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
against
This test assumes that the X_i have a continuous distribution.

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.
Letting T be the number of turning points, then for large n, T is approximately normally distributed with mean /3 and variance /90. The test statistic
is approximately standard normal for large values of n.

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