The maximum likelihood estimations for a type of left ellipsoidal distribution

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R E S E A R C H

Open Access

The maximum likelihood estimations for a

type of left ellipsoidal distribution

Yanling Wang

1*

_{and Yingshan Zhang}

2

*_{Correspondence:}

[email protected] 1_{College of Mathematics and}

Information Science, Henan Normal University, Xinxiang, 453007, China Full list of author information is available at the end of the article

Abstract

In this paper, we prove that if the distribution of ann×mprandom matrixYis a left ellipsoidal distribution with parameter

μ

∗=

μ

(II_m⊗Ip),

∗=m

, andY1,Y2,. . .,Ym

are independent and identical distributions, the maximum likelihood estimations of

μ

,

areY,S2_{if and only if}_Y_∼_N

n×mp(

μ

∗,

∗). IfY1,Y2,. . .,Ymare not independent

and identical distributions, thenYmay be not a normal distribution.

Keywords: left ellipsoidal distribution; normal distribution; maximum likelihood estimation

1 Introduction

LetX= (x,x, . . . ,xn),x,x, . . . ,xnbe independent standard normal distributions. A ran-dom matrixYn×pis said to be a left stochastic ellipsoid matrix if it satisﬁes (see [])

Yn×p=An×nXn×p+μn×p, n×nXn×p d

=Xn×p, EXX=In,

and its distribution, which is denoted byEn×p(μ,), is called a left ellipsoidal distribution, where=AAis reversible, andEis ellipsoid. Besides,ddenotes identical distribution,

SdenotesXX, and alln×n∈o(n) are orthogonal matrices with ordern. Then we have conclusions:

ESkX= , ESk=αkIn,

wherekis an arbitrary integer andkis a constant (see []).

IfXhas the distribution density, the distribution density has the formf(XX), then the distribution density is said to be spherical distribution. If the distribution density ofYis

||–p_f₍_Y_–_μ₎–₍_Y_–_μ₎_,

then the distribution density is called left ellipsoidal distribution contour, and the contour surface is

(Y–μ)–(Y–μ) =D.

Whenp= , it is an ellipsoidal surface. The spectral decomposition ofis, where is an orthogonal matrix,=diag(λ, . . . ,λn),λ>· · ·>λn> .

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If the distribution density ofYis

||–p_f₍_Y_–_μ₎–₍_Y_–_μ₎₌

p

π

np

||–p_exp _–p

tr(Y–μ)

–₍_Y_–_μ₎

,

the elongated vector ofY, which is denoted byY, is normalNnp(μ,_pIp⊗) distribution. For the sake of consistency of symbols, we denote the normal distribution byNn×p(μ,). Classical statistical analysis is built on the basis of normal distribution. However, it is still an important problem whether these graceful properties can also be satisﬁed without the condition that it is a normal distribution.

Since the left ellipsoidal distribution family has much more members than the normal distribution family (in fact, it almost includes all common distributions), a large amount of scholars’ fruitful research shows: on the one hand, the left ellipsoidal distribution as a multivariate normal distribution promotion is ideal; on the other hand, based on the re-search of the left ellipsoidal distribution, we can get many statistics used as solid properties of hypothesis test (see []). It is a trivial idea to extend the properties of the normal dis-tribution family to the left ellipsoidal disdis-tribution family. Nevertheless, that is not always true. In this article, we prove that the maximum likelihood property of normal distribution cannot be extended to the left ellipsoidal distribution.

LetY,Y, . . . ,Ymbe the samples of independent identical distributions, and in the con-dition of normal distribution,

Y= 

m

i=

Yi, S= 

m

i=

(Yi–Y)(Yi–Y)

are the maximum likelihood estimations ofμ,, respectively, ifmis big enough andS

is positive deﬁnite. Whenp= ,S_{is positive deﬁnite with probability  if}_m_>_n_{(see []).}

Additionally, the rank ofS_{does not decrease with the increasing of the columns in}_Y_so

thatS_{is positive deﬁnite when}_m_>_n_{. Now we discuss them in the case of}_m_>_max₍_n_{, ).}

When one focuses on the maximum likelihood estimation, the likelihood equations need to be deduced by the matrix diﬀerential method. Therefore, we add a trivial condition that the distribution density ofEn×p(μ,) is diﬀerentiable (see [–]).

We draw the following conclusions.

Theorem . Let Y,Y, . . . ,Ymbe independent identically distributed,and Yi∼En×p(μ, ).The maximum likelihood estimation ofμ,is Y,Sif and only if Yi∼Nn×p(μ,).

Note that for the proof of Theorem . one can refer to [].

When the variables ofμ∗are not independent, we have the solution as follows.

Theorem . Let Y ∼ En×mp(μ∗,∗), μ∗ =μ(IIm ⊗Ip), and ∗ =m. Besides, Y = (Y,Y, . . . ,Ym),where Yiis an n×p matrix.Then if Y,Y, . . . ,Ymare mutually independent

identically distributed,the maximum likelihood estimations ofμ,are

Y= 

mY

II_m⊗Ip

, S= 

mY(I–PIIm⊗Ip)Y

if and only if Y ∼Nn×mp(μ∗,∗),where PIIm=



mIImIIm,we note it P=PIIm⊗Ipin the proof

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Theorem . Under the conditions of Theorem.,if Y,Y, . . . ,Ym are not independent

identically distributed,and the maximum likelihood estimations ofμ,are

Y= 

mY

II_m⊗Ip

, S= 

mY(I–PIIm⊗Ip)Y _,

then Y may be not a normal distribution.

2 The proof of Theorem 1.2

LetY,Y, . . . ,Ymbe mutually identically independent distributions, andYi∼En×p(μ,). Besides, we assume thatY=Y,Y, . . . ,Ym, andYi=AXi+μ, then

Y=mA



mX

+μII_m⊗Ip

,

n×n



mX

d = 

mX,

E



mX



mX

=I,

where is a random orthogonal matrix with ordern. As a result, Y∼En×mp(μ(IIm⊗ Ip),m).

Now the above proposition can be transformed to a research of the form of the left stochastic ellipsoid distribution with a degraded mean.

Notice that

Y= 

m

i= Yi=



mY(IIm⊗Ip),

S= 

m

i=

(Yi–μ)(Yi–μ)

= 

m

i=

(Yi–Y)(Yi–Y)= 

m(YP–Y)(YP–Y) ₌ 

mY(I–P)Y _,

whereμ=Y,P=PIIm⊗Ip. Therefore, utilizing the single mean degraded left stochastic

ellipsoid to express Theorem ., we can obtain Theorem ..

3 The proof of Theorem 1.3

Based on the above discussion, we get the question: If the condition of mutually identically independent distribution is discarded, can one obtain the same solution with Theorem . fromY∼En×mp(μ·(IIm⊗Ip),m)? In other words, can the proposition be proved or not? LetY∼En×mp(μ·(IIm⊗Ip),m), and then the maximum likelihood estimations ofμ, are

Y= 

mY

II_m⊗Ip

, S= 

mY(I–P)Y _.

Here,P=PIIm⊗Ipif and only ifY∼Nn×mp(μ(II

m⊗Ip),m).

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that the maximum likelihood estimation of the parametersμ,in the left stochastic el-lipsoidEn×mp(μ(IIm⊗Ip),m) isY,S.

We assume thatYhas the distribution density, namely the likelihood function is

L(Y,μ,) =||–mp _f_Y_–_μ_II

m⊗Ip

(m)–Y–μII_m⊗Ip

.

Computing the logarithm diﬀerence of both sides, we can get

dlnL(Y,μ,) = –trGY–μII_m⊗Ip

–·dμ·II_m⊗Ip

–tr

–Y–μII_m⊗Ip

GY–μII_m⊗Ip

–+mp 

–

d,

whereA= (Y–μ(II_m⊗Ip))(m)–(Y–μ(IIm⊗Ip)) = (aij)n×nandA=A.

dlnf(A) = i<j

∂lnf(A) ∂aij

daij

=

i·j

lnf(A)ijdaij

=trG(A)dA.

We deﬁne

G(A)ii=

∂lnf(A) ∂aij

, G(A)ij=  

∂lnf(A) ∂aij

, G=G(A).

LetP=PIIm⊗Ip, and then the maximum likelihood estimation ofμ,, which is denoted byμˆ,ˆ, satisﬁes

II_m⊗Ip

G·P·Y=II_m⊗Ip

G·Y,

Y

(I–P)G(I–P) +P Ip

Y= .

Namely,

PGP=PG,

(I–P)G(I–P) +P

(I–P) = .

Then we can obtain

PGP=PG,

G+P



(I–P) = .

As a result, we can conclude that

G= –P I+CP,

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SinceG=G,CP=PC. Besides, the -order diﬀerential of a likelihood function needs to satisfy:∀dμ,d,d_ln_L₍_Y_,_μ_,_{) < .}

∵ L=||–mp _f₍_A_),

∴ dlnL= –mp

 tr

–_d₊_d_ln_f₍_A_{) = –}mp

 tr

–_d₊_tr_{G d}₍_A_).

Additionally,

dlnL=mp  tr

–_d–_d₊_tr_G_·_d_A₊_tr₍_dG₎_dA

=tr

mp

 +

–_Y_–_μ_II

m⊗Ip

Y–μII_m⊗Ip

–d–d

+trG(IIm⊗Ip)dμ(m)–dμ

II_m⊗Ip

+tr(dG)dA

=trG(IIm⊗Ip)dμ(m)–dμ

II_m⊗Ip

+tr(dG)dA

= –mp  trdμ

₍_m₎–_d_μ₊_tr_C_·_P_(II

m⊗Ip)dμ(m)–dμ

II_m⊗Ip

+tr(dC)·P dA

= –mp  trdμ

₍_m₎–_d_μ₊_tr_II

m⊗Ip

·C·(IIm⊗Ip)·dμ(m)–dμ

+tr(dC)·P dA.

Consequently, we have the conclusion as follows.

LetY∼En×mp(μ·(IIm⊗Ip),m), and then the maximum likelihood estimations ofμ, areY,Sif and only if

() G= –P_I+CP,CP=PC.

() tr(II_m⊗Ip)·C·(IIm⊗Ip)·dμ(m)–dμ+tr(dC)·P dA<mp_ trdμ(m)–dμ ∀dμ,d.

SinceY∼En×mp(μ·(IIm ⊗Ip),m) if and only ifC= ,Ymay not be a normal random matrix in general.

For instance, letC=cI,c= –p_trPA,P=PIIm⊗Ip, andA= (Y–μ·(IIm⊗Ip))(m)–(Y– μ·(II_m⊗Ip)).

Then the distribution density ofYis

L(Y,μ,) =C||–mp _exp _–p



trA+trPA,

whereC= (||–mp _exp{_–p

(trA+trPA)}dY)–.

Since G= –P_I +CP and CP=PC, and (mtrdμ(m)–_d_μ_)(–p

trPA) + (trP dA)×

(–p_trP dA) <  <mp_ trdμ(m)–_d_μ_{so that the maximum likelihood estimations of}_μ_,

areY,S, then the distribution ofYis not a normal distribution.

4 Conclusion

In this paper, we proved that if the distribution of ann×mprandom matrixY is a left ellipsoidal distribution with parameterμ∗=μ(II_m⊗Ip),∗=m, andY,Y, . . . ,Ymare independent and identical distributions, andY∼Nn×mp(μ∗,∗), then the maximum like-lihood estimations ofμ,areY,S_{. If}_Y_,_Y_{, . . . ,}_Y

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distributions, then the maximum likelihood estimations ofμ, areY,S_{. We used the}

matrix diﬀerential method to deduce that only and only ifY∼En×mp(μ·(IIm⊗Ip),m), which needs to satisfy two conditions.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript, read and approved the ﬁnal manuscript.

Author details

1_{College of Mathematics and Information Science, Henan Normal University, Xinxiang, 453007, China.}2_{School of Finance}

and Statistics, East China Normal University, Shanghai, 200062, China.

Acknowledgements

This work is supported by the Foundation and Frontier Technology Research Project of Henan Province of China (102300410264) and the Education Department Fundamental Research Project of Henan Province of China (2010A110010). The authors would like to thank the anonymous referees for several useful interesting comments and suggestions about the paper.

Received: 22 March 2013 Accepted: 15 October 2013 Published:08 Nov 2013 References

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Cite this article as:Wang and Zhang:The maximum likelihood estimations for a type of left ellipsoidal distribution.