% Generated by roxygen2: do not edit by hand
% Please edit documentation in R/confidence_sequences.R
\name{cs_asymptotic}
\alias{cs_asymptotic}
\title{Asymptotic confidence sequence (Appendix C, Eq. 55, Choe & Ramdas 2023)}
\usage{
cs_asymptotic(scores1, scores2, alpha = 0.05, t_star = NULL)
}
\arguments{
\item{scores1}{Numeric vector. Scores for forecaster 1.}

\item{scores2}{Numeric vector. Scores for forecaster 2.}

\item{alpha}{Numeric in (0,1). Significance level. Default: 0.05.}

\item{t_star}{Numeric > 0. Sample size at which CS is tightest.
Default: length(scores1) (tightest at end of sample).}
}
\value{
data.frame with columns t, estimate, lower, upper.
}
\description{
\strong{Asymptotic, not finite-sample}: coverage \verb{>= 1 - alpha} holds only as
\code{t -> infinity}. Valid without requiring bounded score differences —
requires only that \code{hat_delta_t} has finite variance — so it's appropriate
for tick loss and other scoring rules where hard bounds depend on
unbounded realised values. Suitable for large evaluation windows.
}
\details{
\deqn{C_t^A = \hat\Delta_t \pm \sqrt{
  \frac{2(t \sigma^2_t \rho^2 + 1)}{t^2 \rho^2}
  \log\frac{\sqrt{t \sigma^2_t \rho^2 + 1}}{\alpha}}}
where \eqn{\sigma^2_t = \frac{1}{t}\sum_{i=1}^t (\hat\delta_i - \hat\Delta_{i-1})^2}
and \code{rho} is tuned to be tightest at \code{t_star}:
\deqn{\rho(t_{star}) = \sqrt{\frac{2\log(1/\alpha) + \log(1 + 2\log(1/\alpha))}{t_{star}}}}

The running variance estimator uses the predictable mean \verb{hat_Delta_\{t-1\}}
(not the current mean \code{hat_Delta_t}) to maintain predictability, with
\code{hat_Delta_0 := 0}.
}
\examples{
scores1 <- c(-0.4, -0.2, -0.3, -0.1, -0.2)
scores2 <- c(-0.5, -0.3, -0.4, -0.2, -0.3)
cs_asymptotic(scores1, scores2, alpha = 0.05)

}
