Package jline.api.sim

Class Sim_sts_quantile_areas

java.lang.Object
jline.api.sim.Sim_sts_quantile_areas
public final class Sim_sts_quantile_areas extends Object
Standardized time series areas of the batched quantile process.

Splits b*m observations into b nonoverlapping batches of size m and returns the signed standardized time series areas of the quantile-estimation process, the batched quantile estimators, and the three variance-parameter estimators Sim_fquest and Sim_firquest build intervals from.

With yhat_p(j,m) the empirical p-quantile of batch j and yhat_p(j,k) that of its first k observations, the STS process of batch j is 

   T_{j,m}(k/m) = (k/sqrt(m)) (yhat_p(j,m) - yhat_p(j,k)),

its signed area is A_p(w;j,m) = m^-1 sum_k w(k/m) T_{j,m}(k/m), and the three estimators of sigma_p^2 = lim n Var(ytilde_p(n)) are 

   A_p(w;b,m) = b^-1 sum_j A_p(w;j,m)^2                        (STS area)
   N_p(b,m)   = (b-1)^-1 m sum_j (yhat_p(j,m)-ytilde_p(n))^2   (NBQ)
   V_p(w;b,m) = [b A_p(w;b,m) + (b-1) N_p(b,m)] / (2b-1)       (combined)

with ytilde_p(n) the full-sample empirical p-quantile. The first two have limiting chi-square laws on b and b-1 degrees of freedom and are asymptotically independent, so the combined estimator carries 2b-1 degrees of freedom and is about sqrt(2) less variable than either component.

The requirement on a weight function w is that int_0^1 w(t)B(t)dt be standard normal for a standard Brownian bridge B; for a constant w = c that variance is c^2/12, so sqrt(12) is the normalizing choice and the default.

The prefix quantiles are exact order statistics, obtained from a Fenwick tree over the within-batch ranks, so the cost is O(b m log m). On i.i.d. Exp(1) data, where sigma_p^2 = p(1-p)/f(y_p)^2 is exact, both A_p and N_p are unbiased to within 7%.

Port of MATLAB sim_sts_quantile_areas.m.

Reference: C. Alexopoulos, D. Goldsman, A. Lolos, K. D. Dingec, J. R. Wilson, "Steady-State Quantile Estimation Using Standardized Time Series", 2020/2023; A. Lolos et al., Proc. Winter Simulation Conference, 2023, theorems 1 to 3.

Since:
LINE 3.1.0

  • Field Details

    • DEFAULT_WEIGHT

      public static final double DEFAULT_WEIGHT
      Constant STS weight function that normalizes the Brownian bridge area.

  • Method Details

    • sim_sts_quantile_areas

      public static StsQuantileStats sim_sts_quantile_areas(double[] y, int b, int m, double p)
      Computes the STS statistics with the default weight function.
      Parameters:
      y - exactly b*m observations, in time order
      b - batch count, at least 1
      m - batch size, at least 1
      p - quantile probability in (0,1)
      Returns:
      the statistics

    • sim_sts_quantile_areas

      public static StsQuantileStats sim_sts_quantile_areas(double[] y, int b, int m, double p, double weight)
      Computes the STS statistics.
      Parameters:
      y - exactly b*m observations, in time order
      b - batch count, at least 1
      m - batch size, at least 1
      p - quantile probability in (0,1)
      weight - constant STS weight function, nonzero
      Returns:
      the statistics