KSgeneral (version 1.1.1)

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

Description

Computes the complementary cdf \(P(D_{n} \ge q) \equiv P(D_{n} > q)\) at a fixed \(q\), \(q\in[0, 1]\), for the one-sample two-sided Kolmogorov-Smirnov statistic, \(D_{n}\), for a given sample size \(n\), when the cdf \(F(x)\) under the null hypothesis is continuous.

Usage

cont_ks_c_cdf(q, n)

Arguments

q

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

n

the sample size

Value

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

Details

Given a random sample \(\{X_{1}, ..., X_{n}\}\) of size n with an empirical cdf \(F_{n}(x)\), the two-sided 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 cont_ks_c_cdf implements the FFT-based algorithm proposed by Moscovich and Nadler (2017) to compute the complementary cdf, \(P(D_{n} \ge q)\) at a value \(q\), when \(F(x)\) is continuous. This algorithm ensures a total worst-case run-time of order \(O(n^{2}log(n))\) which makes it more efficient and numerically stable than the algorithm proposed by Marsaglia et al. (2003). The latter is used by many existing packages computing the cdf of \(D_{n}\), e.g., the function ks.test in the package stats and the function ks.test in the package dgof. More precisely, in these packages, the exact p-value, \(P(D_{n} \ge q)\) is computed only in the case when \(q = d_{n}\), where \(d_{n}\) is the value of the KS test statistic computed based on a user provided sample \( \{x_{1}, ..., x_{n} \} \). Another limitation of the functions ks.test is that the sample size should be less than 100, and the computation time is \(O(n^{3})\). In contrast, the function cont_ks_c_cdf provides results with at least 10 correct digits after the decimal point for sample sizes \(n\) up to 100000 and computation time of 16 seconds on a machine with an 2.5GHz Intel Core i5 processor with 4GB RAM, running MacOS X Yosemite. For n > 100000, accurate results can still be computed with similar accuracy, but at a higher computation time. See Dimitrova, Kaishev, Tan (2020), Appendix C for further details and examples.

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.

Marsaglia G., Tsang WW., Wang J. (2003). "Evaluating Kolmogorov's Distribution". Journal of Statistical Software, 8(18), 1-4.

Moscovich A., Nadler B. (2017). "Fast Calculation of Boundary Crossing Probabilities for Poisson Processes". Statistics and Probability Letters, 123, 177-182.

Examples

# NOT RUN {
## Compute the value for P(D_{100} >= 0.05)

KSgeneral::cont_ks_c_cdf(0.05, 100)


## Compute P(D_{n} >= q)
## for n = 100, q = 1/500, 2/500, ..., 500/500
## and then plot the corresponding values against q

n <- 100
q <- 1:500/500
plot(q, sapply(q, function(x) KSgeneral::cont_ks_c_cdf(x, n)), type='l')

## Compute P(D_{n} >= q) for n = 141, nq^{2} = 2.1 as shown
## in Table 18 of Dimitrova, Kaishev, Tan (2020)

KSgeneral::cont_ks_c_cdf(sqrt(2.1/141), 141)

# }