Constructs two simultaneous one-sided e-processes for sequentially testing whether forecaster 1 (p) outperforms forecaster 2 (q) or vice versa.
Usage
eprocess(
scores1,
scores2,
alpha = 0.05,
c = 2,
v_opt = 10,
alpha_opt = NULL,
gammas = NULL,
clip_max = 1e+07
)Arguments
- scores1
Numeric vector. Scores S(p_t, y_t) for forecaster 1.
- scores2
Numeric vector. Scores S(q_t, y_t) for forecaster 2.
- alpha
Numeric in (0,1). Significance level. Rejection threshold is 2/alpha for the two-sided test. Default: 0.05.
- c
Numeric > 0. Sub-exponential scale. Must satisfy |hat_delta_i| <= c/2 for all i. For score differences in
[-(b-a), b-a]: c = 2*(b-a). Default: 2 (for Brier score differences in[-1,1]).- v_opt
Numeric > 0. Intrinsic time at which e-process grows fastest. Default: 10 (recommended by CR23).
- alpha_opt
Numeric in (0,1). One-sided alpha used to compute rho. Default: alpha/2 (matches comparecast two-sided convention).
- gammas
Numeric vector or NULL. Predictable centering sequence. If NULL, constructed as lagged running mean.
- clip_max
Numeric. Maximum e-process value before clipping. Default: 1e7 (matches Python comparecast).
Details
The mixture e-process at time t is: $$E_t^{\mathrm{mix}} = m(S_t, \hat{V}_t)$$ where \(S_t = \sum_{i=1}^t \hat{\delta}_i\), \(\hat{V}_t = \sum_{i=1}^t (\hat{\delta}_i - \gamma_i)^2\), and \(m(s, v)\) is the Gamma-Exponential mixture function (Proposition 3, CR23).
VARIANCE PROCESS: The intrinsic time V_hat_t uses NO floor (unlike the EB CS). The GE mixture m(s, v) is well-defined at v=0 (returns 1 when s=0), so no floor is needed. Adding a floor would distort e-values.
SCALE CONVENTION: c is the sub-exponential scale parameter such that
|hat_delta_i| <= c/2. This is the Theorems 2 & 3 convention from CR23.
For Brier score differences in [-1,1]: c = 2.
For Winkler scores (bounded above by 1): c = 2.
LOG-SPACE: E-process values are computed in log-space and clipped before exponentiating to avoid numerical overflow.
Rejection rule
At level alpha: reject \(H_0^w(p, q)\) (conclude p outperforms q)
when e_pq >= 2/alpha; reject \(H_0^w(q, p)\) (conclude q outperforms
p) when e_qp >= 2/alpha. Use eprocess_rejections() to extract the
first crossing time for each.
Examples
scores1 <- c(-0.04, -0.09, -0.01, -0.16)
scores2 <- c(-0.09, -0.16, -0.04, -0.25)
ep <- eprocess(scores1, scores2, alpha = 0.05)
head(ep)
#> t e_pq e_qp log_e_pq log_e_qp
#> 1 1 1.018495 0.9779115 0.01832561 -0.02233605
#> 2 2 1.047660 0.9502583 0.04655865 -0.05102145
#> 3 3 1.059805 0.9381265 0.05808536 -0.06387052
#> 4 4 1.098162 0.9035433 0.09363831 -0.10143124