# R function calling the C++ routines that compute the complementary p-value for a (weighted) two-sample Kolmogorov-Smirnov (KS) test, given an arbitrary positive weight function and arbitrary data samples with possibly repeated observations (i.e. ties) Function calling directly the C++ routines that compute the exact complementary p-value character(0) for the (weighed) two-sample one- or two-sided Kolmogorov-Smirnov statistic, at a fixed character(0), character(0), given the sample sizes `m` and `n`, the vector of weights `w_vec` and the vector `M` containing the number of times each distinct observation is repeated in the pooled sample. ```r KS2sample_c_Rcpp(m, n, kind, M, q, w_vec, tol) ``` ## Arguments - `m`: the sample size of first tested sample. - `n`: the sample size of second tested sample. - `kind`: an integer value (= 1,2 or 3) which specified the alternative hypothesis. When = 1, the test is two-sided. When = 2 or 3, the test is one-sided. See Details for the meaning of the possible values. Other value is invalid. - `M`: an integer-valued vector with character(0) cells, where character(0) denotes the number of distinct values in the ordered pooled sample of tested pair of samples(i.e. character(0)). `M[i]` is the number of times that character(0) is repeated in the pooled sample. A valid `M` must have strictly positive integer values and have the sum of all cells equals to `m+n`. - `q`: numeric value between 0 and 1, at which the p-value character(0) is computed. - `w_vec`: a vector with `m+n-1` cells, giving weights to each observation in the pooled sample. Valid `w_vec` must have `m+n-1` cells and strictly positive value. See Details for the meaning of values in each cell. - `tol`: the value of character(0) for computing character(0), which is equivalent to character(0). Non-positive input (`tol` character(0)) or large input (`tol` character(0)`1e-6`) are replaced by `tol=1e-6`. In cases when `m` and `n` have large least common multiple, a smaller value is highly recommended. ## Details Given a pair of random samples character(0) and character(0) of sizes `m` and `n` with empirical cdfs character(0) and character(0) respectively, coming from some unknown cdfs character(0) and character(0). It is assumed that character(0) and character(0) could be either **continuous**, **discrete** or **mixed**, which means that repeated observations are allowed in the corresponding observed samples. The task is to test the null hypothesis character(0) for all character(0), either against the alternative hypothesis character(0) for at least one character(0), which corresponds to the two-sided test, or against character(0) and character(0) for at least one character(0), which corresponds to the two one-sided tests. The (weighted) two-sample Kolmogorov-Smirnov goodness-of-fit statistics that are used to test these hypotheses are generally defined as: $$ \Delta_{m,n} = \sup |F_{m}(t) - G_n(t)|W(E_{m+n}(t), \textnormal{ to test against the alternative } H_1: F(x)\neq G(x) $$ $$ \Delta_{m,n}^{+} = \sup [F_{m}(t) - G_n(x)]W(E_{m+n}(t)), \textnormal{ to test against the alternative } H_1: F(x)> G(x) $$ $$ \Delta_{m,n}^{-} = \sup [G_n(t) - F_{m}(x)]W(E_{m+n}(t)), \textnormal{ to test against the alternative } H_1: F(x)< G(x), $$ where character(0) is the empirical cdf of the pooled sample character(0), character(0) is a strictly positive weight function defined on character(0). `w_vec[i]` (0character(0)`i`character(0)character(0)) is then equal to character(0)(character(0) is the i-th smallest observation in the pooled sample character(0)). Different value of `w_vec` specifies the weighted Kolmogorov-Smirnov test differently. For example, when `w_vec=rep(1,m+n-1)`, `KS2sample_Rcpp` calculates the p-value of the unweighted two-sample Kolmogorov-Smirnov test, when `w_vec` = `((1:(m+n-1))*((m+n-1):1))^(-1/2)`, it calculates the p-value for the weighted two-sample Kolmogorov-Smirnov test with Anderson-Darling weight character(0). Possible values of `kind` are 1,2 and 3, which specify the alternative hypothesis, i.e. specify the test statistic to be either character(0), character(0) or character(0) respectively. The numeric array `M` specifies the number of **repeated observations** in the pooled sample. For a particular realization of the pooled sample character(0), let there be character(0) distinct values, character(0), in the ordered, pooled sample character(0), where character(0), and where character(0)=`M[i]` is the number of times character(0), character(0) appears in the pooled sample. The calculated complementary p-value is the conditional probability: $$ P(D_{m,n}< q) $$ where character(0) is the two-sample Kolmogorov-Smirnov test statistic defined according to the value of `weight` and `alternative`, for two samples character(0) and character(0) of sizes character(0) and character(0), **randomly drawn from the pooled sample without replacement**, i.e. character(0) is defined on the space character(0) (see further details in `KS2sample`), and character(0). `KS2sample_c_Rcpp` implements an exact algorithm, extending the Fortran 77 subroutine due to Nikiforov (1994), an extended functionality by allowing more flexible choice of weight, as well as for **large sample sizes**. This leads to faster computation time, as well as, relatively high accuracy for very large `m` and `n` (less accurate than `KS2sample_Rcpp`). Compared with other known algorithms, it allows data samples come from **continuous, discrete or mixed distribution**(i.e. ties may appear), and it is more efficient and more generally applicable for **large sample sizes**. This algorithm ensures a total worst-case run-time of order character(0). ## Returns Numeric value corresponding to character(0), given sample sizes `m`, `n`, `M` and `w_vec`. If the value of `m`, `n` are non-positive, or if the length of `w_vec` is not equal to `m+n-1`, then the function returns `-1`, the non-permitted value of `M` or non-permitted value inside `w_vec` returns `-2`, numerically unstable calculation returns `-3`. ## Source Based on the Fortran subroutine by Nikiforov (1994). See also Dimitrova, Jia, Kaishev (2024). ## References Paul L. Canner (1975). "A Simulation Study of One- and Two-Sample Kolmogorov-Smirnov Statistics with a Particular Weight Function". Journal of the American Statistical Association, 70 (349), 209-211. Nikiforov, A. M. (1994). "Algorithm AS 288: Exact Smirnov Two-Sample Tests for Arbitrary Distributions." Journal of the Royal Statistical Society. Series C (Applied Statistics), 43 (1), 265–270. Dimitrina S. Dimitrova, Yun Jia, Vladimir K. Kaishev (2024). "The R functions KS2sample and Kuiper2sample: Efficient Exact Calculation of P-values of the Two-sample Kolmogorov-Smirnov and Kuiper Tests". **submitted** ## Examples ```r ## Computing the unweighted two-sample Kolmogorov-Smirnov test ## Example see in Nikiforov (1994) m <- 120 n <- 150 kind <- 1 q <- 0.1 M <- c(80,70,40,80) w_vec <- rep(1,m+n-1) tol <- 1e-6 KS2sample_c_Rcpp(m, n, kind, M, q, w_vec, tol) kind <- 2 KS2sample_c_Rcpp(m, n, kind, M, q, w_vec, tol) ## Computing the weighted two-sample Kolmogorov-Smirnov test ## with Anderson-Darling weight kind <- 3 w_vec <- ((1:(m+n-1))*((m+n-1):1))^(-1/2) KS2sample_c_Rcpp(m, n, kind, M, q, w_vec, tol) ```

KS2sample_c_Rcpp function