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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_nonpar_test_ks_1sample_user (g08cc)

## Purpose

nag_nonpar_test_ks_1sample_user (g08cc) performs the one sample Kolmogorov–Smirnov distribution test, using a user-specified distribution.

## Syntax

[d, z, p, sx, ifail] = g08cc(x, cdf, ntype, 'n', n)
[d, z, p, sx, ifail] = nag_nonpar_test_ks_1sample_user(x, cdf, ntype, 'n', n)

## Description

The data consists of a single sample of n$n$ observations, denoted by x1,x2,,xn${x}_{1},{x}_{2},\dots ,{x}_{n}$. Let Sn(x(i))${S}_{n}\left({x}_{\left(i\right)}\right)$ and F0(x(i))${F}_{0}\left({x}_{\left(i\right)}\right)$ represent the sample cumulative distribution function and the theoretical (null) cumulative distribution function respectively at the point x(i)${x}_{\left(i\right)}$, where x(i)${x}_{\left(i\right)}$ is the i$i$th smallest sample observation.
The Kolmogorov–Smirnov test provides a test of the null hypothesis H0${H}_{0}$: the data are a random sample of observations from a theoretical distribution specified by you (in cdf) against one of the following alternative hypotheses.
 (i) H1${H}_{1}$: the data cannot be considered to be a random sample from the specified null distribution. (ii) H2${H}_{2}$: the data arise from a distribution which dominates the specified null distribution. In practical terms, this would be demonstrated if the values of the sample cumulative distribution function Sn(x)${S}_{n}\left(x\right)$ tended to exceed the corresponding values of the theoretical cumulative distribution function F0(x)${F}_{0\left(x\right)}$. (iii) H3${H}_{3}$: the data arise from a distribution which is dominated by the specified null distribution. In practical terms, this would be demonstrated if the values of the theoretical cumulative distribution function F0(x)${F}_{0}\left(x\right)$ tended to exceed the corresponding values of the sample cumulative distribution function Sn(x)${S}_{n}\left(x\right)$.
One of the following test statistics is computed depending on the particular alternative hypothesis specified (see the description of the parameter ntype in Section [Parameters]).
For the alternative hypothesis H1${H}_{1}$:
• Dn${D}_{n}$ – the largest absolute deviation between the sample cumulative distribution function and the theoretical cumulative distribution function. Formally Dn = max {Dn + ,Dn}${D}_{n}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left\{{D}_{n}^{+},{D}_{n}^{-}\right\}$.
For the alternative hypothesis H2${H}_{2}$:
• Dn + ${D}_{n}^{+}$ – the largest positive deviation between the sample cumulative distribution function and the theoretical cumulative distribution function. Formally Dn + = max {Sn(x(i))F0(x(i)),0}${D}_{n}^{+}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left\{{S}_{n}\left({x}_{\left(i\right)}\right)-{F}_{0}\left({x}_{\left(i\right)}\right),0\right\}$.
For the alternative hypothesis H3${H}_{3}$:
• Dn${D}_{n}^{-}$ – the largest positive deviation between the theoretical cumulative distribution function and the sample cumulative distribution function. Formally Dn = max {F0(x(i))Sn(x(i1)),0}${D}_{n}^{-}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left\{{F}_{0}\left({x}_{\left(i\right)}\right)-{S}_{n}\left({x}_{\left(i-1\right)}\right),0\right\}$. This is only true for continuous distributions. See Section [Further Comments] for comments on discrete distributions.
The standardized statistic, Z = D × sqrt(n)$Z=D×\sqrt{n}$, is also computed, where D$D$ may be Dn,Dn + ${D}_{n},{D}_{n}^{+}$ or Dn${D}_{n}^{-}$ depending on the choice of the alternative hypothesis. This is the standardized value of D$D$ with no continuity correction applied and the distribution of Z$Z$ converges asymptotically to a limiting distribution, first derived by Kolmogorov (1933), and then tabulated by Smirnov (1948). The asymptotic distributions for the one-sided statistics were obtained by Smirnov (1933).
The probability, under the null hypothesis, of obtaining a value of the test statistic as extreme as that observed, is computed. If n100$n\le 100$, an exact method given by Conover (1980) is used. Note that the method used is only exact for continuous theoretical distributions and does not include Conover's modification for discrete distributions. This method computes the one-sided probabilities. The two-sided probabilities are estimated by doubling the one-sided probability. This is a good estimate for small p$p$, that is p0.10$p\le 0.10$, but it becomes very poor for larger p$p$. If n > 100$n>100$ then p$p$ is computed using the Kolmogorov–Smirnov limiting distributions; see Feller (1948), Kendall and Stuart (1973), Kolmogorov (1933), Smirnov (1933) and Smirnov (1948).

## References

Conover W J (1980) Practical Nonparametric Statistics Wiley
Feller W (1948) On the Kolmogorov–Smirnov limit theorems for empirical distributions Ann. Math. Statist. 19 179–181
Kendall M G and Stuart A (1973) The Advanced Theory of Statistics (Volume 2) (3rd Edition) Griffin
Kolmogorov A N (1933) Sulla determinazione empirica di una legge di distribuzione Giornale dell' Istituto Italiano degli Attuari 4 83–91
Siegel S (1956) Non-parametric Statistics for the Behavioral Sciences McGraw–Hill
Smirnov N (1933) Estimate of deviation between empirical distribution functions in two independent samples Bull. Moscow Univ. 2(2) 3–16
Smirnov N (1948) Table for estimating the goodness of fit of empirical distributions Ann. Math. Statist. 19 279–281

## Parameters

### Compulsory Input Parameters

1:     x(n) – double array
n, the dimension of the array, must satisfy the constraint n1${\mathbf{n}}\ge 1$.
The sample observations, x1,x2,,xn${x}_{1},{x}_{2},\dots ,{x}_{n}$.
2:     cdf – function handle or string containing name of m-file
cdf must return the value of the theoretical (null) cumulative distribution function for a given value of its argument.
[result] = cdf(x)

Input Parameters

1:     x – double scalar
The argument for which cdf must be evaluated.

Output Parameters

1:     result – double scalar
The result of the function.
Constraint: cdf${\mathbf{cdf}}$ must always return a value in the range [0.0,1.0]$\left[0.0,1.0\right]$ and cdf must always satify the condition that cdf(x1)cdf(x2)${\mathbf{cdf}}\left({x}_{1}\right)\le {\mathbf{cdf}}\left({x}_{2}\right)$ for any x1x2${x}_{1}\le {x}_{2}$.
3:     ntype – int64int32nag_int scalar
The statistic to be calculated, i.e., the choice of alternative hypothesis.
ntype = 1${\mathbf{ntype}}=1$
Computes Dn${D}_{n}$, to test H0${H}_{0}$ against H1${H}_{1}$.
ntype = 2${\mathbf{ntype}}=2$
Computes Dn + ${D}_{n}^{+}$, to test H0${H}_{0}$ against H2${H}_{2}$.
ntype = 3${\mathbf{ntype}}=3$
Computes Dn${D}_{n}^{-}$, to test H0${H}_{0}$ against H3${H}_{3}$.
Constraint: ntype = 1${\mathbf{ntype}}=1$, 2$2$ or 3$3$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the array x.
n$n$, the number of observations in the sample.
Constraint: n1${\mathbf{n}}\ge 1$.

None.

### Output Parameters

1:     d – double scalar
The Kolmogorov–Smirnov test statistic ( Dn ${D}_{n}$, Dn + ${D}_{n}^{+}$ or Dn ${D}_{n}^{-}$ according to the value of ntype).
2:     z – double scalar
A standardized value, Z$Z$, of the test statistic, D$D$, without the continuity correction applied.
3:     p – double scalar
The probability, p$p$, associated with the observed value of D$D$, where D$D$ may Dn${D}_{n}$, Dn + ${D}_{n}^{+}$ or Dn${D}_{n}^{-}$ depending on the value of ntype (see Section [Description]).
4:     sx(n) – double array
The sample observations, x1,x2,,xn${x}_{1},{x}_{2},\dots ,{x}_{n}$, sorted in ascending order.
5:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
 On entry, n < 1${\mathbf{n}}<1$.
ifail = 2${\mathbf{ifail}}=2$
 On entry, ntype ≠ 1${\mathbf{ntype}}\ne 1$, 2$2$ or 3$3$.
ifail = 3${\mathbf{ifail}}=3$
The supplied theoretical cumulative distribution function returns a value less than 0.0$0.0$ or greater than 1.0$1.0$, thereby violating the definition of the cumulative distribution function.
ifail = 4${\mathbf{ifail}}=4$
The supplied theoretical cumulative distribution function is not a nondecreasing function thereby violating the definition of a cumulative distribution function, that is F0(x) > F0(y)${F}_{0}\left(x\right)>{F}_{0}\left(y\right)$ for some x < y$x.

## Accuracy

For most cases the approximation for p$p$ given when n > 100$n>100$ has a relative error of less than 0.01$0.01$. The two-sided probability is approximated by doubling the one-sided probability. This is only good for small p$p$, that is p < 0.10$p<0.10$, but very poor for large p$p$. The error is always on the conservative side.

The time taken by nag_nonpar_test_ks_1sample_user (g08cc) increases with n$n$ until n > 100$n>100$ at which point it drops and then increases slowly.
For a discrete theoretical cumulative distribution function F0(x)${F}_{0}\left(x\right)$, Dn = max {F0(x(i))Sn(x(i)),0}${D}_{n}^{-}=\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left\{{F}_{0}\left({x}_{\left(i\right)}\right)-{S}_{n}\left({x}_{\left(i\right)}\right),0\right\}$. Thus if you wish to provide a discrete distribution function the following adjustment needs to be made,
• for Dn + ${D}_{n}^{+}$, return F(x)$F\left(x\right)$ as x$x$ as usual;
• for Dn${D}_{n}^{-}$, return F(xd)$F\left(x-d\right)$ at x$x$ where d$d$ is the discrete jump in the distribution. For example d = 1$d=1$ for the Poisson or binomial distributions.

## Example

```function nag_nonpar_test_ks_1sample_user_example
x = [0.01;
0.3;
0.2;
0.9;
1.2;
0.09;
1.3;
0.18;
0.9;
0.48;
1.98;
0.03;
0.5;
0.07;
0.7;
0.6;
0.95;
1;
0.31;
1.45;
1.04;
1.25;
0.15;
0.75;
0.85;
0.22;
1.56;
0.81;
0.57;
0.55];
ntype = int64(1);
[d, z, p, sx, ifail] = nag_nonpar_test_ks_1sample_user(x, @cdf, ntype)

function [result] = cdf(x)

if x < 0
result = 0;
elseif x > 2
result = 1;
else
result = x/2;
end
```
```

d =

0.2800

z =

1.5336

p =

0.0143

sx =

0.0100
0.0300
0.0700
0.0900
0.1500
0.1800
0.2000
0.2200
0.3000
0.3100
0.4800
0.5000
0.5500
0.5700
0.6000
0.7000
0.7500
0.8100
0.8500
0.9000
0.9000
0.9500
1.0000
1.0400
1.2000
1.2500
1.3000
1.4500
1.5600
1.9800

ifail =

0

```
```function g08cc_example
x = [0.01;
0.3;
0.2;
0.9;
1.2;
0.09;
1.3;
0.18;
0.9;
0.48;
1.98;
0.03;
0.5;
0.07;
0.7;
0.6;
0.95;
1;
0.31;
1.45;
1.04;
1.25;
0.15;
0.75;
0.85;
0.22;
1.56;
0.81;
0.57;
0.55];
ntype = int64(1);
[d, z, p, sx, ifail] = g08cc(x, @cdf, ntype)

function [result] = cdf(x)

if x < 0
result = 0;
elseif x > 2
result = 1;
else
result = x/2;
end
```
```

d =

0.2800

z =

1.5336

p =

0.0143

sx =

0.0100
0.0300
0.0700
0.0900
0.1500
0.1800
0.2000
0.2200
0.3000
0.3100
0.4800
0.5000
0.5500
0.5700
0.6000
0.7000
0.7500
0.8100
0.8500
0.9000
0.9000
0.9500
1.0000
1.0400
1.2000
1.2500
1.3000
1.4500
1.5600
1.9800

ifail =

0

```