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, ..., X_n are independent and identically distributed random variables (iid)

against

H₁: X₁, X₂, ..., X_n are not iid.

This test assumes that the X_i 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, ..., 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 (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.