NNS.SS: NNS Stochastic Superiority

In NNS: Nonlinear Nonparametric Statistics

NNS Stochastic Superiority

Description

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.

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.

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.

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.

Author(s)

Fred Viole, OVVO Financial Systems

References

• Vinod, H.D. and Viole, F. (2020) Arbitrary Spearman's Rank Correlations in Maximum Entropy Bootstrap and Improved Monte Carlo Simulations. \Sexpr[results=rd]{tools:::Rd_expr_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/.

Examples

## Not run: 
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)

## End(Not run)

NNS documentation built on April 10, 2026, 9:10 a.m.