Turning point test

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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 tests the null hypothesis[1]

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


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

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]

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]

has standard normal distribution.


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


  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. 
  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 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. Math. Soc. Fr. 2: 153–4. 
  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. doi:10.1061/(ASCE)0733-9437(1986)112:1(65).