Turning point test
# 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 tests the null hypothesis

    $H_0:$ $X_1,$ $X_2,$ ..., $X_n$ are independent and identically distributed random variables (iid)

against

    $H_1:$ $X_1,$ $X_2,$ ..., $X_n$ are not iid.

### Test statistic

We say i is a turning point if the vector $X_1,$ $X_2,$ ..., $X_i,$ ..., $X_n$ 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.

