On ordered normally distributed vector parameter estimates

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On ordered normally distributed vector parameter estimates

Eric Chaumette, François Vincent, Olivier Besson

To cite this version:

Eric Chaumette, François Vincent, Olivier Besson. On ordered normally distributed vector parameter estimates. Signal Processing, Elsevier, 2015, vol. 115, pp. 20-26. �10.1016/j.sigpro.2015.02.026�.

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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

To link this article: http://dx.doi.org/10.1016/j.sigpro.2015.02.026

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 ^{M T is a}

vector

¹

of M real valued random variables, then θ

^{ð Þ}

^{M ¼} θ

^{ð Þ1}

; θ

^{ð Þ2}

; …; θ

^{ð Þ}

^M

T

denotes the vector of order statistics induced by θ ^where θ

ð Þ1

r θ

ð Þ2

r ⋯r θ

^{ð Þ}

^M [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 θ b

^{ð Þ}

^M induced by a vector θ b ^{of M} estimates formed from a subset of the whole set of P estimates Θ b . 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 A

n;m

or ð Þ A

_n;m

. The n-th coordinate of the column vector a will be denoted by a

n

or ð Þ a

_n

. The matrix/vector transpose is indicated by a superscript T as in A

^T

. For two vectors a and b, a Zb means that a b is positive componentwise. M

R

ð N; P Þ denotes the vector space of real matrices with N rows and P columns. 1

^m:mþ_M ^l

denotes the M-dimensional vector with all components set to 0 except components from m to m þl set to 1. 1

M

denotes the M-dimensional vector with all components set to 1. I

M

AR

^MM

denotes the identity matrix. 1

f gA

denotes the indicator function of the event A. E

_Θ

½ g x ð Þ ¼ R

g x ð Þp x; Θ ^dx denotes the statistical

expectation of the vector of functions gð Þ with respect to x parameterized

by θ ^.

example:

x t Θ ^{¼ A} θ ^{s t þn t} ; Θ ^{T ¼} θ ^T ; s ^T

₁

; …; s ^T _T T

ð1Þ where 1 r t r T, T is the number of independent observations, x t is the vector of samples of size N, M is the number of signal sources, s

t

is the vector of complex amplitudes of the M sources for the tth observation, A θ ^{¼ a} θ

¹

; …; a θ ^{M and}

að Þ is a vector of N parametric functions depending on a single parameter θ ^{, n}

t

are 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 P i AR ^MM : x t Θ ^{¼ A} θ ^{P i ð P i s t Þþn t} ;

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

θ

m

: θ 9 θ

¹

; …; θ ^M ; θ

¹

o⋯ o θ ^M ; and of their estimates as well: θ b 9 θ b

^{ð Þ}

^M . Therefore in the MSE sense, the correct statistical prediction is given by the computation of E _Θ

½ð b θ

^{ð Þ}

^m θ ^{m Þ}

²

; 1 r m r M. 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 θ b . The distribution of θ b 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, 0 o c o 1 [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 Þ}

^{M ¼} θ ^b . However when the condition of opera- tion degrades, distribution spread and/or location bias of each

θ b ^m increase and the hypothesis of resolvable estimates does not hold any longer yielding observation samples for which

θ b

^{ð Þ}

^M a 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 C b _θ , b θ N M μ _{b θ} ; C b _θ

, with p.d.f. denoted p

_NM

ð b θ ; μ _{b θ} ; C b _θ 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 N ^M μ ^{1 M} ; σ

²

¹ ρ ^{I M þ} ρ ^{1 M 1 T} M

, with ρ A ½ 0 ; 1½ . 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 θ b

^{ð Þ}

^M A Perð θ b Þ, where Perð b θ ^{Þ ¼ f} θ ^{b i ¼ P i} θ ^b ; i ¼ 1 ; …; M ! g is the collection of random vectors θ b ^{i corre-} sponding to the M ! different permutations of the components of θ b ^{. Here P} i AR ^MM are permutation matrices with P _i a P _j for all i a j. Let Δ AR

^ð

^M1

^ÞM

be the difference matrix such that Δθ ^¼ θ

²

θ

¹

; θ

³

θ

²

; …; θ ^M θ ^{M1 T} , i.e., the mth row of Δ ^{is d T} m

þ1

d ^T _m , m ¼ 1 ; …; M 1, where d

1

; …; d M

are the M-dimensional unit basis vectors. Let S i ¼ f θ b :

Δ ^b θ i Z 0g where θ b i N ^M μ i ; C i

, μ i ¼ P i μ _{b θ} ^{, C} i ¼ P i C b _θ P ^T _i . Let P D ð Þ be the probability of an event D . As the set of events f g S i ^M _i

_¼^!₁

is a partition of R ^M , whatever the real valued function f ð Þ, by the theorem of total probability we have ; E f ^h θ b

^{ð Þ}

^m ; b θ

^{ð Þ}

^{l i ¼ X M}

^!

i

¼1

E f b θ

^{ð Þ}

^m ; θ b

^{ð Þ}

^{l j} S i

h i

P S ð Þ i

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 ^h θ b

^{ð Þ}

^m ; θ b

^{ð Þ}

^{l i ¼ X M}

^!

i

¼1

E f θ b ⁱ _m ; θ b ⁱ _l ^j S i

h i

P S ð Þ i : ð2Þ

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

E f θ b

^{ð Þ}

^m ; θ b

ð Þ

l

h i

¼ X ^M!

i

¼1

E f θ b i m ; θ b i

l

j U i

h i

P U ð Þ i ð3Þ

where U i ¼ u b i : u b i Z Δμ i

and u i ¼ Δ θ ^{b i} μ i

N ^M1 0 ; Δ ^{C i} Δ ^T

. Then a smart exploitation of (3) (see Appendix for details) yields

E ^h θ b

^{ð Þ}

^{m i ¼ X M}

^!

i

¼1

α ^mji P i þ β ^T mji e i

ð4Þ

E b θ

²ð Þ

m

¼ X ^M

^!

i

¼1

σ

²mji

þ α

²

mji

P i þ2 α ^mji β ^T mji e i þ β ^T mji R i β mji

ð5Þ

E ^h θ b

ðmþlÞ

θ b

^{ð Þm}

^{i ¼ 1} ₂ ^{E b} θ

²ðmþlÞ

þE θ b

²ð Þm

E b θ

ðmþlÞ

θ b

^{ð Þm}

²

ð6Þ

E θ b

ðmþlÞ

θ b

ð Þm

2

¼ 1

^m:mþl1_M1

T

X

^M!

i¼1

Δμ

i

Δμ

i

T

P

i

þ 2e

_i

Δμ

i

T

þR

_i

0

@

1 A 0

@

1 A1

^m:mþl1_M1

ð7Þ P i ¼ P U ð Þ i ; e i ¼ E u i 1

_Ui

; R i ¼ E h u i ð Þ u i ^T 1

_Ui

i

ð8Þ where l Z 1, α ^{mji ¼ d T} m μ i , μ i ¼ P i μ _{b θ} ^{, C i ¼ P i C} _{b θ} ^{P T} i , β mji ¼ ð Δ ^{C i} Δ ^{T Þ}

¹

Δ ^{C i d m ,} σ

²mji

¼ d ^T _m ðC b _θ C i Δ ^T Δ ^{C i} Δ ^T

¹

Δ ^{C i Þ d m .}

Finally, the second order statistical prediction of any L-statistics, that is linear combinations of the vector of order statistics, b z ¼ L θ b

^{ð Þ}

^{M , L} AM

R

ð N ; M Þ, is given by μ _b

_z

^{¼ L} μ _b

_θ

ð ÞM

; C b

z

¼ LC b

_θð ÞM

L

^T

; μ _b

_θ

ð ÞM

¼ E θ b

ð ÞM

h i C b

_{θð ÞM}

¼ E b θ

^{ð ÞM}

θ ^b

^Tð ÞM

E ^h b θ

^{ð ÞM}

^{i E h} θ ^b

^{ð ÞM}

ⁱ

^T

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

^{ð Þ}

^{m i ,} E ^h θ b

ð

m

þlÞ

b θ

^{ð Þ}

^{m i , E} θ ^b

²ð Þ

m

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 _m ^¼

E _Θ ^h θ b

^{ð Þ}

^{m i} θ ^m which may be different from the (a priori) theoretical bias b _Θ θ m ¼ E _Θ ^h θ b ^{m i} θ ^m , leading to the CRB for ordered estimates given by [4,7]

CRB

^b

_Θ θ ^{¼ b} Θ θ ^{b T} Θ θ ^{þ I M þ ∂ b Θ} θ

∂ θ ^T

!

CRB _Θ θ ^{I M þ ∂ b Θ} θ

∂ θ ^T

! T

; ð9Þ

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

^b

_Θ θ ^.

2.2. Implementation

Let D i be the diagonal matrix such that ð Þ D i _m;m ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð Δ ^{C i} Δ ^{T Þ} m;m;

q

, m ¼ 1 ; …; M 1. The computational cost of (4) – (7) can be reduced by reformulating P i , e i , R i (8) as P i ¼ P V ð Þ i ; e i ¼ D i E b v i j V i

; R i ¼ D i E b v i b v ^T _i j V i

h i

D i ; ð10Þ where V i ¼ b v i : b v i Z D _i

¹

Δμ i

n o

and b v i N M

1

ð0 ; D _i

¹

ð Δ ^{C i} Δ ^{T ÞD} i

¹

Þ is a vector of correlated standard normal random variables satisfying P v b i m Z 10

o 10

²³

. Thus, from a numerical point of view, the distribution support of each v b i m can be restricted to the interval ½ 10 ; 10 leading to E v b i m j V i

r 10 P V ð Þ i and E v b i m v b i l j V i

r

10

²

P V ð Þ. Therefore, since i

P V ð Þ i r min

1rmrM1

P v b i m Z D _i

¹

Δμ i

m

n o

;

from a numerical point of view:

if ( m A ½ 1 ; M1 j D _i

¹

Δμ i

m r 10, then:

P V ð Þ ¼ i 0 ; E b v i j V i

¼ 0 ; E b v i b v ^T _i j V i

h i

¼ 0 ;

if 8m A ½ 1 ; M1 ; D _i

¹

Δμ i

m Z 10, then:

P V ð Þ ¼ i 1 ; E b v i j V i ¼ E v b i ¼ 0 ; ptE b v i v b ^T _i j V i

h i

¼ E h v b i v b ^T _i i

¼ D _i

¹

Δ ^C i Δ ^{T D} i

¹

:

In any other case, P V ð Þ, i E b v _i j V i

and E v b _i v b ^T _i j V i

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 ½ 10 ; 10 ^M . 3. The bivariate (two sources) case

Let C 9 C b _θ , μ 9 μ _{b θ} ^{, d} μ ^¼ μ

2

μ

1

, and d θ b ^{¼ u}

¹

^{¼ b} θ

²

θ ^b

¹

^.

As M¼2 then u

2

¼ u

1

, Δμ

1

¼ d μ ^¼ Δμ

2

, P

1

þ P

2

¼ 1, 1

¹₁^:1

¼ 1, e

1

e

2

¼ E u ½ ¼

1

0, R

1

þR

2

¼ E u

²₁

¼ σ

²

d b _θ , Δ ^C

¹

Δ ^{T ¼} Δ ^C

²

Δ ^{T ¼} σ

²

d b _θ ¼ C

_1;1

þC

_2;2

2C

_1;2

. Moreover, if C ¼ σ

²

^{ð ð 1} ρ ^ÞI

²

^þ ρ ¹

²

^{1 T}

2

Þ, then σ _{d b θ} ^¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 σ

²

¹ ρ

q

and a few additional calculations yield

E ^h b θ

^{ð Þ}

^{m i ¼ E h} θ ^{b m i þ ð 1 Þ m h ð Þ} τ σ _{d b θ} ; E θ b

²ð Þ

m

¼ E b θ

²

m

þ 1 ð Þ ^m h ^{ð Þ} τ μ

2

þ μ

1

σ _{d b θ} ; Var ^h θ b

^{ð Þ}

^{m i ¼ Var h} θ ^{b m i} σ

²

d b _θ h ^{ð Þ} τ ^ð τ ^{þh ð Þ} τ ^Þ

MSE ^h θ b

^{ð Þ}

^{m i ¼ Var h} θ ^{b m i} σ

²

d b _θ τ ^{h ð Þ} τ ; ð11Þ

where τ ^{¼ d} μ = σ _{d b θ} ^{, h y ð Þ ¼ E v1} f v

Zy

g h i

y P ð v Z y Þ and MSE ^h θ b

^{ð Þ}

^{m i} 9 E θ b

^{ð Þ}

^m μ m

2

. An interesting feature is the MSE shrinkage factor:

ξ τ ; ρ ^{¼ MSE h b} θ

^{ð Þ}

^{m i}

Var ^h b θ ^{m i ¼ 1 2 1} ρ τ ^{h ð 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 v1 f _v

_Zy

g

h i

2y P ð v Z y Þ ¼ 0, y ¼ τ = ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 1 ρ

q

, that is, after a numerical resolution, y C 0 : 61 and leading to τ C 0 : 863 ffiffiffiffiffiffiffiffiffiffiffi

1 ρ

p , ξ τ ; ρ Z 10 : 2 1 ρ

Z 0 : 6 (see Fig. 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 - 1

^þ

j τ C 1 : 22

. This result is applicable

to any instantiation of (1) for which b θ is normal with C b _θ ¼ σ

²

¹ ρ ^I

2

þ ρ ¹

2

1 ^T

₂

, 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

Fig. 1. MSE shrinkage factor ξ τ; ð ρ Þ versus τ ¼ dμ=σ

_d

b

_θ

for ρ ¼ 1; 0:5; 0; 0:5; 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

²

^þ ρ ¹

²

^{1 T}

2

[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 _{d b θ} must be inferior to 10

¹⁵⁼²⁰

C 6 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 _{d b θ} ≪ 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 ð M ¼ 2 Þ of equal power with opposite frequencies where the observation model (1) is deterministic: a θ ^{T ¼ 1} ; e ^{j2 πθ} ; …;

e ^{j2 π}

^ð

^N

^1Þ

^θ Þ, N¼8, T¼2, d θ ^{¼ 1} = 12N, C

n_t

¼ I

2

, C

s_t

¼ SNR = N

1þ

¹₈

I

2

1

8

1

2

1 ^T

₂

Þ 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

Fig. 2. Average empirical and theoretical MSE (11) of order MLEs of two closely spaced frequencies versus SNR.

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. 3 displays the empirical and theoretical MSE shrinkage factor (12) averaged over the two frequencies as well. The empirical MSE are assessed with 10

⁴

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 SNR Z 30 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

Fig. 3. Average empirical and theoretical MSE shrinkage factor (12) of order MLEs of two closely spaced frequencies versus SNR.

4. Conclusion

In this communication we have provided a second- order statistical prediction of 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

;

i ¼ ðð θ b ^{i Þ} m ; u b ^T _i Þ ^T , u b i ¼ Δ ^{ð b} θ ⁱ μ i Þ N M

1

ð0 ; Δ ^{C i} 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

_NM

b ξ

m;i

; μ

m;i

; C

_m;i

¼ p

_N1

b θ

i

m

jb u

_i

; μ

mji

; σ

²mji

p

_NM1

u b

_i

; 0 ; Δ ^C

i

Δ

^T

where μ mji ¼ d ^T _m μ i þC i Δ ^T Δ ^{C i} Δ ^T

¹

^{u b i and} σ

²mji

¼ d ^T _m C b _θ C i Δ ^T Δ ^C i Δ ^T

¹

Δ ^C i

d m . Let U i ¼ u b i : u b i Z Δμ i

,

then whatever the real valued function f ð Þ : E f θ b ⁱ _m ^j S i

h i

¼ E f b θ ⁱ _m ^j U i

h i

¼ E E f ^h b θ ⁱ _m ^{jb u i i j} U i

h i

In the particular cases where f θ b

^{ð Þ}

^{m ¼} θ ^{b k}

ð Þ

m ; k A f 1 ; 2 g : E ^h θ b ⁱ _m ^{j u b i i ¼} μ mji ¼ α ^{mji þ} β ^T mji u b i

E θ b ⁱ

²

_m ^{jb u i ¼} σ

²mji

þ μ

²

mji ¼ σ

²mji

þ α

²

mji

þ2 α ^mji β ^T mji u b i þ β ^T mji u b i b u ^T _i β ^mji

where α ^{mji ¼ d T} m μ i , β mji ¼ Δ ^{C i} Δ ^T

¹

Δ ^{C i d m , and (2) can}

finally be expressed as E ^h θ b

^{ð Þ}

^{m i ¼ X M}

^!

i

¼1

α ^mji P U ð Þþ i β ^T mji E u b i 1

_Ui

E b θ

²ð Þ

m

¼ X ^M!

i

¼1

σ

²mji

þ α

²

mji

P U ð Þþ2 i α ^mji β ^T mji E u b _i 1

_Ui

þ β ^T mji E u b i u b i T 1

_Ui

h i

β mji

ð13Þ

From (2), 8m A ½ 1 ; M , 8ljmþl A ½ 1 ; M : E θ b

ð

m

þlÞ

θ b

^{ð Þ}

^m

²

^{¼ X M}

^!

i

¼1

E θ b ⁱ _mþl θ ^{b i} _m

²

^j S i

" #

P S ð Þ i

Additionally, 8l Z 1jmþl A ½ 1 ; M : b θ ⁱ _m

_þl

^b θ ⁱ _m ^{¼ 1 m} M1

^:

^mþl

¹

T

Δ ^b θ ⁱ

therefore b θ ⁱ _m

_þl

^b θ ⁱ _m

2

¼ 1 ^m:m _M1

^þl1

T

Δ ^b θ ⁱ Δ ^b θ ^T i 1 ^m:mþl _M1

¹

Finally

E b θ

^ð

^mþl

^Þ

^b θ

^{ð Þ}

^m

²

^{¼ X M}

^!

i

¼1

1 ^m _M1

^:

^mþl

¹

T

E Δ θ ^{b i} Δ θ ^{b T} i j S i

1 ^m:m _M

₁^þl1

P S ð Þ i ð14Þ

Last, as E Δ θ ^{b i} Δ θ ^{b T} i j S i

¼ E b u i u b ^T _i j U i

h i

þE u b i j U i Δμ i

T

þ Δμ i E u b i j U i

T

þ Δμ i Δμ i

T

an equivalent expression of (14) is (7).

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