Wald Wolfowitz Test

For one sample Wald Wolfowitz test we consider the time series

Wald Wolfowitz test is used to test the randomness of a distribution, by taking the data in the given order and marking with + the data greater than the specified value called border (median by default), and with – the data less than the border.

A "run" of a sequence is a maximal non-empty segment of the sequence consisting of adjacent equally marked elements. For example, the sequence "++++−−−+++−−++++++−−−−" consists of six runs, three of which consist of +'s and the others of −'s.

Under the null hypothesis, the number of runs in a sequence of length N is a random variable whose conditional distribution given the observation of n₊ positive values and n_- negative values (n = n₊ + n_-) is approximately normal, with:

• mean

• variance

These parameters do not depend on the "fairness" of the process generating the elements of the sequence. Only on the assumption that the elements are independent and identically distributed. If the number of runs is significantly higher or lower than expected, the hypothesis of statistical independence of the elements may be rejected.

• Tail.Both. In the alternative hypothesis data are not independent or not identically distributed.

• Tail.Left. In the alternative hypothesis there are too few groups of identical signs to be arbitrary by chance (signs tend to cluster).

• Tail.Right. In the alternative hypothesis there are too many groups of identical signs to be arbitrary by chance (signs tend to mix).

Note
Decision based on P-value and significance level comparison is more exact.

Test statistic is calculated as follows:

The following constructors create an instance of WaldWolfowitzTest class.

The class provides the following methods:

The class provides the following properties:

The example of WaldWolfowitzTest class usage:

 1using System;
 2using FinMath.LinearAlgebra;
 3using FinMath.Statistics.HypothesisTesting;
 4using FinMath.Statistics.Distributions;
 5
 6namespace FinMath.Samples
 7{
 8    class WaldWolfowitzTestSample
 9    {
10        static void Main()
11        {
12            //Create an instance of normal distribution.
13            Normal distr = new Normal(0, 1);
14
15            // Generate random series.
16            Vector series = Vector.Random(100, distr);
17
18            // Create an instance of WaldWolfowitzTest.
19            WaldWolfowitzTest test = new WaldWolfowitzTest(0.05, Tail.Both);
20            test.Update(series, 0);
21
22            Console.WriteLine("Test Result:");
23            // Test decision
24            Console.WriteLine($"  The null hypothesis failed to be rejected: {test.Decision}");
25            // The statistic of WaldWolfowitzTest test.
26            Console.WriteLine($"  Statistics = {test.Statistics:0.000}");
27            // The p-value of the test statistic.
28            Console.WriteLine($"  P-Value = {test.PValue:0.000}");
29
30        }
31    }
32}

Reference