Turning point test: Difference between revisions
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===Test statistic=== |
===Test statistic=== |
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We say ''i'' is a turning point if the vector ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''i''</sub>, ..., ''X''<sub>''n''</sub> is not monotonic at index ''i''. |
We say ''i'' is a turning point if the vector ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''i''</sub>, ..., ''X''<sub>''n''</sub> is not monotonic at index ''i''. The number of turning points is the number of maxima and minima in the series.<ref>{{cite doi|10.1093/biomet/59.3.680}}</ref> |
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Let ''T'' be the number of turning points then for large ''n'', ''T'' is approximately [[normal distribution|normally distributed]] with mean (2''n'' − 4)/3 and variance (16''n'' − 29)/90. The test statistic<ref>{{cite doi|10.1007/978-94-007-1861-6_4}}</ref> |
Let ''T'' be the number of turning points then for large ''n'', ''T'' is approximately [[normal distribution|normally distributed]] with mean (2''n'' − 4)/3 and variance (16''n'' − 29)/90. The test statistic<ref>{{cite doi|10.1007/978-94-007-1861-6_4}}</ref> |
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Revision as of 11:47, 17 June 2013
In statistical hypothesis testing, a turning point test is a statistical test of the independence of a series of random variables.[1][2][3]
Statement of test
The turning point tests the null hypothesis[1]
- H0: X1, X2, ... Xn are independent and identically distributed random variables
against
- H1: X1, X2, ... Xn are not iid.
Test statistic
We say i is a turning point if the vector X1, X2, ..., Xi, ..., Xn is not monotonic at index i. The number of turning points is the number of maxima and minima in the series.[4]
Let T be the number of turning points then for large n, T is approximately normally distributed with mean (2n − 4)/3 and variance (16n − 29)/90. The test statistic[5]
has standard normal distribution.
References
- ^ a b Le Boudec, Jean-Yves (2010). Performance Evaluation Of Computer And Communication Systems (PDF). EPFL Press. pp. 136–137. ISBN 978-2-940222-40-7.
- ^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1007/b97391, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with
|doi=10.1007/b97391instead. - ^ Kendall, Maurice George (1973). Time series. Griffin. ISBN 0852642202.
- ^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1093/biomet/59.3.680, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with
|doi=10.1093/biomet/59.3.680instead. - ^ Attention: This template ({{cite doi}}) is deprecated. To cite the publication identified by doi:10.1007/978-94-007-1861-6_4, please use {{cite journal}} (if it was published in a bona fide academic journal, otherwise {{cite report}} with
|doi=10.1007/978-94-007-1861-6_4instead.
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