mixed_ks_c_cdf {KSgeneral}

Computes the complementary cumulative distribution function of the two-sided Kolmogorov-Smirnov statistic when the cdf under the null hypothesis is mixed

Description

Computes the complementary cdf, P(D_{n} \ge q) at a fixed q, q\in[0, 1], of the one-sample two-sided Kolmogorov-Smirnov statistic, when the cdf F(x) under the null hypothesis is mixed, using the Exact-KS-FFT method expressing the p-value as a double-boundary non-crossing probability for a homogeneous Poisson process, which is then efficiently computed using FFT (see Dimitrova, Kaishev, Tan (2020)).

Usage

mixed_ks_c_cdf(q, n, jump_points, Mixed_dist, ..., tol = 1e-10)

Arguments

q

numeric value between 0 and 1, at which the complementary cdf P(D_{n} \ge q) is computed

n

the sample size

jump_points

a numeric vector containing the points of (jump) discontinuity, i.e. where the underlying cdf F(x) has jump(s)

Mixed_dist

a pre-specified (user-defined) mixed cdf, F(x), under the null hypothesis.

...

values of the parameters of the cdf, F(x) specified (as a character string) by Mixed_dist.

tol

the value of \epsilon that is used to compute the values of A_{i} and B_{i}, i = 1, ..., n, as detailed in Step 1 of Section 2.1 in Dimitrova, Kaishev and Tan (2020) (see also (ii) in the Procedure Exact-KS-FFT therein). By default, tol = 1e-10. Note that a value of NA or 0 will lead to an error!

Details

Given a random sample \{X_{1}, ..., X_{n}\} of size n with an empirical cdf F_{n}(x), the Kolmogorov-Smirnov goodness-of-fit statistic is defined as D_{n} = \sup | F_{n}(x) - F(x) | , where F(x) is the cdf of a prespecified theoretical distribution under the null hypothesis H_{0}, that \{X_{1}, ..., X_{n}\} comes from F(x).

The function mixed_ks_c_cdf implements the Exact-KS-FFT method, proposed by Dimitrova, Kaishev, Tan (2020) to compute the complementary cdf P(D_{n} \ge q) at a value q, when F(x) is mixed. This algorithm ensures a total worst-case run-time of order O(n^{2}log(n)).

We have not been able to identify alternative, fast and accurate, method (software) that has been developed/implemented when the hypothesized F(x) is mixed.

Value

Numeric value corresponding to P(D_{n} \ge q).

References

Dimitrina S. Dimitrova, Vladimir K. Kaishev, Senren Tan. (2020) "Computing the Kolmogorov-Smirnov Distribution When the Underlying CDF is Purely Discrete, Mixed or Continuous". Journal of Statistical Software, 95(10): 1-42. doi:10.18637/jss.v095.i10.

Examples

# Compute the complementary cdf of D_{n}
# when the underlying distribution is a mixed distribution
# with two jumps at 0 and log(2.5),
# as in Example 3.1 of Dimitrova, Kaishev, Tan (2020)

## Defining the mixed distribution

Mixed_cdf_example <- function(x)
{
     result <- 0
     if (x < 0){
         result <- 0
     }
     else if (x == 0){
         result <- 0.5
     }
     else if (x < log(2.5)){
         result <- 1 - 0.5 * exp(-x)
     }
     else{
         result <- 1
     }

     return (result)
 }

KSgeneral::mixed_ks_c_cdf(0.1, 25, c(0, log(2.5)), Mixed_cdf_example)


## Not run: 
## Compute P(D_{n} >= q) for n = 5,
## q = 1/5000, 2/5000, ..., 5000/5000
## when the underlying distribution is a mixed distribution
## with four jumps at 0, 0.2, 0.8, 1.0,
## as in Example 2.8 of Dimitrova, Kaishev, Tan (2020)

n <- 5
q <- 1:5000/5000

Mixed_cdf_example <- function(x)
{
  result <- 0
  if (x < 0){
    result <- 0
  }
  else if (x == 0){
    result <- 0.2
  }
  else if (x < 0.2){
    result <- 0.2 + x
  }
  else if (x < 0.8){
    result <- 0.5
  }
  else if (x < 1){
    result <- x - 0.1
  }
  else{
    result <- 1
  }

  return (result)
}

plot(q, sapply(q, function(x) KSgeneral::mixed_ks_c_cdf(x, n,
     c(0, 0.2, 0.8, 1.0), Mixed_cdf_example)), type='l')


## End(Not run)