The Pearson type VII family Kurtosis

pdf pearson type vii distribution excess kurtosis of infinity (red); 2 (blue); , 0 (black)



log-pdf pearson type vii distribution excess kurtosis of infinity (red); 2 (blue); 1, 1/2, 1/4, 1/8, , 1/16 (gray); , 0 (black)

the effects of kurtosis illustrated using parametric family of distributions kurtosis can adjusted while lower-order moments , cumulants remain constant. consider pearson type vii family, special case of pearson type iv family restricted symmetric densities. probability density function given by

f
(
x
;
a
,
m
)
=



Γ
(
m
)


a



π



Γ
(
m
−
1

/

2
)





[
1
+


(


x
a


)


2


]


−
m


,



{\displaystyle f(x;a,m)={\frac {\gamma (m)}{a\,{\sqrt {\pi }}\,\gamma (m-1/2)}}\left[1+\left({\frac {x}{a}}\right)^{2}\right]^{-m},\!}

where scale parameter , m shape parameter.

all densities in family symmetric. kth moment exists provided m > (k + 1)/2. kurtosis exist, require m > 5/2. mean , skewness exist , both identically zero. setting a = 2m − 3 makes variance equal unity. free parameter m, controls fourth moment (and cumulant) , hence kurtosis. 1 can reparameterize



m
=
5

/

2
+
3

/


γ

2




{\displaystyle m=5/2+3/\gamma _{2}}

,




γ

2




{\displaystyle \gamma _{2}}

excess kurtosis defined above. yields one-parameter leptokurtic family 0 mean, unit variance, 0 skewness, , arbitrary non-negative excess kurtosis. reparameterized density is

g
(
x
;

γ

2


)
=
f

(
x
;

a
=


2
+


6

γ

2






,

m
=


5
2


+


3

γ

2




)

.



{\displaystyle g(x;\gamma _{2})=f\left(x;\;a={\sqrt {2+{\frac {6}{\gamma _{2}}}}},\;m={\frac {5}{2}}+{\frac {3}{\gamma _{2}}}\right).\!}

in limit




γ

2


→
∞


{\displaystyle \gamma _{2}\to \infty }

1 obtains density

g
(
x
)
=
3


(
2
+

x

2


)


−


5
2




,



{\displaystyle g(x)=3\left(2+x^{2}\right)^{-{\frac {5}{2}}},\!}

which shown red curve in images on right.

in other direction




γ

2


→
0


{\displaystyle \gamma _{2}\to 0}

1 obtains standard normal density limiting distribution, shown black curve.

in images on right, blue curve represents density



x
↦
g
(
x
;
2
)


{\displaystyle x\mapsto g(x;2)}

excess kurtosis of 2. top image shows leptokurtic densities in family have higher peak mesokurtic normal density, although conclusion valid select family of distributions. comparatively fatter tails of leptokurtic densities illustrated in second image, plots natural logarithm of pearson type vii densities: black curve logarithm of standard normal density, parabola. 1 can see normal density allocates little probability mass regions far mean ( has thin tails ), compared blue curve of leptokurtic pearson type vii density excess kurtosis of 2. between blue curve , black other pearson type vii densities γ2 = 1, 1/2, 1/4, 1/8, , 1/16. red curve again shows upper limit of pearson type vii family,




γ

2


=
∞


{\displaystyle \gamma _{2}=\infty }

(which, strictly speaking, means fourth moment not exist). red curve decreases slowest 1 moves outward origin ( has fat tails ).