non-central chi-squared random variable

non-central chi-squared random variable

Let

X

1

,

…

,

X

n

be IID

random variables

, each with the

standard normal distribution

. Then, for any

𝝁

∈

ℝ

n

, the random variable

X

defined by

X

=

∑

i

=

1

n

(

X

i

+

μ

i

)

2

is called a

non-central chi-squared random variable

.
Its

distribution

depends only on the number of

degrees of freedom

n

and non-centrality parameter

λ

≡

∥

𝝁

∥

. This is denoted by

χ

2

⁢

(

n

,

λ

)

and has

moment generating function

M

X

⁡

(

t

)

≡

𝔼

⁢

[

e

t

⁢

X

]

=

(

1

-

2

⁢

t

)

-

n

2

⁢

exp

⁡

(

λ

⁢

t

1

-

2

⁢

t

)

,

(1)

which is defined for all

t

∈

ℂ

with real part less than

1

/

2

.
More generally, for any

n

,

λ

≥

0

, not necessarily

integers

, a random variable has the

non-central chi-squared distribution

,

χ

2

⁢

(

n

,

λ

)

, if its moment generating function is given by (

1

).

A non-central chi-squared random variable for any

n

,

λ

≥

0

can be constructed as follows. Let

Y

be a (

central

) chi-squared

variable

with degree

n

,

Z

1

,

Z

2

,

…

be standard normals, and

N

have the

Poisson

⁢

(

λ

/

2

)

distribution. If these are all

independent

then

X

≡

Y

+

∑

k

=

1

2

⁢

N

Z

k

2

.

has the

χ

2

⁢

(

n

,

λ

)

distribution. Correspondingly, the

probability density function

for

X

is

f

X

⁢

(

x

)

=

∑

k

=

0

∞

λ

k

2

k

⁢

k

!

⁢

e

-

λ

/

2

⁢

f

n

+

2

⁢

k

⁢

(

x

)

,

(2)

where

x

>

0

and

f

k

is the probability density of the

χ

(

k

)

2

distribution.
Alternatively, this can be expressed as

f

X

⁢

(

x

)

=

1

2

⁢

e

-

(

x

+

λ

)

/

2

⁢

(

x

/

λ

)

n

/

4

-

1

/

2

⁢

I

n

/

2

-

1

⁢

(

λ

⁢

x

)

.

where

I

ν

is a modified Bessel function of the first kind,

I

ν

⁢

(

x

)

=

∑

k

=

0

∞

(

x

/

2

)

ν

+

2

⁢

k

k

!

⁢

Γ

⁢

(

ν

+

k

+

1

)

.

Figure 1:

Densities

of the non-central chi-squared distribution

χ

2

⁢

(

n

,

λ

)

.

Remarks

1.

χ

2

⁢

(

n

,

λ

)

has mean

n

+

λ

and

variance

2

⁢

n

+

4

⁢

λ

.

2.

χ

2

⁢

(

n

,

0

)

=

χ

(

n

)

2

. The (central)

chi-squared random variable

is a special case of the non-central chi-squared random variable, when the non-centrality parameter

λ

=

0

.

3.

(The reproductive

property

of

chi-squared distributions

). If

Z

1

,

…

,

Z

m

are non-central chi-squared random variables such that each

Z

i

∼

χ

2

⁢

(

n

i

,

λ

i

)

, then their total

Z

=

∑

Z

i

is also a non-central chi-squared random variable with distribution

χ

2

⁢

(

∑

n

i

,

∑

λ

i

)

.

4.

If

n

>

0

then the

χ

2

⁢

(

n

,

λ

)

distribution is restricted to the domain

(

0

,

∞

)

with probability density function (

2

). On the other hand, if

n

=

0

, then there is also an atom at

0

,

ℙ

(

X

=

0

)

=

lim

t

→

-

∞

M

X

(

t

)

=

e

-

λ

/

2

.

5.

If

𝒙

is a multivariate normally distributed

n

-dimensional

random vector

with distribution

𝑵

⁢

(

𝝁

,

𝑽

)

where

𝝁

is the mean vector and

𝑽

is the

n

×

n

covariance matrix

. Suppose that

𝑽

is

singular

, with

k

= rank of

V

<

n

. Then

𝒙

𝐓

⁢

𝑽

-

⁢

𝒙

is a non-central chi-squared random variable, where

𝑽

-

is a

generalized inverse

of

𝑽

. Its distribution has

k

degrees of freedom with non-centrality parameter

λ

=

𝝁

𝐓

⁢

𝑽

-

⁢

𝝁

.

Title

non-central chi-squared random variable

Canonical

name

NoncentralChisquaredRandomVariable

Date of creation

2013-03-22 14:56:16

Last modified on

2013-03-22 14:56:16

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

11

Author

CWoo (3771)

Entry type

Definition

Classification

msc 62E99

Classification

msc 60E05

Synonym

non-central chi-squared distribution

Defines

non-centrality parameter

Generated on Sat Feb 10 12:04:30 2018 by

LaTeXML