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Fixed-lambda e-process with predictable bounds (Proposition 7)

Source: R/predictable_bounds.R
eprocess_predictable.Rd

Constructs a valid e-process when score difference bounds vary over time but are predictable (known at time i-1 before observing hat_delta_i).

Usage

eprocess_predictable(
  scores1,
  scores2,
  c_seq,
  lambda = NULL,
  alpha = 0.05,
  gammas = NULL,
  clip_max = 1e+07,
  strict = FALSE
)

Arguments

scores1

Numeric vector. Scores for forecaster 1.

scores2

Numeric vector. Scores for forecaster 2.

c_seq

Numeric vector. Predictable bound sequence (c_i), same length as scores1. Must satisfy |scores1[i]-scores2[i]| <= c_i/2 and c_i > 0 for all i.

lambda

Numeric in [0, 1/c_0). Betting parameter. Must be strictly less than 1/c_0 where c_0 = max(c_seq). If NULL, uses the recommended default lambda = 0.5/c_0.

alpha

Numeric in (0,1). Significance level for rejection rule. Default: 0.05. Not used in computation, only for API consistency. Pass the same value to predictable_rejections() when evaluating rejection.

gammas

Numeric vector or NULL. Predictable centering sequence. If NULL, constructed as lagged running mean.

clip_max

Numeric. Maximum e-process value. Default: 1e7.

strict

Logical. If TRUE, any violation of the bound condition at any time point will stop execution with an error. If FALSE (default), a warning is issued but the e-process is still computed. Note that violations invalidate the e-process guarantee, so strict = TRUE is recommended for formal inference.

Value

data.frame with columns: t, e_pq, e_qp, log_e_pq, log_e_qp, c_seq, lambda_used

Details

The e-process is computed as: $$\log E_t(\lambda) = \sum_{i=1}^t \Bigl[\lambda\,\hat{\delta}_i - \psi_{E,c_i}(\lambda)\,(\hat{\delta}_i - \gamma_i)^2\Bigr]$$

where $$\psi_{E,c}(\lambda) = \frac{-\log(1 - c\lambda) - c\lambda}{c^2}$$ is evaluated at each step with the current \(c_i\).

LAMBDA CHOICE: lambda = 0.5/c_0 is a conservative default that stays well within the valid domain [0, 1/c_0). For better power, lambda can be tuned to the expected signal size, but must never reach 1/c_0.

VALIDITY CHECK: The function verifies |hat_delta_i| <= c_i/2 at each step and warns if violated. Violations invalidate the e-process guarantee.

Predictability

The bound sequence c_seq (and the centering sequence gammas) must be predictable: c_i is fixed and known at time i - 1, before scores1[i]/scores2[i] (and hence hat_delta_i) are observed — formally, \(c_i\) is \(\mathcal{F}_{i-1}\)-measurable. A bound chosen after seeing hat_delta_i (e.g. derived from the realised data range) invalidates the e-process guarantee, even if it numerically satisfies |hat_delta_i| <= c_i/2.

Examples

scores1 <- c(0.10, 0.20, 0.15, 0.25)
scores2 <- c(0.05, 0.10, 0.10, 0.20)
c_seq <- rep(1, length(scores1))
ep <- eprocess_predictable(scores1, scores2, c_seq = c_seq)
head(ep)
#>   t     e_pq      e_qp   log_e_pq    log_e_qp c_seq lambda_used
#> 1 1 1.024820 0.9748391 0.02451713 -0.02548287     1         0.5
#> 2 2 1.076844 0.9268480 0.07403426 -0.07596574     1         0.5
#> 3 3 1.103971 0.9038549 0.09891355 -0.10108645     1         0.5
#> 4 4 1.131857 0.8814913 0.12385990 -0.12614010     1         0.5