Quantiles of the Kolmogorov distribution

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Numbers

$p$

$\kappa_p$

1/200:

0.4170896319324850867905501134898356351263428701323292368475594033482161657864024836196763238068937417

1/100:

0.4410276985179293684219030143755841048900419285820892798291537637848956368732150219072779862858093404

1/40:

0.4806349324173608645324561804098553822330685284775644930727627545719316793125235149596417634717764585

1/20:

0.5196103791686225390998625781948264958366503793254604646139524807922427752393812065336404737663171858

1/10:

0.5711732651063401631889351384398059817148678480284728030235355554604416646321997534464062436155701831

1/4:

0.6764476915028200829972356030586177409014383743383662966750912699038189170880504697352991195339570036

1/2:

0.8275735551899076901138270828889768075843727832232029452002281054669822747682290224084911530683945353

3/4:

1.019184720253685836068625002918171811407390553076526621883722292413331907446629736257112854140923781

9/10:

1.223847870217082388278217372606174346655027540577328363121881530427405122098915947366647575965697021

19/20:

1.358098639322550604285013101166668089106032828500128014891996683507475082635893711388711194536899969

39/40:

1.480206857424922658790194136332041330012352838424966200027089272760558478979639139967994758260332393

99/100:

1.627623611518950346526002560706360796410198983962770088946293644092396076427703365804815961881217672

199/200:

1.730818380345404861621169426004570017886738918004768566057535531165511238798676049587849871222204636

999/1000:

1.949474603504375273324313494108617533285058140848713669418727644979931407141835163599267784019875271

Definition

For the Kolmogorov distribution [2], with cumulative distribution function $F(x)=1-2\sum_{j=1}^{\infty}(-1)^{j-1}e^{-2j^2x^2}$ for $x>0$ and $F(x)=0$ for $x\leq0$, this table gives the lower-tail quantile $\kappa_p=\inf\{x:F(x)\geq p\}$.

Parameters

$p$

— lower-tail probability ($0<p<1$)

Formulas

(1)

For $x>0$, the equivalent positive series is $F(x)=\frac{\sqrt{2\pi}}{x}\sum_{j=1}^{\infty}\exp\!\left(-\frac{(2j-1)^2\pi^2}{8x^2}\right)$ [1].

(2)

$F(\kappa_p)=p$, and $\kappa_p<\kappa_q$ whenever $0<p<q<1$.

(3)

$1-F(\kappa_{1-\alpha})=\alpha$ for $0<\alpha<1$.

(4)

For the two-sided statistic $D_n$ defined in the comments, $\lim_{n\to\infty}\Pr\{\sqrt n D_n>\kappa_{1-\alpha}\}=\alpha$. Thus the asymptotic level-$\alpha$ rejection threshold for $D_n$ is $\kappa_{1-\alpha}/\sqrt n$ [2].

Comments

(5)

The table stores lower-tail quantiles. If $Q^{\mathrm{upper}}(\alpha)$ denotes an upper-tail critical value, then $\kappa_p=Q^{\mathrm{upper}}(1-p)$. For example, $p=19/20$ is the probability $0.95$, so its entry is the asymptotic critical value at significance level $0.05$.

(6)

Let $G_n$ be the empirical cumulative distribution function of $n$ independent observations from a fixed continuous distribution with cumulative distribution function $G$, and let $D_n=\sup_t|G_n(t)-G(t)|$. The distribution of $\sqrt{n}D_n$ tends to the Kolmogorov distribution as $n$ tends to infinity [2]. The entries are quantiles of this limiting distribution; finite-sample critical values, and tests with parameters estimated from the observations, require their corresponding sampling distributions.

(7)

The statistic $D_n$ already measures deviations in both directions. The asymptotic level-$\alpha$ critical value for $\sqrt nD_n$ is $\kappa_{1-\alpha}$, with no division of $\alpha$ by two. The distinction from the one-sided statistic is described in [1].

(8)

The probabilities are exact rationals in lowest terms. The selection includes the median, the quartiles, conventional significance levels and their complements, together with the upper $0.1\%$ critical value.

(9)

The symbol $\kappa_p$ distinguishes these quantiles from Khinchin's means $K_p$, which concern continued fractions.

Programs

(P1)

Python

import mpmath as mp
mp.mp.dps = 130
p = mp.mpf(19) / 20
mp.findroot(lambda x: mp.jtheta(4, 0, mp.exp(-2*x*x)) - p,
            (mp.mpf('0.25'), mp.mpf(4)), solver='bisect',
            maxsteps=600, tol=mp.mpf('1e-120'))

(P2)

Python

# Machine precision, using the lower-tail probability.
from scipy.stats import kstwobign
kstwobign.ppf(0.95)

References

[1]

Paul van Mulbregt, Computing the Cumulative Distribution Function and Quantiles of the limit of the Two-sided Kolmogorov-Smirnov Statistic (2018), equations (3) and (16). (arXiv)

Links

[2]

SciPy: limiting distribution of the scaled two-sided Kolmogorov-Smirnov statistic

Similar tables

Values of the Jacobi theta constant $\theta_4(0,q)$ — For $x>0$, the Kolmogorov cumulative distribution function is $F(x)=\theta_4(0,e^{-2x^2})$.

Quantiles of the standard normal distribution — uses the same lower-tail probability convention for normal critical values

Quantiles of the chi-squared distribution — gives critical values for chi-squared tests using the same probability convention

Quantiles of Student's $t$-distribution — gives Student critical values at the same selected probabilities

Quantiles of the $F$-distribution — gives central F critical values using the same lower-tail convention

Differential entropies of continuous probability distributions — gives another numerical characteristic of standard continuous distributions

Kullback-Leibler divergences between probability distributions — gives divergences between distributions, while these entries locate probability thresholds within a distribution

Data properties

Entries are of type: real number

How well the digits are known: proven

How they were obtained:

The generator bisects exact rational brackets with 461-bit Arb arithmetic, increasing the precision if a sign is undecidable. Every cumulative probability includes the alternating-series remainder bound $2\exp(-2(N+1)^2x^2)$ after $N$ terms.

The returned ball encloses the whole final bracket, whose width is less than $10^{-112}$; every returned ball retained at least 110 decimal digits before rounding to 100. Independently, 560-bit MPFI interval arithmetic evaluates the positive series in (1) with a geometric bound on its omitted terms. It certifies opposite signs of $F(x)-p$ at the two endpoints represented by every stored decimal. The checks also test strict monotonicity, the upper-tail conversion, a special-value control, and rejection of shifted intervals. All 14 values agree with mpmath at the stored precision and with SciPy at machine precision.

Table is complete: no (it holds the 14 probabilities $p\in\{1/200,1/100,1/40,1/20,1/10,1/4,1/2,3/4,9/10,19/20,39/40,99/100,199/200,999/1000\}$, selected for the median, quartiles, conventional significance levels and their complements, and the upper $0.1\%$ critical value)

parameter selection: The median uses $p=1/2$ and the quartiles use $p=1/4,3/4$. Upper-tail significance levels $\alpha\in\{1/10,1/20,1/40,1/100,1/200\}$ use $p=1-\alpha$; their complementary probabilities supply the corresponding lower-tail thresholds. The additional $p=999/1000$ gives the upper $0.1\%$ critical value. This is the shared selection of conventional probability levels used for classical statistical quantiles. (Unknown key)