hat{σ}_{t,T}^{-1} ≤ft| ∑_{s = 1}^t X_s - \frac{t}{T}∑_{s = 1}^T \right|

A more general version is

T^{-1/2} \max_{t_T ≤q t ≤q T - t_T} \hat{σ}_{t,T}^{-1} ≤ft(\frac{t}{T} ≤ft(\frac{T - t}{T}\right)\right)^{τ} ≤ft| ∑_{s = 1}^t X_s - \frac{t}{T}∑_{s = 1}^T \right|

The parameter kn corresponds to the trimming parameter t_T and the parameter tau corresponds to τ.

See \insertCitehorvathricemiller19CPAT for more details.

If both estimate and get_all_vals are FALSE, the value of the test statistic; otherwise, a list that contains the test statistic and the other values requested (if both are TRUE, the test statistic is in the first position and the estimated change point in the second)

\insertAllCited

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CPAT:::stat_Vn(rnorm(1000))
CPAT:::stat_Vn(rnorm(1000), kn = function(n) {0.1 * n}, tau = 1/2)
CPAT:::stat_Vn(rnorm(1000), use_kernel_var = TRUE, bandwidth = "nw", kernel = "bo")