KolmogorovTwoSample (JMSL Numerical Library (jmsl) 2021.0.0 API)

java.lang.Object
- com.imsl.stat.KolmogorovTwoSample

All Implemented Interfaces:

Serializable
```
public class KolmogorovTwoSample
extends Object
implements Serializable
```
Performs a Kolmogorov-Smirnov two-sample test.
Class KolmogorovTwoSample computes Kolmogorov-Smirnov two-sample test statistics for testing that two continuous cumulative distribution functions (CDF's) are identical based upon two random samples. One- or two-sided alternatives are allowed. Exact p-values are computed for the two-sided test when $nm \le 104$, where n is the number of non-missing X observations and m the number of non-missing Y observation.

Let $F_n(x)$ denote the empirical CDF in the X sample, let $G_m(y)$ denote the empirical CDF in the Y sample and let the corresponding population distribution functions be denoted by $F(x)$ and $G(y)$, respectively. Then, the hypotheses tested by KolmogorovTwoSample are as follows: $$ \begin{array}{ll} H_0:~ F(x) = G(x) & H_1:~F(x) \ne G(x) \\ H_0:~ F(x) \ge G(x) & H_1:~F(x) \lt G(x) \\ H_0:~ F(x) \le G(x) & H_1:~F(x) \gt G(x) \end{array} $$ The test statistics are given as follows: $$ \begin{array}{rl} D_{mn} & = \max(D_{mn}^{+}, D_{mn}^{-}) \\ D_{mn}^{+} & = \max_x(F_n(x)-G_m(x)) \\ D_{mn}^{-} & = \max_x(G_m(x)-F_n(x)) \end{array} $$ Asymptotically, the distribution of the statistic $$ Z = D_{mn} \sqrt{\frac{mn}{m+n}} $$ converges to a distribution given by Smirnov (1939).

Exact probabilities for the two-sided test are computed when $nm \le 104$, according to an algorithm given by Kim and Jennrich (1973). When $nm \gt 104$, the very good approximations given by Kim and Jennrich are used to obtain the two-sided p-values. The one-sided probability is taken as one half the two-sided probability. This is a very good approximation when the p-value is small (say, less than 0.10) and not very good for large p-values.

See Also:

Example, Serialized Form

Constructor Summary

Constructors
Constructor and Description

KolmogorovTwoSample(double[] x, double[] y)
Constructs a two sample Kolmogorov-Smirnov goodness-of-fit test.

Constructors
Constructor and Description
`KolmogorovTwoSample(double[] x, double[] y)` Constructs a two sample Kolmogorov-Smirnov goodness-of-fit test.

Method Summary

All Methods Instance Methods Concrete Methods
Modifier and Type	Method and Description
`double`	`getMaximumDifference()` Returns $D^{+}$, the maximum difference between the theoretical and empirical CDF's.
`double`	`getMinimumDifference()` Returns $D^{-}$, the minimum difference between the theoretical and empirical CDF's.
`int`	`getNumberMissingX()` Returns the number of missing values in the `x` sample.
`int`	`getNumberMissingY()` Returns the number of missing values in the `y` sample.
`double`	`getOneSidedPValue()` Probability of the statistic exceeding D under the null hypothesis of equality and against the one-sided alternative.
`double`	`getTestStatistic()` Returns $D = \max(D^{+}, D^{-})$.
`double`	`getTwoSidedPValue()` Probability of the statistic exceeding D under the null hypothesis of equality and against the two-sided alternative.
`double`	`getZ()` Returns the normalized D statistic without the continuity correction applied.

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

- Constructor Detail
  - KolmogorovTwoSample
```
public KolmogorovTwoSample(double[] x,
                           double[] y)
```
    Constructs a two sample Kolmogorov-Smirnov goodness-of-fit test.
    
    Parameters:
    
    x - is an array containing the observations from the first sample.
    
    y - is an array containing the observations from the second sample.
- Method Detail
  - getTestStatistic
```
public double getTestStatistic()
```
    Returns $D = \max(D^{+}, D^{-})$.
    
    Returns:
    
    The value D.
  - getMaximumDifference
```
public double getMaximumDifference()
```
    Returns $D^{+}$, the maximum difference between the theoretical and empirical CDF's.
    
    Returns:
    
    The value $D^{+}$.
  - getMinimumDifference
```
public double getMinimumDifference()
```
    Returns $D^{-}$, the minimum difference between the theoretical and empirical CDF's.
    
    Returns:
    
    The value $D^{-}$.
  - getZ
```
public double getZ()
```
    Returns the normalized D statistic without the continuity correction applied.
    
    Returns:
    
    the value Z
  - getOneSidedPValue
```
public double getOneSidedPValue()
```
    Probability of the statistic exceeding D under the null hypothesis of equality and against the one-sided alternative. An exact probability is computed if the number of observation is less than or equal to 80, otherwise an approximate probability is computed.
    
    Returns:
    
    the one-sided probability.
  - getTwoSidedPValue
```
public double getTwoSidedPValue()
```
    Probability of the statistic exceeding D under the null hypothesis of equality and against the two-sided alternative. This probability is twice the probability, $p_1$, reported by getOneSidedPValue, (or 1.0 if $p_1 \ge 1/2$). This approximation is nearly exact when $p_1 \lt 0.1$.
    
    Returns:
    
    the two-sided probability.
  - getNumberMissingX
```
public int getNumberMissingX()
```
    Returns the number of missing values in the x sample.
    
    Returns:
    
    The number of missing values in x.
  - getNumberMissingY
```
public int getNumberMissingY()
```
    Returns the number of missing values in the y sample.
    
    Returns:
    
    The number of missing values in y.

Modifier and Type	Method and Description
`double`	`getMaximumDifference()` Returns \(D^{+}\), the maximum difference between the theoretical and empirical CDF's.
`double`	`getMinimumDifference()` Returns \(D^{-}\), the minimum difference between the theoretical and empirical CDF's.
`int`	`getNumberMissingX()` Returns the number of missing values in the `x` sample.
`int`	`getNumberMissingY()` Returns the number of missing values in the `y` sample.
`double`	`getOneSidedPValue()` Probability of the statistic exceeding D under the null hypothesis of equality and against the one-sided alternative.
`double`	`getTestStatistic()` Returns \(D = \max(D^{+}, D^{-})\).
`double`	`getTwoSidedPValue()` Probability of the statistic exceeding D under the null hypothesis of equality and against the two-sided alternative.
`double`	`getZ()` Returns the normalized D statistic without the continuity correction applied.

Class KolmogorovTwoSample

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Constructor Detail

KolmogorovTwoSample

Method Detail

getTestStatistic

getMaximumDifference

getMinimumDifference

getZ

getOneSidedPValue

getTwoSidedPValue

getNumberMissingX

getNumberMissingY