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{{Short description|Statistical test}}
In [[statistical hypothesis testing]], a '''turning point test''' is a statistical test of the independence of a series of random variables.<ref name="boudec">{{cite book | title = Performance Evaluation Of Computer And Communication Systems | first = Jean-Yves | last = Le Boudec | isbn = 978-2-940222-40-7 | year=2010 | publisher = EPFL Press | url = http://infoscience.epfl.ch/record/146812/files/perfPublisherVersion.pdf |pages=136-137}}</ref><ref>{{cite doi|10.1007/b97391}}</ref><ref>{{cite book | title = Time series | first= Maurice George | last = Kendall | isbn = 0852642202 | publisher = Griffin | year = 1973}}</ref>
{{Orphan|date=June 2024}}

In [[statistical hypothesis testing]], a '''turning point test''' is a statistical test of the independence of a series of random variables.<ref name="boudec">{{cite book | title = Performance Evaluation Of Computer And Communication Systems | first = Jean-Yves | last = Le Boudec | isbn = 978-2-940222-40-7 | year = 2010 | publisher = [[EPFL Press]] | url = http://infoscience.epfl.ch/record/146812/files/perfPublisherVersion.pdf | pages = 136–137 | url-status = dead | archive-url = https://web.archive.org/web/20131012014413/http://infoscience.epfl.ch/record/146812/files/perfPublisherVersion.pdf | archive-date = 2013-10-12 }}</ref><ref>{{Cite book | editor1-last = Brockwell | editor1-first = Peter J | editor2-first = Richard A | doi = 10.1007/b97391 | title = Introduction to Time Series and Forecasting | series = Springer Texts in Statistics | year = 2002 | isbn = 978-0-387-95351-9 | editor2-last = Davis }}</ref><ref>{{cite book | title = Time series | first= Maurice George | last = Kendall | author-link = Maurice Kendall | isbn = 0852642202 | publisher = Griffin | year = 1973}}</ref> [[Maurice Kendall]] and Alan Stuart describe the test as "reasonable for a test against cyclicity but poor as a test against trend."<ref name="davison" /><ref>{{cite book| last1 = Kendall | first1 = M. G. | author-link1 = Maurice Kendall | last2 = Stuart | first2 = A. | year = 1968 | title = The Advanced Theory of Statistics, Volume 3: Design and Analysis, and Time-Series | edition = 2nd | location = London | publisher = Griffin | isbn = 0-85264-069-2 | pages = 361–2}}</ref> The test was first published by [[Irénée-Jules Bienaymé]] in 1874.<ref name="davison" /><ref>{{cite journal|last=Bienaymé|first=Irénée-Jules|author-link = Irénée-Jules Bienaymé| year = 1874 | title = Sur une question de probabilités | url = http://archive.numdam.org/ARCHIVE/BSMF/BSMF_1873-1874__2_/BSMF_1873-1874__2__153_1/BSMF_1873-1874__2__153_1.pdf | journal = [[Bull. Soc. Math. Fr.]] | volume = 2 | pages = 153–4|doi=10.24033/bsmf.56 }}</ref>


==Statement of test==
==Statement of test==


The turning point tests the null hypothesis<ref name="boudec" />
The turning point test is a test of the null hypothesis<ref name="boudec" />


:''H''<sub>0</sub>: ''X''<sub>1</sub>, ''X''<sub>2</sub>, ... ''X''<sub>''n''</sub> are [[independent and identically distributed random variables]]
:''H''<sub>0</sub>: ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''n''</sub> are [[independent and identically distributed random variables]] (iid)


against
against


:''H''<sub>1</sub>: ''X''<sub>1</sub>, ''X''<sub>2</sub>, ... ''X''<sub>''n''</sub> are not iid.
:''H''<sub>1</sub>: ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''n''</sub> are not iid.

This test assumes that the ''X''<sub>i</sub> have a ''continuous'' distribution (so adjacent values are almost surely never equal).<ref name="davison">{{Cite journal|author1-link=Chris Heyde | last1 = Heyde | first1 = C. C. |author2-link=Eugene Seneta | last2 = Seneta | first2 = E. | doi = 10.1093/biomet/59.3.680 | title = Studies in the History of Probability and Statistics. XXXI. The simple branching process, a turning point test and a fundamental inequality: A historical note on I. J. Bienaymé | journal = Biometrika | volume = 59 | issue = 3 | pages = 680 | year = 1972 }}</ref>


===Test statistic===
===Test statistic===


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 name="davison"/>


Let ''T'' be the number of turning points then for large ''n'', ''T'' is approximately [[normal distribution|normally distributed]] with mean (2''n''&nbsp;−&nbsp;4)/3 and variance (16''n''&nbsp;−&nbsp;29)/90. Therefore the [[p-value]] is<ref>{{cite doi|10.1007/978-94-007-1861-6_4}}</ref>
Letting ''T'' be the number of turning points, then for large ''n'', ''T'' is approximately [[normal distribution|normally distributed]] with mean (2''n''&nbsp;−&nbsp;4)/3 and variance (16''n''&nbsp;−&nbsp;29)/90. The test statistic<ref>{{Cite book | last1 = Machiwal | first1 = D. | last2 = Jha | first2 = M. K. | doi = 10.1007/978-94-007-1861-6_4 | chapter = Methods for Time Series Analysis | title = Hydrologic Time Series Analysis: Theory and Practice | pages = 51 | year = 2012 | isbn = 978-94-007-1860-9 }}</ref>


⚫
:<math>z =\frac{T - \frac{2n-4}{3}}{\sqrt{\frac{16n-29}{90}}}</math>
:<math>p = 2 \left(

1 - N_{(0,1)} \left(
is approximately standard normal for large values of n.
⚫
\frac{\left| T - \frac{2n-4}{3} \right|}{\sqrt{\frac{16n-29}{90}}}

\right) \right).</math>
==Applications==

The test can be used to verify the accuracy of a fitted [[time series]] model such as that describing [[irrigation]] requirements.<ref>{{Cite journal | last1 = Gupta | first1 = R. K. | last2 = Chauhan | first2 = H. S. | doi = 10.1061/(ASCE)0733-9437(1986)112:1(65) | title = Stochastic Modeling of Irrigation Requirements | journal = Journal of Irrigation and Drainage Engineering | volume = 112 | pages = 65–76 | year = 1986 }}</ref>


==References==
==References==
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{{Reflist}}
{{Reflist}}


⚫
[[Category:Time series statistical tests]]
{{statistics-stub}}

⚫
[[Category:Statistical tests]]

Latest revision as of 04:44, 27 June 2024

In statistical hypothesis testing, a turning point test is a statistical test of the independence of a series of random variables.[1][2][3] Maurice Kendall and Alan Stuart describe the test as "reasonable for a test against cyclicity but poor as a test against trend."[4][5] The test was first published by Irénée-Jules Bienaymé in 1874.[4][6]

Statement of test

[edit]

The turning point test is a test of the null hypothesis[1]

H0: X1, X2, ..., Xn are independent and identically distributed random variables (iid)

against

H1: X1, X2, ..., Xn are not iid.

This test assumes that the Xi have a continuous distribution (so adjacent values are almost surely never equal).[4]

Test statistic

[edit]

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]

Letting 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[7]

is approximately standard normal for large values of n.

Applications

[edit]

The test can be used to verify the accuracy of a fitted time series model such as that describing irrigation requirements.[8]

References

[edit]
  1. ^ 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. Archived from the original (PDF) on 2013-10-12.
  2. ^ Brockwell, Peter J; Davis, Richard A, eds. (2002). Introduction to Time Series and Forecasting. Springer Texts in Statistics. doi:10.1007/b97391. ISBN 978-0-387-95351-9.
  3. ^ Kendall, Maurice George (1973). Time series. Griffin. ISBN 0852642202.
  4. ^ a b c d Heyde, C. C.; Seneta, E. (1972). "Studies in the History of Probability and Statistics. XXXI. The simple branching process, a turning point test and a fundamental inequality: A historical note on I. J. Bienaymé". Biometrika. 59 (3): 680. doi:10.1093/biomet/59.3.680.
  5. ^ Kendall, M. G.; Stuart, A. (1968). The Advanced Theory of Statistics, Volume 3: Design and Analysis, and Time-Series (2nd ed.). London: Griffin. pp. 361–2. ISBN 0-85264-069-2.
  6. ^ Bienaymé, Irénée-Jules (1874). "Sur une question de probabilités" (PDF). Bull. Soc. Math. Fr. 2: 153–4. doi:10.24033/bsmf.56.
  7. ^ Machiwal, D.; Jha, M. K. (2012). "Methods for Time Series Analysis". Hydrologic Time Series Analysis: Theory and Practice. p. 51. doi:10.1007/978-94-007-1861-6_4. ISBN 978-94-007-1860-9.
  8. ^ Gupta, R. K.; Chauhan, H. S. (1986). "Stochastic Modeling of Irrigation Requirements". Journal of Irrigation and Drainage Engineering. 112: 65–76. doi:10.1061/(ASCE)0733-9437(1986)112:1(65).
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