sim_vonneumann.h File Reference

Von Neumann ratio test for randomness of a sequence. More...

#include <cmath>
#include <cstddef>
#include <vector>
#include "line/api/sim/sim_dist.h"
#include "line/api/sim/sim_types.h"
#include "line/num/number.h"
#include "line/util/error.h"

Include dependency graph for sim_vonneumann.h:

Go to the source code of this file.

Classes
struct	line::sim::VonNeumannResult< T >
	Outcome of the von Neumann randomness test. More...

Namespaces
namespace	line
namespace	line::sim

Functions
template<class T>
VonNeumannResult< T >	line::sim::sim_vonneumann (const std::vector< T > &x, double alpha=0.05)
	Von Neumann ratio test for randomness of a sequence.

Detailed Description

Von Neumann ratio test for randomness of a sequence.

Port of matlab/src/api/sim/sim_vonneumann.m. The statistic is the ratio of the mean square successive difference to the variance, ratio = sum_{i=1}^{b-1} (x_{i+1}-x_i)^2 / sum_{i=1}^{b} (x_i - xbar)^2, with b the number of observations. Under the null hypothesis that x is i.i.d. normal the ratio has mean 2 and variance 4(b-2)/((b-1)(b+1)), and (ratio-2)/sd is asymptotically standard normal, so the two-sided p-value is 2(1-Phi(|z|)). Serial correlation of either sign moves the ratio away from 2: positive correlation shrinks the successive differences and pushes the ratio below 2, negative correlation pushes it above.

The null mean and variance above were confirmed by Monte Carlo over b = 10, 16, 24, 32, 50 to within 0.3% in the reference.

The test is TWO-SIDED and its rejection is used as a stopping rule by the QUEST procedures, so alpha here is a stage significance (0.30 by default in sim_fquest, decaying during warmup) and not the interval's coverage level.

Reference: J. von Neumann, "Distribution of the Ratio of the Mean Square Successive Difference to the Variance", Ann. Math. Statist. 12(4), 1941; L. C. Young, "Randomness in Ordered Sequences", Ann. Math. Statist. 12, 1941.

Definition in file sim_vonneumann.h.