Turning point test

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

Statement of test
The turning point test is a test of the null hypothesis


 * 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).

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.

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


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

is approximately standard normal for large values of n.

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