What default theoretical distribution does Mathematica use in PearsonChiSquareTest?

Question

I used Mathematica's function PearsonChiSquareTest to test whether the data were drawn from a normal distribution.

As in the reference, I expected Mathematica to compare the result to a normal distribution, by running: PearsonChiSquareTest[data]. As I wanted to extract the test statistics from properties, I was "forced" to name the distribution I wanted to compare it to, namely the normal distribution: PearsonChiSquareTest[data, NormalDistribution[],"PValue"].
What is surprising is that the PValue for the first case is different from the second case where I use the Normal Distribution explicitly. Anyone knows what's the difference here?

data = RandomVariate[NormalDistribution[], 10^4];

PearsonChiSquareTest[data]

(*
==> 0.3248593157
*)

PearsonChiSquareTest[data, NormalDistribution[]]

(*
==> 0.4704917701
*)

Answer 1

According to the documentation you linked, difference is:

• PearsonChiSquareTest[data, NormalDistribution[]] tests whether the data is distributed according to a normal distribution of mean 0 and standard deviation 1.
• PearsonChiSquareTest[data] is actually more flexible, and checks whether your data is distributed according to any normal distribution.

The difference becomes flagrant with data from a different distribution:

data = RandomVariate[NormalDistribution[1, 0.1], 10^4];
PearsonChiSquareTest[data]
(* Out[33]= 0.419701 *)

PearsonChiSquareTest[data, NormalDistribution[]]
(* Out[34]= 2.712072730217051*10^-23123 *)

---

In short, the following two statements are equivalent:

PearsonChiSquareTest[data]
PearsonChiSquareTest[data, NormalDistribution[μ, Σ]]

where the second one uses symbolic parameters in the distribution. You can retrieve information about the test by doing:

PearsonChiSquareTest[data, NormalDistribution[μ, Σ], "HypothesisTestData"]

and exploring the different properties of this object. All properties can be listed by:

PearsonChiSquareTest[data, NormalDistribution[μ, Σ], "HypothesisTestData"]["Properties"]

and the fitted distribution is recovered with:

PearsonChiSquareTest[data, NormalDistribution[μ, Σ], "HypothesisTestData"]["FittedDistribution"]

Enjoy!