NNS Stochastic Superiority

Computes stochastic superiority between two numeric vectors as the empirical probability that an observation from x exceeds an observation from y, with optional tie adjustment and optional confidence intervals via maximum entropy bootstrap.

Usage

NNS.SS(
  x,
  y,
  confidence.interval = FALSE,
  reps = 999,
  ci = 0.95,
  rho = 1
)

Arguments

x: a numeric vector.
y: a numeric vector.
confidence.interval: logical; FALSE (default) returns only the empirical stochastic superiority measures. Set to TRUE to compute bootstrap confidence intervals for p_star.
reps: numeric; number of maximum entropy bootstrap replicates used when confidence.interval = TRUE. Default is 999.
ci: numeric in $(0, 1)$; confidence level used for the bootstrap interval when confidence.interval = TRUE. Default is 0.95.
rho: numeric; dependence target passed to NNS.meboot. Default is 1.

Value

If confidence.interval = FALSE, returns a list containing:

p_gt: empirical probability that x > y.
p_tie: empirical probability that x = y.
p_star: tie-adjusted stochastic superiority probability.

If confidence.interval = TRUE, returns a list containing:

p_gt: empirical probability that x > y.
p_tie: empirical probability that x = y.
p_star: tie-adjusted stochastic superiority probability.
lower: lower confidence bound for p_star.
upper: upper confidence bound for p_star.
ci: confidence level used.
reps: number of bootstrap replicates used.
boot_vals: bootstrap replicate values of p_star.

Details

NNS.SS returns: $$P(X > Y),$$ the tie probability $$P(X = Y),$$ and the tie-adjusted stochastic superiority measure $$P^* = P(X > Y) + \frac{1}{2} P(X = Y).$$

When confidence.interval = TRUE, confidence bounds for P^* are computed from NNS.meboot bootstrap replicates using LPM.VaR and UPM.VaR with degree = 0.

Missing values are removed from both x and y using stats::na.omit. The empirical estimates are computed via a fast sorted comparison routine rather than explicit pairwise expansion of all x-y combinations.

For continuous data, p_tie will typically be zero, so p_star and p_gt will be identical up to numerical precision. For discrete data, p_star provides the standard tie-adjusted superiority measure.

When confidence.interval = TRUE, the interval is constructed from the empirical bootstrap distribution of p_star, where $\alpha = 1 - ci$. The lower bound is obtained from LPM.VaR evaluated at $\alpha / 2$, and the upper bound is obtained from UPM.VaR evaluated at $\alpha / 2$, both with degree = 0.

Note

This function measures stochastic superiority as a pairwise exceedance probability. This is distinct from first-, second-, or third-degree stochastic dominance; see NNS.FSD, NNS.SSD, and NNS.TSD for dominance testing.

References

Vinod, H.D. and Viole, F. (2020) Arbitrary Spearman's Rank Correlations in Maximum Entropy Bootstrap and Improved Monte Carlo Simulations. doi:10.2139/ssrn.3621614
Viole, F. and Nawrocki, D. (2013) Nonlinear Nonparametric Statistics: Using Partial Moments. ISBN: 1490523995, 2nd edition: https://ovvo-financial.github.io/NNS/book/.

Author

Fred Viole, OVVO Financial Systems

Examples

if (FALSE) { # \dontrun{
set.seed(123)
x <- rnorm(200, mean = 0.4, sd = 1)
y <- rnorm(200, mean = 0.0, sd = 1)

# Empirical stochastic superiority
NNS.SS(x, y)

# With confidence intervals
NNS.SS(x, y, confidence.interval = TRUE, reps = 999, ci = 0.95)

# Discrete example with ties
x <- sample(1:5, 100, replace = TRUE)
y <- sample(1:5, 100, replace = TRUE)
NNS.SS(x, y)
} # }