By default, univariate data is compared to

NormalDistribution:

The parameters have been estimated from the data:

Multivariate data is compared to

MultinormalDistribution by default:

The parameters of the test distribution are estimated from the data if not specified:

Specified parameters are not estimated:

Maximum likelihood estimates are used for unspecified parameters of the test distribution:

PearsonChiSquareTest effectively compares the observed and expected histograms:

The data is binned into approximately

bins that are equiprobable under

:

Under

, each bin will contain an equal number of points:

Observed histograms for when

is true and false, respectively:

The degrees of freedom are equal to the number of non-empty bins minus one:

One degree of freedom is removed for each parameter that is estimated from the data:

If the parameters are unknown,

PearsonChiSquareTest corrects the degrees of freedom:

No correction is applied when the parameters are specified:

The fitted distribution is equivalent but the degrees of freedom and

-value are corrected:

The Pearson

statistic asymptotically follows

ChiSquareDistribution under

:

Independent marginal densities are assumed in tests for multivariate goodness of fit:

The test statistic is identical when independence is assumed: