# Computes the complementary cumulative distribution function of the two-sided Kolmogorov-Smirnov statistic when the cdf under the null hypothesis is mixed 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)). ```r 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. ## Returns 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 ```r # 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) ```

mixed_ks_c_cdf function