Turning point test

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

The turning point tests the null hypothesis^[1]

H₀: X₁, X₂, ... X_n are independent and identically distributed random variables

against

H₁: X₁, X₂, ... X_n are not iid.

Test statistic

We say i is a turning point if the vector X₁, X₂, ..., X_i, ..., X_n 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^[7]

z={\frac {T-{\frac {2n-4}{3}}}{\sqrt {\frac {16n-29}{90}}}}

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/b97391 instead.
^ Kendall, Maurice George (1973). Time series. Griffin. ISBN 0852642202.
^ ^a ^b ^c 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/59.3.680 instead.
^ Kendall, M. G.; Stuart, A. (1968). The Advanced Theory of Statistics. Vol. 3 (2nd edition ed.). London: Griffin. pp. 361–2. {{cite book}}: |edition= has extra text (help)
^ Bienaymé, Irénée-Jules (1874). "Sur une question de probabilités" (PDF). Bull. Math. Soc. Fr. 2: 153–4.
^ 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_4 instead.

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[boudec-1] Le Boudec, Jean-Yves (2010). Performance Evaluation Of Computer And Communication Systems (PDF). EPFL Press. pp. 136–137. ISBN 978-2-940222-40-7.

[2] 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/b97391 instead.

[3] Kendall, Maurice George (1973). Time series. Griffin. ISBN 0852642202.

[davison-4] 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/59.3.680 instead.

[5] Kendall, M. G.; Stuart, A. (1968). The Advanced Theory of Statistics. Vol. 3 (2nd edition ed.). London: Griffin. pp. 361–2. {{cite book}}: |edition= has extra text (help)

[6] Bienaymé, Irénée-Jules (1874). "Sur une question de probabilités" (PDF). Bull. Math. Soc. Fr. 2: 153–4.

[7] 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_4 instead.

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