ieee transactions on signal processing, vol. 58, no. 4...

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 4, APRIL 2010 1967

On the Invariance, Coincidence, and StatisticalEquivalence of the GLRT, Rao Test, and Wald Test

Antonio De Maio, Senior Member, IEEE, Steven M. Kay, Fellow, IEEE, and Alfonso Farina, Fellow, IEEE

Abstract—Three common techniques to discriminate between al-ternatives in a binary hypothesis testing problem are: the general-ized likelihood ratio test (GLRT), the Rao test, and the Wald test. Inthis paper, we investigate some characteristics of the correspondingdecision statistics and provide their expressions for some problemsof particular interest in statistical signal processing. First of all, wefocus on the invariance of the Rao and Wald tests with respect totransformations leaving the testing problem unaltered. Then, weintroduce necessary and sufficient conditions in order for their de-cision statistics to coincide with twice the logarithm of the GLRTstatistic. Finally, we present some detection problems, usually en-countered in practical signal processing applications, where the de-cision variables of the three quoted tests are equivalent, namely re-lated by strictly monotonic transformations.

Index Terms—Detection, GLRT, invariance, Rao test, Wald test.

I. INTRODUCTION

T ESTING binary hypotheses usually arises in several sta-tistical signal processing research fields such as commu-

nication, radar, remote sensing, biomedicine and seismology.The optimality principle adopted to discriminate between thetwo alternatives is the Neyman–Pearson criterion based on themaximization of the detection probability (power of the test)with a constraint on the false alarm probability (type-I error).A Uniformly Most Powerful (UMP) detector rarely exists. It isthus of interest to focus on suboptimal tests such as the gen-eralized likelihood ratio test (GLRT) [1], Rao test [2] and theWald test [3], which are referred to in statistical literature ontesting of hypotheses as the holy trinity. All these tests are equiv-alent to the first-order of asymptotics, but differ to some ex-tent in the second-order properties. Indeed, the aforementionedstatistics can be deemed as different measures of distance be-tween the null and the alternative hypothesis. Moreover, amongthem, the GLRT has been the most commonly employed instatistical signal processing, even if it shares no known opti-mality properties for a finite number of observations. Neverthe-

Manuscript received September 02, 2009; accepted November 02, 2009. Firstpublished December 31, 2009; current version published March 10, 2010. Theassociate editor coordinating the review of this manuscript and approving it forpublication was Dr. Biao Chen.

A. De Maio is with Dipartimento di Ingegneria Biomedica, Elettronica edelle Telecomunicazioni, Università degli Studi di Napoli “Federico II”, I-80125Napoli, Italy (e-mail: [email protected]).

S. M. Kay is with the Department of Electrical and Computer Engineering,University of Rhode Island, Kingston, RI 02881 USA (e-mail: [email protected]).

A. Farina is with Selex Sistemi Integrati, via Tiburtina Km.12.4, I-00131,Roma, Italy (e-mail: [email protected]).

Digital Object Identifier 10.1109/TSP.2009.2039728

less, since there is no particular a priori reason to exploit theGLRT rather than the others, during the last three decades, ex-amples of Rao and Wald tests applied to practical signal pro-cessing detection problems have started to appear in open lit-erature. For instance, in [4], an adaptive detector based on theRao criterion is devised to discriminate the presence of a de-terministic signal, with unknown amplitude, in Gaussian noisewith unknown but structured (AR parameterized) spectra. In [5]and [6], the Rao and Wald tests are respectively derived to de-tect a signal with unknown amplitude in homogeneous Gaussiandisturbance with unknown covariance matrix. The two afore-mentioned tests and the GLRT for the same problem [7] are notequivalent and lead to different performance levels. In [8], theRao test has been applied to the problem of radar space–timeadaptive processing (STAP), while in [9], the Rao and Wald testsare devised with reference to adaptive detection of distributedtargets in non-Gaussian clutter. Finally, in [10], the equivalenceof the GLRT, Rao test, and Wald test is proved for the detectionof a signal, whose amplitude is unknown, in partially homoge-neous environment.

In this paper, we provide some insights into the relevant prop-erties of the three tests and derive their general expressions forsome classical models usually encountered in signal processingapplications. Specifically, we first discuss the invariance prop-erties of the Rao and Wald tests with respect to transformationsgroups leaving the hypothesis testing problem invariant. In thiscontext, we show that the Rao test is always invariant while theWald test requires some mild technical conditions in order to en-sure invariance. Then, we derive necessary and sufficient condi-tions for the coincidence of the Rao, Wald, and twice the loga-rithm of the GLRT decision statistics. Interestingly, the problemof detecting a subspace signal in the presence of subspace inter-ference and Gaussian noise with known variance [11], complieswith those requirements. Moreover, with reference to detectionproblems without nuisance parameters, coincidence arises whenthe probability density function (pdf) of the data belongs tothe exponential family. Finally, we analyze problems where thethree decision statistics are not coincident but can be statisticallyequivalent, namely related to the same statistic by monotonictransformations.

The paper is organized as follows. In Section II, we deal withthe problem formulation; in Section III, we discuss the invari-ance of the tests; and in Sections IV and V, we address, respec-tively, coincidence and equivalence issues, also with referenceto some common models for statistical signal processing. Fi-nally, in Section VI, we draw conclusions and outline some pos-sible future research tracks.

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1968 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 4, APRIL 2010

II. PROBLEM FORMULATION

Let us consider the hypothesis testing problem

(1)

where is the parameter vector, is-dimensional (vector of main parameters), is-dimensional (vector of nuisance parameters), and

is the parameter space.Let be the vector of the observations whose

components are continuous random variables with joint proba-bility distribution . Let be the corre-sponding pdf, assumed to exist and continuously dif-ferentiable with respect to .

Three decision criteria, usually exploited to discriminate be-tween the two alternatives, are the GLRT, Rao test, and Wald test[12], [13]. Their decision statistics are reported in (2), shown atthe bottom of the page, where denotes the gradient op-erator with respect to (column vector), is the transposeof the argument, and are the maximum-like-lihood (ML) estimates of under and , respectively,

is the ML estimate of under . denotes theFisher information matrix (FIM), assumed nonsingular, whichcan be partitioned as

where is -dimensional, and is thematrix formed by the first rows and colums of .

The aforementioned tests are asymptotically equivalent (as) and the Rao test is usually the easiest to compute,

because it only requires ML estimates under the hypoth-esis. However, Rao and Wald tests can only be applied whenthe unknown parameters range in a continuous set becausethe computation of the decision statistics requires derivativeoperations. On the contrary, the GLRT can also be applied withreference to problems where the parameters range in discretesets; in this last case the computation of the GLRT decisionstatistic usually involves a combinatorial optimizationproblem.

Finally, it is worth pointing out that there exist generalizedversions of the Rao test and Wald test based on suitable estima-tors of the FIM and/or resorting to parameter estimates which

are not ML. These are usually exploited in econometric applica-tions (see, for instance, [14] for the analytic expression of somegeneralized Rao and Wald decision statistics).

III. INVARIANCE ISSUES

Denote by a group of transformations (assumed continu-ously differentiable) of the sample space onto itself. Accordingto [15, p. 283], problem (1) remains invariant under if the dis-tributions remain in the same family and the parameter spacesare preserved. Moreover, if the problem is invariant underthen the class of induced transformations in the parameterspace form a group . Assume further that the generic trans-formation can be expressed as

(3)

and is continuously differentiable with Jacobian

(4)

where means block-diagonal,

and

A density relationship, which is a necessary and sufficient con-dition to ensure invariance of problem (1) under the group ,has been derived in [16] and [17], i.e.,1

(5)

where is theJacobian of the transformation, the determinant of asquare matrix, and is the modulus of a complex number(which degenerates into the absolute value if the argument isreal). It implies that

(6)

1In the following, �� we denote by �� the corre-sponding induced transformation in the parameter space.

(2)

DE MAIO et al.: ON THE INVARIANCE, COINCIDENCE, AND STATISTICAL EQUIVALENCE OF THE GLRT, RAO TEST, AND WALD TEST 1969

where and are the inverse transformations of andrespectively (which exist because is a group), and

By definition, a statistic is invariant under a group of trans-formations if and . Itis known that the GLRT decision statistic is invariant under agroup of transformations which leaves the hypothesis testingproblem invariant [15], [16]. Now, in order to gain some insightsinto the invariance properties of the Rao and Wald decision sta-tistics, it is first necessary to introduce the following two tech-nical Lemmas.

Lemma 1: If problem (1) remains invariant under the groupof transformations , then

(7)

Lemma 2: Equivariant property of the ML estimate. Ifproblem (1) remains invariant under the group of transforma-tions , then

The Proof of Lemma 1 is given in Appendix, whereas that ofLemma 2 can be found in [17, p. 258, Prop. 7.11].

Lemma 1 and (4) imply that

(8)

and

(9)

Exploiting the above lemmas, we can prove the following the-orem showing the invariance of the Rao statistic.

Theorem 1: If problem (1) remains invariant under the groupof transformations , then the Rao test decision statistic is in-variant under .

Proof: Let . From(6),

Now, resorting to Lemmas 1 and 2 and using (9), the equalitychain, shown in the equation at the bottom of the page, holdstrue. The invariance is thus proved.

The next theorem provides a necessary and sufficient condi-tion in order for the Wald statistic be invariant.

Theorem 2: Assume that problem (1) remains invariant underthe group of transformations . The Wald test is invariant under

if and only if (iff) there exists a orthogonal matrixsuch that

(10)

Proof: First of all, notice that the following equality holdstrue from (9)

(11)

Then, the right-hand side of (11) is equal to

iff (10) is true.The next corollary, based on Theorem 2, gives a useful suffi-

cient condition for the invariance of the Wald test.


Corollary 1: Assume that problem (1) remains invariantunder the group of transformations . If ( denotesthe vector with all zero components) and the group , inducedin the parameter space, is such that is a linear transforma-tion, i.e.,

(12)

with the transformation matrix, then the Wald teststatistic is invariant under .

Proof: It is sufficient to observe that, for the case at hand,the condition of Theorem 2 is verified with equal to the iden-tity matrix. In fact,

An important remark is now necessary. The condition ofCorollary 1 holds true in many signal processing detectionproblems (especially regarding radar). In fact, the unknownparameter is quite often related to the mean of a Gaussianvector. Hence, since transformations leaving invariant Gaussianstatistics are affine, quite often the induced group in theparameter space is composed of linear transformations. Asa consequence, in many practical situations, the Wald testleads to an invariant decision rule. Here, we mention somedetection problems, involving additive Gaussian disturbance(with unknown nuisance parameters such as the variance or thecovariance matrix), where the condition of Corollary 2 holdstrue: generalized linear model [12], [19]–[24], and [25].

A. The Case of No Nuisance Parameters

In this subsection, we specialize the results of Section III toscenarios where there are no nuisance parameters, namely wefocus on the hypothesis testing problem

(13)

The GLRT, Rao test, and Wald test decision statistics for thiscase are shown in (14) at the bottom of the page. It is not difficultto show, in the light of the results of Section III, that the GLRTand the Rao test statistics are invariant with respect to transfor-

mation groups leaving (13) invariant. Moreover, the Wald deci-sion statistic is invariant iff

In particular, if is linear and , the Wald statistic isinvariant. This last claim can be also justified as a special caseof Theorem 1 in [18], which provides a sufficient condition inorder for a general Wald statistic, as defined in (1) of [18], beinvariant.

Nevertheless, Corollary 1 (and its particularization whenthere are not nuisance parameters) provides only a sufficientcondition which, as shown in the following example, is notnecessary.

Example. Testing the Parameter in a Bernoulli Population.Let , be independent and identically distributedBernoulli random variables, that take on values 0 and 1, withcommon parameter . We are interested in thefollowing hypothesis test:

(15)

with . The joint probability mass function (pmf) ofcan be written as

where is the sample mean.Let and observe

that problem (15) is invariant under . In fact

namely the pmf is invariant and the transformation induced inthe parameter space preserves the original partition

and .Evidently, this problem does not comply with the assump-

tions of Corollary 1 (particularized to the case of no nuisanceparameters) because . Nevertheless, the Wald criterionstill leads to an invariant decision rule. To prove this claim, weevaluate the Wald statistic

(14)


where is the ML estimate of under .Moreover, the Fisher Information is given by

The Wald statistic can be thus computed as

and shares the invariance property since

IV. COINCIDENCE OF , RAO, AND

WALD DECISION STATISTICS

This section is aimed at establishing conditions in order forthe decision statistics of the Rao and Wald tests to coincide with

. Specifically, in Theorem 3, a necessary and suffi-cient condition for coincidence is provided, while, in Theorem4, a useful sufficient condition is given. The proofs of both thetheorems are in the Appendix.

Theorem 3: Let us focus on problem (1). A necessary anda sufficient condition for the equality of the Rao test decisionstatistic, , and the Wald test decision statisticis

(16)

and

(17)

, where is a orthogonal matrix.Theorem 4: Let us focus on problem (1). A sufficient condition

for the equality of the Rao test decision statistic, , andthe Wald test decision statistic is

(18)

and

(19)

A remark is necessary: if (18) holds true then

and, as a consequence by using (18)

which can be equivalently written as

where .Now, notice that:• since (19) must hold for all , must be

functionally independent of the first argument ;• if , (19) implies

(constant matrix).As a consequence,

with . In this case, the condition for theequality can be formulated as follows.

Corollary 2: If , a sufficient condition forthe equality of the Rao test decision statistic, , andthe Wald test decision statistic is

(20)


with and ,

(21)

Example. Detection of a subspace signal in subspace interfer-ence plus Gaussian noise of known level. Let us consider thefollowing detection problem:

(22)

where and are known full rank real matricessuch that and the subspaces spanned by the columnsof and are linearly independent, is a zero-mean real

-dimensional Gaussian vector with covariance matrix (known), is the identity matrix, , and .For this problem,

with is the Euclidean norm of a real vector. Thus, exploiting[26, p. 17, eq. 3.31], the FIM is given by

(23)

and, as a consequence

where is the projector onto the sub-space spanned by the columns of and .

Now, since

(24)

we have that

(25)

Let us consider the difference at the exponent of the last memberin (25); it can be written as

where, in the first equality, we have exploited the relationship

, whereas, in the

second, we have used . Itfollows that (16) holds true.

Notice also that

which is tantamount to saying that (17) is satisfied. Hence, in-voking Theorem 3, we can claim that and the deci-sion statistics of the Rao and Wald tests are equal . As amatter of fact, the three decision statistics reduce to

(26)

with . For , (26)returns the GLRT statistic derived in [11, p. 2153, eq. 7.3], i.e.,

A. The Case of No Nuisance Parameters

The results of Section IV can be specialized to situationswhere there are no nuisance parameters. Specifically, the fol-lowing theorem, whose proof is in Appendix, holds true

Theorem 5: Let us focus on problem (13). A necessary andsufficient condition for the equality of the Rao test decisionstatistic, , and the Wald test decision statistic is

(27)

and .It is interesting to recast condition (27) of Theorem 5 into an

equivalent form. To this end, notice that from (27)


and thus

which, after some standard algebra, gives

(28)

where and are defined in Corollary 2. Otherwisestated, must be an element of the exponential familywith natural parameter and sufficient statistic

. Moreover, (28) impliesthat must be Gaussian. In fact, the moment generatingfunction (MGF) of is

(29)

where is the set of complex numbers, denotes statisticalexpectation and in the second equality we let .This is the MGF of a Gaussian vector with mean and covari-ance matrix , i.e., .

An important consequence of this last finding is that undercoincidence the distribution of the resulting decision statistic

is a central chi-square random variable with degrees offreedom under , and a noncentral chi-square random vari-able with degrees of freedom and noncentrality parameter

under . It is worth pointing out that asymptotically the afore-mentioned result on the distribution of the Rao, , andWald statistics holds under more general conditions than thoseof Theorem 5 [12]. Nevertheless, what is interesting to observeis that, under coincidence (i.e., under the assumptions of The-orem 5), the decision statistic is chi-square distributed indepen-dently of the available number of samples.

Example. Detection of a subspace signal (linear model) inwhite Gaussian noise of known level. Let us consider the fol-lowing detection problem:

(30)

where is the vector with the observations, the usefulsignal component complies with the linear model isa full rank real matrix of size , and is areal Gaussian vector with known covariance matrix

. This is a very common model in signal processing applica-tions [12]. The data pdf, under , can be written as

or, equivalently, as

Since the FIM is given by and, the data pdf complies with the hypotheses of

Theorem 5. As a consequence, , the Rao and Walddecision statistics all collapse into

V. STATISTICAL EQUIVALENCE OF GLRT, RAO TEST, AND

WALD TEST

Sometimes, it happens that and the decision statis-tics of the Rao and Wald tests are related by strictly monotonictransformations. This means that, even if there is not the equalitydiscussed in the previous section, the GLRT, Rao, and Waldtests are statistically equivalent (in the sense that they lead to re-ceivers ensuring the same performance). Two examples of suchsituation can be found in [10] and [28]. Here, we provide twomore detection problems (of particular interest in radar signalprocessing) where the statistical equivalence may arise.

Example 1. Detection of an unknown signal in complexGaussian noise with unknown covariance matrix. We assumethat data are collected from sensors and denote by thecomplex vector of the samples where the presence of theuseful signal is sought (primary data). We also suppose that asecondary data set , is available, that each ofsuch snapshots does not contain any useful signal, and exhibitsthe same covariance structure as the primary data.


The detection problem to be solved can be formulated interms of the following binary hypotheses test:

(31)

where denotes the unknown useful signal vector.As to the noise vectors, we assume that and ’s,

, are independent, complex, zero mean, Gaussian vec-tors with covariance matrices given by

(32)

where denotes conjugate transpose. Moreover, and ’spossess the circular property usually associated with I and Qpairs of a wide-sense stationary process. For this situation, theGLRT is known in statistical literature as the complex Hotelling

test. In the signal processing context, it has been derived in[24] and can be expressed as

(33)

where and is the detection threshold to beset according to the desired value of the false alarm probability.This receiver also turns out to be the UMP Invariant (UMPI)test according to the group of transformations defined in [24].Moreover, the induced group of transformations in the param-eter space complies with the condition of Theorem 2. It followsthat the Rao and the Wald decision statistics must be invariant.In the following, we derive the Rao, and Wald tests for problem(31) and show that they are statistically equivalent to the GLRT(33).

Rao Test Design. In order to evaluate the Rao test decisionstatistic, we denote by the following:

• and the vectors whose components are the real andthe imaginary parts of the elements of respectively;

• a -dimensional real vector;• an -dimensional real column vector of

nuisance parameters, with the one-to-one operatorwhich provides the column vector containing, in a givenorder, the real and the imaginary part of the off-diagonalentries together with the diagonal elements of ;

• a -dimensional real vector.The data pdf under the hypothesis is

where denotes the trace of the argument. As to the blocksof the FIM, it can be shown, applying Slepian–Bang formula[29, p. 927, eq. 8.34], that

and

with a matrix of zeros and andthe real and the imaginary parts of the argument. The block di-agonal form of the FIM implies that .Moreover,

(34)

(35)

and

(36)

Observing that

(37)

and plugging , (34), (35), and (36) into the general ex-pression of the Rao test, after some algebra, yields

(38)

or, equivalently,

(39)

Exploiting the matrix inversion lemma [27, p. 19]

test (39) can be rewritten as

(40)

which is statistically equivalent to

with the suitable modification of the threshold in (40). Thislast representation proves the coincidence of the Rao test and theGLRT.

Wald Test Design. Exploiting the block diagonal structure ofthe FIM, the Wald test can be written as

(41)


Moreover,

(42)

Hence, plugging and (42) into (41), we get

(43)

This last expression proves the equivalence of the Wald test andthe GLRT.

Example 2. Detection of a subspace signal in subspace inter-ference and Gaussian noise with unknown covariance parame-ters. Let us focus on the detection problem (22) with ,

modeled as a zero mean real -dimensional Gaussian vectorwith covariance matrix , and a -dimensional realvector of unknown parameters. We assume that the joint MLestimates of , , and exist and are unique (problem wellposed). The pdf of is given by

(44)

Maximizing it over , and yields the ML estimates under, i.e.,

where

and

Moreover, letting in (44) and maximizing over and, we get the ML estimates of the unknowns under , i.e.,

As to the FIM, it shares a block diagonal structure, namely, where

(45)

It follows that

Previous formulae imply that, for the present problem,, the Rao , and Wald decision

statistics are given, respectively, by

It is seen that the three test statistics will in general be differentand also that they are not in general functions of each other.Hence, the detection performance for finite data records will bedifferent. We next examine some special cases where coinci-dence or equivalence arises:

1) known with .In this case, the three decision statistics end up coincidenti.e.,

with

2) with unknown.


In this case, the ML estimates of under and are,respectively

(46)

As a consequence,

It is evident that the three tests are statistically equiva-lent because and

. Otherwise stated, they are monotonicfunctions of .

3) Asymptotically .In this situation, the three tests will be the same. This is be-cause when the unconstrained and the constrainedML estimate of will be nearly identical. Hence, we willhave , and

Consideration. It is worth pointing out that in the two refer-ences [10], [28], and for the special case 2 of the previously ad-dressed detection problem [11], there exists a UMPI test whichis equivalent to the GLRT, Rao, and Wald tests. Actually, the ex-istence of the UMPI receiver is not sufficient for the statisticalequivalence of the three tests. The following example refers toa situation where the UMPI test exists but the three tests are notstatistically equivalent.

Example 3. Detection of a rank-one signal with unknown butpositive amplitude in subspace interference plus Gaussian noiseof known level. Let us focus on the problem

(47)

where is an -dimensional real vector, ,is a known full rank real matrix such that and thesubspaces spanned by the columns of and are linearly inde-pendent, is a zero mean real -dimensional Gaussian vectorwith covariance matrix ( is assumed known).

For this problem, the ML estimates of and are given by[11, p. 249]

(48)

where . Moreover, the GLRT is UMPIand can be written as

(49)

with the detection threshold. In order to derive the Rao andWald tests, we observe that the FIM is given by (23) with inplace of . It follows that

Further,

and the Rao test is equivalent to

Finally, the Wald test is statistically equivalent to (49).

VI. CONCLUSION

In this paper, we have investigated some properties con-cerning the GLRT, Rao test, and Wald test. First of all, wehave studied the invariance of the Rao and Wald statisticswith respect to transformations leaving the hypothesis testingproblem invariant. We have shown that the Rao test alwaysturns to be invariant while the Wald test, under very mildtechnical conditions, ensures invariance. This is a very nicefeature shared by the GLRT, Rao test, and Wald test.

In the second part of the work, we have addressed coinci-dence issues. Specifically, we have provided necessary andsufficient conditions in order for the decision statistics of theRao and Wald tests to coincide with twice the logarithm ofthe GLRT statistic. Interestingly, with reference to hypothesistesting problems without nuisance parameters, coincidencearises when the pdf of the observables belongs to the expo-nential family. In the last part of the work, we have focusedon the study of the statistical equivalence of the tests. In thiscontext, we have provided some examples where equivalencearises. These examples refer to detection problems commonlyencountered in statistical signal processing literature such asthe detection of subspace signals in the presence of subspaceinterference and Gaussian noise with unknown covariance


parameters. For the considered models, we have also given theexpression of the three test statistics.

Among the open research tracks, we are currently investi-gating the relationship between the three tests when multidi-mensional signal models are in force, as well as the Rao andWald tests when a nuisance parameter is unidentified under thenull hypothesis.

APPENDIX

Lemma 1:Proof: Let . From (6)

As a consequence,

(50)

Multiplying both the sides of (50) for and exploiting(6) yields

(51)

Integrating both the sides of the previous equality leads to

which is equivalent to (7).Theorem 3:

Proof: The condition is sufficient. From (16),

(52)

Moreover, from (17)

(53)

and the Rao statistic can be written as

(54)

As to the Wald statistic, (by definition) it is given by

(55)

and the sufficiency of (16) and (17) is thus proved.The condition is necessary. The equality between the Wald

statistic and implies that

(56)

, namely condition (16) holds true.The equality between the Rao and Wald statistics holds iff

there exists a orthogonal matrix such that

(57)

. This is equivalent to (17).Theorem 4:

Proof: The condition is sufficient. From (18),

(58)

which, exploiting (19), leads to

(59)

Moreover, from (18)

(60)


and the Rao statistic can be written as

(61)

which, using (19), gives (59). The Wald statistic is given by (55)and the use of (19) in (55), gives (59).

Theorem 5:Proof: The condition is sufficient. From (27)

(62)

Moreover,

and the Rao statistic becomes

(63)

The Wald decision statistic by definition still coincides with(63). The sufficiency of (27) is thus proved.

The condition is necessary. The equality between the Waldstatistic and requires that

(64)

Hence, for

and the Rao statistic is given by

(65)

Furthermore, the equality between the Rao and Wald statisticsholds iff

namely iff

The proposition is thus proved.

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Antonio De Maio (S’01–AM’02–M’03–SM’07)was born in Sorrento, Italy, on June 20, 1974. Hereceived the Dr.Eng. degree (with honors) and thePh.D. degree in information engineering, both fromthe University of Naples Federico II, Naples, Italy,in 1998 and 2002, respectively.

From October to December 2004, he was a visitingresearcher at the U.S. Air Force Research Laboratory,Rome, NY. From November to December 2007, hewas a visiting researcher at the Chinese Universityof Hong Kong, Hong Kong. Currently, he is an As-

sociate Professor at the University of Naples Federico II. His research interestlies in the field of statistical signal processing, with emphasis on radar detec-tion, convex optimization applied to radar signal processing, and multiple-ac-cess communications.

Steven M. Kay (F’89) was born in Newark, NJ, onApril 5, 1951. He received the B.E. degree from theStevens Institute of Technology, Hoboken, NJ, in1972, the M.S. degree from Columbia University,New York, in 1973, and the Ph.D. degree from theGeorgia Institute of Technology, Atlanta, in 1980,all in electrical engineering.

From 1972 to 1975, he was with Bell Laboratories,Holmdel, NJ, where he was involved with transmis-sion planning for speech communications and simu-lation and subjective testing of speech processing al-

gorithms. From 1975 to 1977, he attended the Georgia Institute of Technologyto study communication theory and digital signal processing. From 1977 to1980, he was with the Submarine Signal Division, Portsmouth, RI, where he en-gaged in research on autoregressive spectral estimation and the design of sonarsystems. He is presently a Professor of Electrical Engineering at the Univer-sity of Rhode Island, Kingston, and a consultant to numerous industrial con-cerns, the Air Force, the Army, and the Navy. As a leading expert in statisticalsignal processing, he has been invited to teach short courses to scientists andengineers at government laboratories, including NASA and the CIA. He haswritten numerous journal and conference papers and is a contributor to severaledited books. He is the author of the textbooks Modern Spectral Estimation(Prentice-Hall, 1988), Fundamentals of Statistical Signal Processing, Vol. I:Estimation Theory (Prentice-Hall, 1993), Fundamentals of Statistical SignalProcessing, Vol. II: Detection Theory (Prentice-Hall, 1998), and Intuitive Prob-ability and Random Processes using MATLAB (Springer, 2005). His current in-terests are spectrum analysis, detection and estimation theory, and statisticalsignal processing.

Dr. Kay is a member of Tau Beta Pi and Sigma Xi. He has been a distinguishedlecturer for the IEEE Signal Processing Society. He has been an Associate Ed-itor for the IEEE SIGNAL PROCESSING LETTERS and the IEEE TRANSACTIONS ON

SIGNAL PROCESSING. He has received the IEEE Signal Processing Society Edu-cation Award “for outstanding contributions in education and in writing schol-arly books and texts� � �.” He has recently been included on a list of the 250 mostcited researchers in the world in engineering.

Alfonso Farina (F’00) received the Ph.D. degree inelectronic engineering from the University of Rome(I) in 1973.

In 1974, he joined Selenia, now SELEX-SistemiIntegrati, where he as been a Manager since May1988. He was Scientific Director in the Chief Tech-nical Office. Today he is Director of the Analysisof Integrated Systems Unit. In his professional life,he has provided technical contributions to detection,signal, data, image processing, and fusion for themain radar systems conceived, designed, and de-

veloped in the company. He has provided leadership in many projects—alsoconducted in the international arena—in surveillance for ground and navalapplications, in airborne early warning and in imaging radar. Since 1979, hehas also been Professore Incaricato of radar techniques at the University ofNaples; in 1985, he was appointed Associate Professor. He is the author ofmore than 450 peer reviewed technical publications and the author of booksand monographs: Radar Data Processing (Vols. 1 and 2) (translated in Russianand Chinese), 1985–1986; Optimised Radar Processors, 1987; and AntennaBased Signal Processing Techniques for Radar Systems, 1992. He has writtenthe chapter on “ECCM techniques” in the Radar Handbook (2nd ed., 1990, and3rd ed., 2008), edited by Dr. M. I. Skolnik of the Naval Research Laboratory.

Dr. Farina has been session chairman at many international radar conferences.He has lectured at universities and research centers in Italy and abroad; he alsofrequently gives tutorials at the International Radar Conferences on signal, data,and image processing for radar; in particular on multisensor fusion, adaptivesignal processing, space–time adaptive processing (STAP), and detection. In1987, he received the Radar Systems Panel Award of IEEE AESS for develop-ment of radar data processing techniques. He is the Italian representative at theInternational Radar Systems Panel of IEEE-AESS. He is the Italian industrialrepresentative (Panel Member at Large) at the SET (Sensor and Electronic Tech-nology) of RTO (Research Technology Organisation) of NATO. He has beenin the BoD of the International Society for Information Fusion (ISIF). He hasbeen the Executive Chair of the International Conference on Information Fu-sion (Fusion) 2006, Florence, July 10–13, 2006. He was nominated Fellow ofIEEE with the following citation: “For development and application of adaptivesignal processing methods for radar systems.” Recently he has been nominatedinternational fellow of the Royal Academy of Engineering (U.K.); this fellow-ship was presented to him by HRH Prince Philip, the Duke of Edinburgh. He isa referee of numerous publications submitted to several journals of the IEEE,IEE, Elsevier, etc. He has also cooperated with the Editorial Board of the IEEElectronics & Communication Engineering Journal (ECEJ). More recently, hehas served as a member in the Editorial Board of Signal Processing (Elsevier).He has also been the co-Guest Editor of the Signal Processing (Elsevier) Spe-cial Issue on New Trends and Findings in Antenna Array Processing for Radar,September 2004. He is the corecipient of the following Best Paper Awards: en-titled to B. Carlton, of IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC

SYSTEMS for the 2001 and 2003; also of the International Conference on Fu-sion 2005. He has been the leader of the team that received the 2002 AMS CEOaward for Innovation Technology. He has also been the corecipient of the AMSRadar Division award for Innovation Technology in 2003. Moreover, he hasbeen the corecipient of the 2004 AMS CEO award for Innovation Technology.He has been the leader of the team that has won in 2004 the first prize awardfor Innovation Technology of Finmeccanica, Italy. This award context has seenthe submission of more than 320 projects. This award has been set for the firsttime in 2004. In September 7, 2006, he received the Annual European GroupTechnical Achievement Award 2006 by the European Association for Signal,Speech and Image Processing, with the citation: “For development and appli-cation of adaptive signal processing technique in practical radar systems.” InOctober 2006, he was on the team that received the annual Innovation Tech-nology award of Selex-SI for “an emulator of an integrated system for bordercontrol surveillance.” He was appointed member of the Editorial Boards of theIET Radar, Sonar and Navigation and the Signal, Image, and Video Processing(SIVP) journals. He has been the General Chairman of the IEEE Radar Confer-ence 2008, Rome, May 26–30, 2008 (www.radarcon2008.org). He is a fellowof the Royal Academy of Engineering and fellow of the IET). He has recentlybeen nominated a Fellow of EURASIP, with the following citation: “For contri-butions to radar system design, signal, data and image processing, data fusionand particularly for the development of innovative algorithms for deploymentinto practical radar systems.” He is also the recipient of the 2010 IEEE Dennis J.Picard Gold Medal for Radar Technologies and Applications with the followingcitation: “For continuous, innovative, theoretical and practical contributions toradar systems and adaptive signal processing techniques.”

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