RAFE-Stein-Corrected Mean — mu_rafe

Positive-part James–Stein shrinkage of a forecast mean toward a target, in the $\hat\Sigma^{-1/2}$-whitened space.

Usage

mu_rafe_stein(
  mu_hat,
  Sigma_hat,
  kappa = NULL,
  target = c("grandmean", "zero"),
  T_obs = NULL
)

Arguments

mu_hat: Forecast mean.
Sigma_hat: Forecast covariance. Must be a covariance, never a precision matrix; the function inverts it internally.
kappa: Optional shrinkage intensity in [0, 1], where 0 leaves mu_hat untouched and 1 collapses it onto the target. If NULL, the James–Stein intensity is computed from T_obs.
target: Shrinkage target. One of "grandmean" (default; shrinks toward the cross-sectional mean of mu_hat) or "zero".
T_obs: Number of observations the forecast was estimated from. Needed only when kappa is NULL.

Value

A numeric vector of corrected means (same length as mu_hat), with the applied intensity attached as attribute kappa.

Details

Writing $\hat m := \hat\Sigma^{-1/2}\hat\mu$ and $m_0 := \hat\Sigma^{-1/2}\mu_0$ for the whitened forecast and target, the James–Stein map of Salcher, Stöckl & Hanke is $$JS(z) = m_0 + \left(1 - \frac{N-2}{T\,\|z - m_0\|_2^2}\right)_{+}(z - m_0),$$ and the corrected mean is $\tilde\mu = \hat\Sigma^{1/2} JS(\hat m)$. Because $\|\hat m - m_0\|_2^2 = (\hat\mu - \mu_0)^\top \hat\Sigma^{-1} (\hat\mu - \mu_0)$, this is algebraically a linear shrinkage in the raw space whose intensity is computed in the risk metric: $\tilde\mu = (1 - \kappa)\hat\mu + \kappa\mu_0$.

Supply T_obs to use the James–Stein intensity, or kappa to fix the intensity directly (which is what sep_tune() and joint_trafe_tune() search over). Exactly one of the two is required.

References

Stöckl, S., Salcher, L., & Hanke, M. Post-Optimal Moment Correction for Mean-Variance Portfolios. Working paper.

Examples

set.seed(1)
R <- matrix(rnorm(60 * 5, mean = 0.01, sd = 0.05), 60, 5)
mu_hat    <- colMeans(R)
Sigma_hat <- stats::cov(R)

# James-Stein intensity, implied by the sample size:
mu_rafe_stein(mu_hat, Sigma_hat, T_obs = 60)
#> [1] 0.01167921 0.01167921 0.01167921 0.01167921 0.01167921
#> attr(,"kappa")
#> [1] 1

# Fixed intensity:
mu_rafe_stein(mu_hat, Sigma_hat, kappa = 0.5)
#> [1] 0.013530016 0.013634742 0.009676794 0.008699989 0.012854528
#> attr(,"kappa")
#> [1] 0.5