arl_op

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arl_op(lam, cl, op_dgp, reps=10_000; chart_choice, d=1, ced=false, ad=100)

Function to compute the average run length (ARL) for ordinal patterns using the EWMA statistic. The function implements the test statistics by Weiss and Testik (2023), who use a pattern length of 3.

lam::Float64 Smoothing parameter for the EWMA statistic.
cl::Float64 Control limit for the EWMA statistic.
op_dgp::Union{IC, AR1, MA1, MA2, TEAR1, AAR1, QAR1} DGP.
reps::Int64 Number of replications.
chart_choice::Int
1. \(\widehat{H}^{(d)}=-\sum_{k=1}^{m!} \hat{p}_k{ }^{(d)} \ln \hat{p}_k{ }^{(d)}\)
2. \(\widehat{H}_{\mathrm{ex}}^{(d)}=-\sum_{k=1}^{m!}\left(1-\hat{p}_k{ }^{(d)}\right) \ln \left(1-\hat{p}_k{ }^{(d)}\right)\)
3. \(\widehat{\Delta}^{(d)}=\sum_{k=1}^{m!}\left(\hat{p}_k^{(d)}-1 / m!\right)^2\)
4. \(\hat{\beta}^{(d)}=\hat{p}_6^{(d)}-\hat{p}_1^{(d)}\)
5. \(\hat{\tau}^{(d)}=\hat{p}_6^{(d)}+\hat{p}_1^{(d)}-\frac{1}{3}\)
6. \(\hat{\delta}^{(d)}=\hat{p}_4^{(d)}+\hat{p}_5^{(d)}-\hat{p}_3^{(d)}-\hat{p}_2^{(d)}\)
The patterns are categorized as follows:

\(\qquad p_1 = (3,2,1); \quad p_2=(3,1,2); \quad p_3 = (2,3,1);\)

\(\qquad p_4 = (1,3,2); \quad p_5 = (2,1,3); \quad p_ 6 = (1,2,3)\)
d::Union{Int,Vector{Int}}=1: Delay vector. Default is 1. A vector would denote the indices of the observations to use. For example, d = [1, 3, 4] would denote the first, third, and fourth observations.
ced::Bool=false: Use conditional expected delay? Default is false.
ad::Int=100: Number of iterations for ced.

# Compute initial values via function cl_op()
 if j == 1 || j == 2
      cl_init = quantile(stat_op(data, lam[i], j)[1], 0.01)                
  else
      cl_init = quantile(stat_op(data, lam[i], j)[1], 0.99)
end 

# Run function
arl_op(0.1, cl_init, IC(Normal(0, 1)), 10_000; chart_choice=1, d=1, ced=false, ad=100)