In[]:=
CompoundExpression[
]
​​deploy
Sat 15 Jul 2023 16:08:40
https://mathematica.stackexchange.com/questions/287607/grouping-expressions-by-their-o-rate-of-growth-around-0
​

Grouping distributions by asymptotics

In[]:=
pairs={​​{"Kumaraswamy(2,3)",KumaraswamyDistribution[2,3]},​​{"Weibull(1/2,2)",WeibullDistribution[1/2,2]},​​{"ArcSin",ArcSinDistribution[]},​​{"Bates(3)",BatesDistribution[3]},​​{"Triangle",TriangularDistribution[]},​​{"Uniform",UniformDistribution[]},​​{"Logistic",LogisticDistribution[]},​​{"Extreme Value",ExtremeValueDistribution[]},​​{"Frechet(2,1,0)",FrechetDistribution[2,1,0]},​​{"Erlang(2,2)",ErlangDistribution[2,2]},​​{"Gamma(1,2)",GammaDistribution[1,2]},​​{"InverseGamma",InverseGammaDistribution[1,2]},​​{"F-ratio(2,2)",FRatioDistribution[2,2]},​​{"Gumbel",GumbelDistribution[]},​​{"Beta(1/2,1)",BetaDistribution[1/2,1]},​​{"Marchenko-Pastur",MarchenkoPasturDistribution[1]},​​{"Semicircle",WignerSemicircleDistribution[2]},​​{"LogNormal",LogNormalDistribution[0,1]},​​{"Cauchy",CauchyDistribution[]},​​{"Pareto(1,2)",ParetoDistribution[1,2]},​​{"Normal",NormalDistribution[]},​​{"Student-T",StudentTDistribution[1]},​​{"ChiSquared",ChiSquareDistribution[1]},​​{"Chi",ChiDistribution[1]},​​{"Exponential",ExponentialDistribution[1]},​​{"Inverse Normal",InverseGaussianDistribution[2,1]}​​};​​trunc[dist_]:=TruncatedDistribution[{0,∞},dist];​​​​getGrowth[dist_]:=Assuming[0<x<1/1000,Asymptotic[Refine@CDF[dist,x],x->0]];​​trunc[dist_]:=TruncatedDistribution[{0,∞},dist];​​square[dist_]:=TransformedDistribution[x^2,xdist];​​​​pairs2={First[#],getGrowth@trunc@Last@#}&/@pairs;
In[]:=
order[pair1_,pair2_]:=AsymptoticGreater[Last@pair1,Last@pair2,x->0,Direction->"FromAbove"];TableForm@Sort[pairs2,order]
Out[]//TableForm=
ChiSquared
2
π
x
Marchenko-Pastur
2
x
π
Beta(1/2,1)
x
ArcSin
2
x
π
Weihbul(1/2,2)
x
2
Exponential
x
Chi
2
π
x
Student-T
2x
π
Normal
2
π
x
Cauchy
2x
π
Semicircle
2x
π
Gumbel
x
F-ratio(2,2)
x
Gamma(1,2)
x
2
Extreme Value
x
-1+
Logistic
x
2
Uniform
x
Inverse Normal
-
2
(-2+x)
8x

2
π
x
-
2
x
16
-

2
x
16
Erlang(2,2)
2
2
x
Triangle
2
2
x
Kumaraswamy(2,3)
3
2
x
Bates(3)
9
3
x
2
LogNormal
-
-
1
2
2
Log[x]

2π
Log[x]
InverseGamma
-2/x

Frechet(2,1,0)
-
1
2
x

Pareto(1,2)
0

Extra distributions

In[]:=
getGrowthPDF[dist_]:=Assuming[0<x<1/1000,Asymptotic[Refine@PDF[dist,x],x->0]];​​Print["showing PDF"];​​pairs={​​{"Erlang(1,2)",ErlangDistribution[1,2]},​​{"Kumaraswamy(2,3)",KumaraswamyDistribution[2,3]},​​{"Weibull(1/2,2)",WeibullDistribution[1/2,2]},​​{"Power(2,3)",PowerDistribution[2,3]},​​{"Power(1,2)",PowerDistribution[1,2]},​​{"Power(4,1)",PowerDistribution[4,1]},​​{"NBeta(2,3,1)",NoncentralBetaDistribution[2,3,1]},​​{"NBeta(1,2,1/2)",NoncentralBetaDistribution[1,2,1/2]},​​{"Nbeta(1/2,1,.7)",NoncentralBetaDistribution[1/2,1,7/10]},​​{"Voigt(1,1/2)",VoigtDistribution[1,1/2]},​​{"Voigt(1,1)",VoigtDistribution[1,1]},​​{"Rayleigh(1/2)",RayleighDistribution[1/2]},​​{"Rayleigh(1)",RayleighDistribution[1]},​​{"Rayleigh(2)",RayleighDistribution[2]},​​{"MaxwellDistribution(1/2)",MaxwellDistribution[1/2]},{"MaxwellDistribution(3/4)",MaxwellDistribution[3/4]},​​{"MaxwellDistribution(3/2)",MaxwellDistribution[3/2]},​​{"F-ratio(1,1)",FRatioDistribution[1,1]},​​{"F-ratio(1,2)",FRatioDistribution[1,2]},​​{"F-ratio(2,1)",FRatioDistribution[2,1]},​​{"Fisher-Z(1/2,3)",FisherZDistribution[1/2,3]},​​{"Fisher-Z(3/2,3)",FisherZDistribution[3/2,3]},​​{"Fisher-Z(10,3)",FisherZDistribution[10,3]},​​{"Hotelling(1,30)",HotellingTSquareDistribution[1,30]},​​{"Hotelling(2,30)",HotellingTSquareDistribution[2,30]},​​{"Hotelling(3,30)",HotellingTSquareDistribution[3,30]}​​​​};​​pairs2={First[#],getGrowthPDF@trunc@Last@#}&/@pairs;​​order[pair1_,pair2_]:=AsymptoticGreater[Last@pair1,Last@pair2,x->0,Direction->"FromAbove"];TableForm@Sort[pairs2,order]​​
showing PDF
Out[]//TableForm=
Hotelling(1,30)
9694845
15
2
67108864
x
F-ratio(1,2)
1
2
2
x
F-ratio(1,1)
1
π
x
Nbeta(1/2,1,.7)
1
2
7/20

x
Weibull(1/2,2)
1
2
2
x
Hotelling(2,30)
29
60
Fisher-Z(10,3)
-
32484375
3
13
19307236-1+BetaRegularized
10
13
,5,
3
2

Fisher-Z(3/2,3)
-
8
2
Gamma
9
4

9
1/4
3
π
-1+BetaRegularized
1
3
,
3
4
,
3
2
Gamma
3
4

Fisher-Z(1/2,3)
-
24
6
π
Gamma
7
4

7
3/4
7
-1+BetaRegularized
1
7
,
1
4
,
3
2
Gamma
1
4

F-ratio(2,1)
1
Voigt(1,1)
2
π
-
2
π
Erf
1
2

Voigt(1,1/2)
2
2

2
π
-2
2

2
π
Erf[
2
]
NBeta(1,2,1/2)
2
1/4

Power(4,1)
4
Erlang(1,2)
2
Hotelling(3,30)
4524261
15
2
x
33554432
Rayleigh(2)
x
4
Rayleigh(1)
x
Rayleigh(1/2)
4x
NBeta(2,3,1)
12x

Power(1,2)
2x
Kumaraswamy(2,3)
6x
MaxwellDistribution(3/2)
8
27
2
π
2
x
MaxwellDistribution(3/4)
64
27
2
π
2
x
MaxwellDistribution(1/2)
8
2
π
2
x
Power(2,3)
24
2
x

Big Table

Missing pairs

Slow distributions