On ordered normally distributed vector parameter estimates

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Eprints ID: 16657

To cite this version:

Chaumette, Eric and Vincent, François and Besson, Olivier On

ordered normally distributed vector parameter estimates. (2015) Signal Processing, vol.

115. pp. 20-26. ISSN 0165-1684

On ordered normally distributed vector parameter estimates

Eric Chaumette, François Vincent, Olivier Besson

University of Toulouse-ISAE, Department of Electronics, Optronics and Signal, 10 Avenue Edouard Belin, 31055 Toulouse, France

Keywords:

Multivariate normal distribution Order statistics

Mean square error

Maximum likelihood estimator

a b s t r a c t

The ordered values of a sample of observations are called the order statistics of the sample and are among the most important functions of a set of random variables in probability and statistics. However the study of ordered estimates seems to have been overlooked in maximum-likelihood estimation. Therefore it is the aim of this communication to give an insight into the relevance of order statistics in maximum-likelihood estimation by providing a second-order statistical prediction of ordered normally distributed estimates. Indeed, this second-order statistical prediction allows to refine the asymptotic perfor-mance analysis of the mean square error (MSE) of maximum likelihood estimators (MLEs) of a subset of the parameters. A closer look to the bivariate case highlights the possible impact of estimates ordering on MSE, impact which is not negligible in (very) high resolution scenarios.

1. Introduction

The ordered values of a sample of observations are called the order statistics of the sample: if

θ

¼

θ

1;

θ

2; …;

θ

MT is a

vector1 _{of M real valued random variables, then}

_θ

M ð Þ¼

θ

ð Þ1;

θ

ð Þ2; …;

θ

ð ÞM

T

denotes the vector of order statistics induced by

θ

where

θ

ð Þ1r

θ

ð Þ2r⋯r

θ

ð ÞM [1,2]. Order statis-tics and extremes (smallest and largest values) are among the

most important functions of a set of random variables in probability and statistics. There is natural interest in studying the highs and lows of a sequence, and the other order statistics help in understanding the concentration of probability in a distribution, or equivalently, the diversity in the population represented by the distribution. Order statistics are also useful in statistical inference, where estimates of parameters are often based on some suitable functions of the order statistics vector (robust location estimates, detection of outliers, cen-sored sampling, characterizations, goodness of fit, etc.). How-ever the study of ordered estimates seems to have been

overlooked in maximum-likelihood estimation[4], which is

at first sight a little bit surprising. Indeed, if x denotes the

random observation vector and p x
_;

Θ

 denotes the

prob-ability density function (p.d.f.) of x depending on a vector of P

real unknown parameters

Θ

¼

_Θ

1; …;

Θ

P



to be estimated, then many estimation problems in this setting lead to estima-tion algorithms yielding ordered estimates b

θ

ð ÞM induced by a vector b

θ

of M estimates formed from a subset of the whole set

of P estimates b

Θ

. Among all various possible instances of

this setting, the most studied in signal processing is that of separating the components of data formed from a linear superposition of individual signals and noise (nuisance). For the sake of illustration, let us consider the following simplified

http://dx.doi.org/10.1016/j.sigpro.2015.02.026

E-mail addresses:eric.chaumette@isae.fr(E. Chaumette),

francois.vincent@isae.fr(F. Vincent),olivier.besson@isae.fr(O. Besson).

1

The notational convention adopted is as follows: italic indicates a scalar quantity, as in a; lower case boldface indicates a column vector quantity, as in a; upper case boldface indicates a matrix quantity, as in A. The n-th row and m-th column element of the matrix A will be denoted by An;mor Að Þ_n;m. The n-th coordinate of the column vector a will be denoted by an or að Þn. The matrix/vector transpose is indicated by a

superscript T as in AT_{. For two vectors a and b, aZb means that ab is}

positive componentwise. MRðN; PÞ denotes the vector space of real

matrices with N rows and P columns. 1m:m þ lM denotes the M-dimensional

vector with all components set to 0 except components from m to m þ l set to 1. 1Mdenotes the M-dimensional vector with all components set to

1. IMARMM denotes the identity matrix. 1f gA denotes the indicator function of the event A. E_Θ½g xð Þ ¼Rg xð Þp x;
Θdx denotes the statistical expectation of the vector of functions gð Þ with respect to x parameterized byθ.

example:

xt


Θ

¼ A


θ

stþnt;

Θ

T¼

θ

T; sT1; …; sTT

 T

ð1Þ

where 1rt rT, T is the number of independent observations,

xtis the vector of samples of size N, M is the number of signal

sources, st is the vector of complex amplitudes of the M

sources for the tth observation, A


θ

¼ a


_θ

1 ; …; a

θ

Mand að Þ is a vector of N parametric functions depending on a single parameter

θ

, ntare Gaussian complex circular noises

indepen-dent of the M sources. Since(1)is invariant over permutation of signal sources, i.e. for any permutation matrix PiARMM: xt


Θ

¼ A



θ

PiðPistÞþnt;

it is well known that (1) is an ill-posed unidentifiable

estimation problem which can be regularized, i.e. trans-formed into a well-posed and identifiable estimation pro-blem, by imposing the ordering of the unknown parameters

θm

:

θ

9

θ

1; …;

θ

M



;

θ

1o⋯o

θ

M; and of their estimates as well: b

θ

9 b

θ

ð ÞM. Therefore in the MSE sense, the correct

statistical prediction is given by the computation of E_Θ

½ðb

_θ

ð Þm

θ

mÞ2; 1rmrM. Unfortunately, the correct statistical prediction cannot be obtained from scratch since most of the available results in the open literature on order statistics[1–

3]have been derived the other way round, i.e. they request

the knowledge of the distribution of b

θ

. The distribution of b

θ

can be obtained from a priori information on the problem at hand or may have been derived in some regions of operation of the observation model. For instance, it has been known for a while that, under reasonably general conditions on the observation model[4,7], the ML estimates are asymptotically Gaussian distributed when the number of independent observation tends to infinity. Additionally, if the observation model is linear Gaussian as in(1), some additional asymptotic regions of operation yielding Gaussian MLEs have also been identified: at finite number of independent observations[5–

9] or when the number of samples and the number of

independent observations increase without bound at the

same rate, i.e. N; T-1, N=T-c, 0oco1[10]. Nevertheless

a close look at the derivations of these results reveals an implicit hypothesis: the asymptotic condition of operation considered yields resolvable estimates[11,12], what prevents from estimates re-ordering. Therefore, under this implicit hypothesis b

θ

ð ÞM ¼ b

θ

. However when the condition of opera-tion degrades, distribuopera-tion spread and/or locaopera-tion bias of each b

θ

m increase and the hypothesis of resolvable estimates does

not hold any longer yielding observation samples for which b

θ

ð ÞMa b

θ

[11,12].

Therefore it is the aim of this communication to give an insight into the relevance of order statistics in maximum-likelihood estimation by providing a second-order statistical prediction of ordered normally distributed estimates. This second-order statistical prediction allows to refine the asymp-totic performance analysis of the MSE of MLEs of a subset

θ

of the parameters set

Θ

.2_{Indeed, in the setting of a multivariate}

normal distribution with mean vector

μ

_b

θ and covariance

matrix _Cb

θ, b

θ

 NM

μ

bθ; Cbθ

 

, with p.d.f. denoted

p_NMðb

_θ

;

μ

_{bθ; Cbθ}Þ, the most general statistical characterization, i. e. including distribution and moments, have been derived for an exchangeable multivariate normal random vector[1,13,14],

that is a normal distribution with a common mean

μ

, a

common variance

σ

2 _{and a common correlation coefficient}

ρ

: b

θ

 NM

μ

1M;

σ

2
1 

ρ

IMþ

ρ

1M1TM

 

 

, with

ρ

A 0; 1½½ . If the focus is on distribution, then the most general result has been released recently in[3]where the exact distribution of linear combinations of order statistics (L-statistics) [15] of arbitrary dependent random variables has been derived (see also[16]for the joint distribution of order statistics in a set of univariate or bivariate observations). In particular,[3]examines the case where the random variables have a joint elliptically contoured distribution and the case where the random vari-ables are exchangeable. Arellano-Vallea and Genton[3] inves-tigate also the particular L-statistics that simply yield a set of order statistics, and study their joint distribution. Unfortu-nately, general derivations of closed form expressions for moments and cumulants of L-statistics were beyond the scope

of [3] and were left for future research. However in the

particular case of a multivariate normal distribution, it is possible to obtain closed forms for first and second order moments of its order statistics directly, i.e. without explicitly computing the order statistics distribution (see Section 2). These closed forms not only generalize the earlier work from the exchangeable case to the general case providing a second-order statistical prediction of L-statistics from multivariate normal distribution but are also required to characterize the MSE of normally distributed vector parameter estimates. Indeed, since it is always assumed that

θ

has distinct

compo-nents, any sensible estimation technique of

θ

must preserve

this resolvability requirement and yield distinct mean values, leading to asymptotically non-exchangeable multivariate nor-mal random vector.

2. Second-order statistical prediction of ordered normally distributed estimates

First, note that b

θ

ð ÞMAPerðb

θ

Þ, where Perðb

θ

Þ ¼ fb

θ

i¼ Pi

θ

b; i ¼ 1; …; M!g is the collection of random vectors b

θ

i

corre-sponding to the M! different permutations of the components

of b

θ

. Here PiARMM are permutation matrices with PiaPj for all iaj. Let

Δ

ARðM  1ÞM _{be the difference matrix such} that

Δθ

¼

_θ

2

θ

1;

θ

3

θ

2; …;

θ

M

θ

M  1
T , i.e., the mth row of

Δ

is dTm þ 1d T m, m ¼ 1; …; M 1, where d1; …; dM

are the M-dimensional unit basis vectors. Let Si¼ fb

θ

:

Δ

b

_θ

_{iZ0g where b}

_θ

_{i NM}

_μ

i; Ci



,

μ

i¼ Pi

μ

b_θ, Ci¼ Pi Cb_θPTi.

Let P Dð Þ be the probability of an event D. As the set of

eventsf gSi Mi ¼ 1! is a partition ofRM, whatever the real valued function fð Þ, by the theorem of total probability we have; E f b

θ

ð Þm; b

θ

ð Þl   h i ¼X M! i ¼ 1 E f b

θ

ð Þm; b

θ

ð Þl   jSi h i P Sð Þi 2

Note that these results are also applicable to other estimators, such as M-estimators, Bayesian estimators (MAP, MMSE), as long as their distribution is normal.

that is E f b

θ

ð Þm; b

θ

ð Þl   h i ¼X M! i ¼ 1 E f

θ

bi   m; b

θ

i   l   jSi h i P Sð Þ:i ð2Þ

However, from a computational point of view, it is wiser to express(2)as E f b

θ

ð Þm; b

θ

ð Þl   h i ¼XM! i ¼ 1 E f

θ

bi   m; b

θ

i   l   jUi h i P Uð Þi ð3Þ where Ui¼ ubi: buiZ 

Δμ

i  and ui¼

Δ

θ

bi

μ

i    NM  1 0;

Δ

Ci

Δ

T  

. Then a smart exploitation of(3)(see Appendix

for details) yields E b

θ

ð Þm h i ¼X M! i ¼ 1

α

mjiPiþ

β

T mjiei   ð4Þ E b

θ

2ð Þm  ¼X M! i ¼ 1

σ

2 mjiþ

α

2 mji   Piþ2

α

mji

β

T mjieiþ

β

T mjiRi

β

mji   ð5Þ E bθðm þ lÞθbð Þm h i ¼1 2 E bθ 2 m þ l ð Þ  þE bθ2ð Þm  E bθðm þ lÞ bθð Þm  2   ð6Þ E θbðm þ lÞ bθð Þm  2  ¼ 1m:m þ l  1 M  1  T XM! i ¼ 1 ΔμiΔμi
T Piþ 2eiΔμi
T_þR i 0 @ 1 A 0 @ 1 A1m:m þ l  1 M  1 ð7Þ Pi¼ P Uð Þ; ei i¼ E ui1Ui   ; Ri¼ E uið ÞuiT1Ui h i ð8Þ where lZ1,

α

mji¼ dTm

μ

i,

μ

i¼ Pi

μ

_b θ, Ci¼ PiCbθP T i,

β

mji¼ ð

_Δ

Ci

Δ

T Þ 1

_Δ

_C idm,

σ

2_mji¼ dTmðCb_θCi

Δ

T

Δ

Ci

Δ

T   1

Δ

CiÞ dm. Finally, the second order statistical prediction of any L-statistics, that is linear combinations of the vector of order statistics,bz ¼ Lb

θ

ð ÞM, LAMRðN; MÞ, is given by

μ_bz¼ Lμb_θð ÞM; Cbz¼ LCbθð ÞM LT_; μ_bθ M ð Þ ¼ E bθð ÞM h i Cbθð ÞM ¼ E bθð ÞMθb T M ð Þ  E bθð ÞM h i E bθð ÞM h iT    

and can be computed from expressions(4)–(7) of E b

θ

ð Þm

h i , E b

θ

ðm þ lÞb

θ

ð Þm h i , E b

θ

2ð Þm 

when the L-statistics derive from multivariate normal distribution.

2.1. Cramér–Rao bound for ordered normally distributed

estimates

Let CRB_Θ


θ

denote the Cramér–Rao bound (CRB) for

unbiased estimates of

θ

[4,7]. Then (4) allows for the

computation of the ordered estimates bias b_Θ


θ



m¼

E_Θ

θ

bð Þm

h i



θ

m which may be different from the (a priori)

theoretical bias
b_Θ


θ

_m¼ E_Θ

θ

bm

h i



θ

m, leading to the

CRB for ordered estimates given by[4,7]

CRBb_Θ


θ

¼ b_Θ


_θ

bT_Θ


θ

þ IMþ∂ b_Θ


θ

∂

θ

T !  CRB_Θ


_θ

IMþ∂ b_Θ


θ

∂

θ

T !T ; ð9Þ

which may be different from the (a priori) theoretical CRBb_Θ


θ

.

2.2. Implementation

Let Di be the diagonal matrix such that ð ÞDim;m¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð

_Δ

Ci

Δ

T Þ_m;m; q

, m ¼ 1; …; M 1. The computational cost of

(4)–(7) can be reduced by reformulatingPi, ei, Ri (8)as Pi¼ P Vð Þ; ei i¼ DiEbvijVi   ; Ri¼ DiE bvibv T ijVi h i Di; ð10Þ where Vi¼ bvi: bviZ Di 1

Δμ

i n o and bvi NM  1ð0; Di 1 ð

_Δ

Ci

Δ

T ÞD 1

i Þ is a vector of correlated standard normal

random variables satisfyingP bvi

 m

 _Z10



o10 23_{. Thus,} from a numerical point of view, the distribution support of each bvi



m can be restricted to the interval ½10; 10

leading to E bvi
 mjVi    _{r10P V}ð Þ and E_i bvi
 m bvi
 ljVi    _r 102 P Vð Þ. Therefore, sincei P Vð Þri min 1r m r M  1 P bvi
 mZ  D  1 i

Δμ

i   m   n o ; from a numerical point of view:

 if (mA 1; M 1½ j D 1 i

Δμ

i   mr 10, then: P Vð Þ ¼ 0; E bvi ijVi   ¼ 0; E bvibv T ijVi h i ¼ 0;  if 8mA 1; M 1½ ; D 1 i

Δμ

i   mZ10, then: P Vð Þ ¼ 1; E bvi  ijVi¼ E bvi ¼ 0; ptE bvibv T ijVi h i ¼ E bvibv T i h i ¼ D 1 i

Δ

Ci

Δ

T   Di 1:

In any other case, P Vð Þ, Ei bvijVi

  and E bvibv T ijVi h i can be

computed by resorting to algorithms proposed by Genz[17]

for numerical evaluation of multivariate normal

distribu-tions and moments over domains included in  10½ ; 10M

. 3. The bivariate (two sources) case

Let C_9Cb θ,

μ

9

μ

bθ, d

μ

¼

μ

2

μ

1, and db

θ

¼ u1¼ b

θ

2 b

θ

1. As M¼2 then u2¼ u1,

Δμ

1¼ d

μ

¼ 

Δμ

2, P1þP2¼ 1, 111:1¼ 1, e1e2¼ E u½  ¼ 0, R1 1þR2¼ E u21   ¼

_σ

2 dbθ,

Δ

C1

Δ

T ¼

Δ

C2

Δ

T ¼

_σ

2 dbθ¼ C1;1þC2;22C1;2. Moreover, if C ¼

σ

2_{ðð1 }

ρ

ÞI2þ

ρ

121T2Þ, then

σ

dbθ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2

σ

2
_{1 }

ρ

 q

and a few additional calculations yield E b

θ

ð Þm h i ¼ E b

_θ

m h i þ 1ð Þm hð Þ

τ

_σ

db_θ; E b

θ

2ð Þm  ¼ E b

_θ

2m  þ 1ð Þm hð Þ

τ

_μ

2þ

μ

1


σ

_dbθ; Var b

θ

ð Þm h i ¼ Var b

θ

m h i 

σ

2 dbθhð Þ

τ

ð

τ

þhð Þ

τ

Þ

MSE b

θ

ð Þm h i ¼ Var b

_θ

m h i 

_σ

2 dbθ

τ

hð Þ;

τ

ð11Þ where

τ

¼ d

_μ

=

σ

_dbθ, h yð Þ ¼ E v1fvZ yg h i

yP vZyð Þ and

MSE b

θ

ð Þm h i 9E b

θ

ð Þm 

μ

m  2  . An interesting feature is the MSE shrinkage factor:

ξ τ

;

ρ

¼MSE b

θ

ð Þm

h i

Var b

θ

m

h i ¼ 12 1

ρ



τ

hð Þ:

τ

ð12Þ

The computation of the derivatives vector∂

ξ τ

;

ρ

=∂

τ

;

ρ



shows that, for a given

ρ

,

ξ τ

;

ρ

admits a global minimum

which is the solution of E v1fvZ yg

h i

2yP vZyð Þ ¼ 0, y ¼

τ

=qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 1 

ρ

, that is, after a numerical resolution, yC0:61 and leading to

τ

C0:863pffiffiffiffiffiffiffiffiffiffiffi1 

ρ

,

ξ τ

;

ρ

Z10:2

1 

ρ



Z0:6 (seeFig. 1). Therefore the minimum of

ξ τ

;

ρ

 is 0.6 ( 2.2 dB) and is reached for couples

τ

;

ρ

belonging to the set

τ

;

ρ

-1þ_j

_τ

C1:22



. This result is applicable

to any instantiation of (1) for which b

θ

is normal with

Cb_θ¼

σ

2
_{1 }

ρ

_I

2þ

ρ

121T2

 

, as for example, the asymptotic

−50 −45 −40 −35 −30 −25 −20 −15 −10 −5 0 5 10 15 20 −2.3 −2.2 −2.1 −2 −1.9 −1.8 −1.7 −1.6 −1.5 −1.4 −1.3 −1.2 −1.1 −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 (dB) τ (dB)

Mean Square Error shrinkage factor

ρ = −1 ρ = −0.5 ρ = 0 ρ = 0.5 ρ = 1

behaviour of the MLEs of the direction of arrival (DoA) of two sources of equal power impinging on a uniform linear array (ULA) with symmetric DoAs relative to boresight, or two cisoids (tones) of equal power with opposite frequen-cies, whatever the observation model is conditional or

unconditional[5]. Indeed in these two cases CRB_Θ


θ

¼

σ

2
_{1 }

_ρ

_I

2þ

ρ

121T2

 

[4,5,7]and a priori, a violation of the CRB not attributable to the variance of the empirical MSE

is expected in some scenarios, unless the ad hoc CRB(9)is

computed. However, a closer look to Fig. 1 reveals that

τ

¼ d

_μ

=

σ

_dbθ must be inferior to 1015=20C6 in order to

exhibit a measurable MSE shrinkage effect. As the normality of estimates and convergence to the CRB is obtained only

in asymptotic conditions where

σ

dbθ≪1, this means that d

θ

¼ d

_μ

≪1 as well, that is the scenario considered is a (very) high resolution scenario. To highlight the possible impact of estimates ordering on MSE in (very) high resolution scenarios,

we consider the estimation of two tones M ¼ 2ð Þ of equal

power with opposite frequencies where the observation model (1) is deterministic: a


θ

T_{¼ 1; e}
j2πθ_{; …; e}j2πðN  1Þθ_Þ, N¼8, T¼2, d

θ

¼ 1=12N, Cnt¼ I2, Cst¼ SNR=N
 1 þ1 8
 I2
1

8121T2Þ where SNR is the signal to noise ratio measured at the

20 25 30 35 40 45 50 55 60 65 −75 −70 −65 −60 −55 −50 −45 −40 −35 −30 −25 −20 −15

Signal to Noise Ratio (dB)

dB

Mean Square Error of MLEs

Simu CRB Theo

output of the frequency matched filter. Fig. 2 displays the

empirical and theoretical MSE of ordered estimates (11)

averaged over the two frequencies (since equal by symmetry) and the associated unbiased CRB.Fig. 3displays the empirical

and theoretical MSE shrinkage factor(12)averaged over the

two frequencies as well. The empirical MSE are assessed with 104_{Monte-Carlo trials and a frequency step}

_δθ

_{¼ 1=12N}
1

64. As expected there is a perfect match between the theoretical results and the empirical results in the asymptotic region

(SNR Z 55 dB) where Gaussianity is reached by MLEs,

highlighting the non-negligible impact ( 0.6 dB) of estimates ordering on high resolution scenarios. As the SNR values enter

the threshold area, the MLEs start to explore the side lobe peaks of the matched filter producing outliers and loosing

gradually their Gaussian distribution [4]. Because of this

phenomenon, as the SNR values enter the threshold area, the results derived in this section must be regarded as an approximation which accuracy is somewhat difficult to quan-tify in general. Indeed, it would require to recompute the second-order statistical prediction of ordered estimates taking into account the probability of outlier[4], which is far beyond the scope of this communication. We can simply notice, but without generality, that in our example, the approximation is

quite good for SNRZ30 dB.

20 25 30 35 40 45 50 55 60 65 −2 −1 0 1 2 3 4 5 6 7 8 9 10

Signal to Noise Ratio (dB)

dB

Ratio Mean Square Error/CRB of MLEs

Simu Theo

4. Conclusion

In this communication we have provided a second-order statistical prediction of second-ordered normally distributed estimates which allows to refine the asymptotic

perfor-mance analysis of the MSE of MLEs of a subset

θ

of the

parameters set

Θ

. Analytical expressions of the MSEs of

the whole parameters set

Θ

after re-ordering of the subset

θ

should be the next topic to be addressed.

Appendix As the vector b

ξ

m;i¼ ððb

θ

iÞm; bu T iÞ T_{, bu} i¼

Δ

ðb

θ

i

μ

iÞ  NM  1ð0;

Δ

Ci

Δ

T

Þ, is a M-dimensional normal vector result-ing from a bijective affine transformation of b

θ

i, then we

have[1]

p_NMbξm;i;μm;i; Cm;i¼ pN1 bθi

  mjbui;μmji;σ 2 mji   p_{NM  1} bui; 0;ΔCiΔ T   where

μ

mji¼ d T m

μ

iþCi

Δ

T

Δ

Ci

Δ

T   1 bui  and

σ

2 mji¼ d T m Cb_θCi

Δ

T

_Δ

Ci

Δ

T   1

Δ

Ci  dm. Let Ui¼ ubi: buiZ 

Δμ

i  , then whatever the real valued function f ð Þ:

E f

θ

bi   m   jSi h i ¼ E f b

_θ

i   m   jUi h i ¼ E E f b

_θ

i   m   jbui h i jUi h i

In the particular cases where f b

θ

ð Þm

  ¼ b

θ

kð Þm; kA 1; 2f g: E

θ

bi   mjbui h i ¼

_μ

mji¼

α

mjiþ

β

T mjiubi E

θ

bi  2 mjbui  ¼

_σ

2 mjiþ

μ

2 mji¼

σ

2mjiþ

α

2 mji þ2

_α

mji

β

T mjibuiþ

β

T mjiubibu T i

β

mji

where

α

mji¼ dTm

μ

i,

β

mji¼

Δ

Ci

Δ

T   1

Δ

Cidm, and (2) can finally be expressed as E b

θ

ð Þm h i ¼X M! i ¼ 1

α

mjiP Uð Þþi

β

T mjiEubi1Ui     E b

θ

2ð Þ_m  ¼XM! i ¼ 1

σ

2 mjiþ

α

2 mji   P Uð Þþ2i

α

mji

β

T mjiEubi1Ui    þ

β

TmjiE ubi
bui T 1_Ui h i

β

mji  ð13Þ From(2), 8 mA 1; M½ , 8ljmþlA 1; M½ : E

θ

bðm þ lÞ b

θ

ð Þm  2  ¼X M! i ¼ 1 E

θ

bi   m þ l b

θ

i   m  2 jSi " # P Sð Þi Additionally, 8 lZ1jmþlA 1; M½ : b

θ

i   m þ l b

θ

i   m¼ 1 m:m þ l  1 M  1  T

Δ

b

_θ

_i therefore b

θ

i   m þ l b

θ

i   m  2 ¼ 1m:m þ l  1 M  1  T

Δ

b

_θ

_i

_Δ

b

_θ

T i1m:m þ l  1M  1 Finally E b

θ

ðm þ lÞ b

θ

ð Þm  2  ¼X M! i ¼ 1 1mM  1:m þ l  1  T  E

Δ

θ

bi

Δ

θ

b T ijSi  1m:m þ l  1M  1 P Sð Þi ð14Þ Last, as E

Δ

θ

bi

Δ

θ

b T ijSi  ¼ E buibu T ijUi h i þEbuijUi

Δμ

i
T þ

_Δμ

iE buijUi  T þ

_Δμ

i

Δμ

i
T an equivalent expression of(14)is(7). References

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