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1 Sgna Processng 142 (218) Contents sts avaabe at ScenceDrect Sgna Processng ourna homepage: Sepan-Bangs-type formuas the reated Msspecfed Cramér-Rao Bounds for Compex Eptcay Symmetrc dstrbutons Abdemaek Mennad a, Stefano Fortunat b, d,, Mohammed Nab E Korso c, Arezk Youns a, Abdehak M. Zoubr d, Aexre Renaux e a aboratore Mcro-Onde et Radar, Ecoe Mtare Poytechnque, P.O. Box 17, Bord E Bahr 16111, Agera b Dpartmento d Ingegnera de Informazone, Unvesty of Psa, Psa 56122, Itay c Pars Ouest Unversty, EME EA4416, 5 rue de Sèvres, Ve d Avray 9241, France d Sgna Processng Group, Technsche Unverstät Darmstadt Merckstr. 25, Darmstadt 64283, Germany e Unversté Pars-Sud, Orsay, Dpt Physcs aboratory of Sgnas Systems (2S) Modesaton Estmaton Group, France a r t c e n f o a b s t r a c t Artce hstory: Receved 26 Apr 217 Revsed 5 Juy 217 Accepted 24 Juy 217 Avaabe onne 25 Juy 217 Keywords: Sepan-Bangs formua Msspecfed Cramér-Rao Bound Compex Eptcay Symmetrc dstrbuton Msspecfed modes Msmatch In ths paper, Sepan-Bangs-type formuas for Compex Eptcay Symmetrc dstrbuted (CES) data vectors n the presence of mode msspecfcaton are provded. The basc Sepan-Bangs (SB) formua has been ntroduced n the array processng terature as a convenent compact representaton of the Fsher Informaton Matrx (FIM) for parameter estmaton under (parametrc) Gaussan data mode. Extendng recent resuts on ths topc, n ths paper, we provde a new generazaton of the cassca SB formua to parametrc estmaton probems nvovng non-gaussan, heavy-taed, CES dstrbuted data n the presence of mode msspecfcaton. Moreover, we show that our proposed formuas encompass the speca cases of the SB formua for CES dstrbutons under perfect mode specfcaton, the SB formuas n the presence of msspecfed Gaussan modes, the SB formua for the estmaton of the scatter matrx of a set of CES dstrbuted data under msspecfcaton of the densty generator. 217 Esever B.V. A rghts reserved. 1. Introducton The asymptotc performance anayss of an estmaton agorthm mosty rees on two smpfed assumptons: ) the data are assumed to be Gaussan dstrbuted ) the data mode used to derve the estmaton agorthm s supposed to be correcty specfed, that s the probabty densty functon (pdf) assumed to derve an estmator of the parameters of nterest the true pdf that statstcay characterzes the data are exacty the same. Athough these assumptons guarantee the possbty to perform eegant performance assessment, e.g. by evauatng the Cramér-Rao Bound (CRB) for the estmaton probem at h /or by obtanng a cosed form expresson for the Mean Square Error (MSE) of a gven estmator, the everyday engneerng practce ceary cas the hypotheses ) ) nto queston. Regardng the Gaussan mode assumpton, arge-scae measurement campagns the subsequent statstca anayss of the data gathered from a pethora of engneerng appcatons, e.g. outdoor/ndoor mobe communcatons, radar/sonar systems or magnetc resonance mag- Correspondng author at: Dpartmento d Ingegnera de Informazone, Unvesty of Psa, 56122, Psa, Itay. E-ma address: stefano.fortunat@et.unp.t (S. Fortunat). ng (MRI), have hghghted the mpusve, heavy-taed behavour of the observatons 1. These expermenta evdences have motvated the need to go beyond the Gaussan mode deveop new statstca modes abe to better characterze the data. One of the more fexbe genera non-gaussan mode s represented by the set of the Compex Eptcay Symmetrc (CES) dstrbutons 2, aso caed Mutvarate Eptcay Contoured dstrbutons 3. CES dstrbutons encompasses the compex Gaussan, the Generazed Gaussan a the Compound Gaussan (CG) dstrbutons, such as the compex t -dstrbuton the K -dstrbuton, as speca cases. The pdf of a CES dstrbuted N -dmensona rom vector x C N s competey characterzed by the mean vaue γ, the scatter (or shape) matrx by a rea vaued functon w (t) : R + R, caed the densty generator,.e. x CES N ( γ,, w ) 2,3. The CES dstrbutons have been used n a varety of appcatons, n partcuar n the radar array sgna processng feds. Other expermenta evdences revea recurrng voatons of the matched mode assumpton, that s the cam of a perfect match between the assumed the true data mode. The mathematca bases of a forma theory of the parameter estmaton under mode msspecfcaton has been frsty deveoped by statstcans as uber 4, Whte 5 Vuong 6 recenty redscovered by the Sgna Processng (SP) communty 7 9 apped to a va / 217 Esever B.V. A rghts reserved.

2 A. Mennad et a. / Sgna Processng 142 (218) rety of we-known engneerng probems: to Drecton-of-Arrva (DoA) estmaton n array MIMO processng 7,1, to covarance/scatter matrx estmaton n CES dstrbuted data 8,11,12, to radar-communcaton systems coexstence 13 to waveform parameter estmaton n the presence of uncertanty n the propagaton mode 14, ust to name a few. Ths bref dscusson ceary hghghts the need to overtake both the Gaussan the matched mode assumptons whe assessng the (asymptotc) performance of an estmator. As extensvey dscussed n the SP terature, one of the man too for the performance assessment s the CRB that provdes a ower bound to the MSE achevabe by any unbased estmator for a gven estmaton probem (see e.g. 15). Under the matched mode assumpton, the CRB can be evauated as the nverse of the Fsher Informaton Matrx (FIM), then havng a convenent easy way to evauate the FIM woud be of great practca utty. To ths end, n array processng appcatons, the ceebrated Sepan-Bangs (SB) formua has been ntroduced. Deveoped n the semna works of Sepan 16 Bangs 17, the SB formua provdes a usefu compact expresson of the FIM for vector parameter estmaton under both Gaussan matched mode assumptons 15, Chapter 3, Appendx 3C. Specfcay, et θ R d be a d -dmensona, determnstc parameter vector et x = x =1 wth x C N, be a set of ndependent (possby) compex rom vectors, usuay caed snapshots, representng the avaabe observatons. If we assume that each snapshot foows a (compex) Gaussan parametrc mode, such that x CN (γ(θ), (θ)), then the FIM for the estmaton of θ can be expressed by means of the SB formua. The frst generazaton of the SB formua to a non-gaussan, but st perfecty matched, data mode has been proposed by Besson Abramovch n 18. Specfcay, Besson Abramovch derved a compact expresson for the FIM for the estmaton of θ when each snapshot x s characterzed by a parametrc CES dstrbuton,.e. x CES N ( γ ( θ), ( θ), w ). Note that the functona form of the parametrzed mean vaue γ ( θ) s aowed to vary from snapshot to snapshot, whe the covarance matrx s assumed to be constant. Ceary, snce the Gaussan mode beongs to the CES cass, ths generazed SB formua coapses to the cassca one f the data are Gaussan dstrbuted. The second mportant step ahead has been made by Rchmond orowtz n 7 then by Parker Rchmond n 14 where the cassca, Gaussan-based, SB formua has been extended to estmaton probems under mode msspecfcaton,.e. when the assumed parametrc Gaussan mode, say CN (γ(θ), (θ)), coud dffer from the true (possby non parametrc) one, ndcated as CN (μ, ). In other words, we aow the assumed parametrc mean vaue γ( θ) the assumed parametrc covarance matrx ( θ) to dffer from the true μ for every possbe vaue of the parameter vector θ,.e. CN (γ(θ), (θ)) = CN (μ, ), θ. It s worth to underne that n the estmaton framework under mode msspecfcaton, the FIM oses ts cassca statstca sense t has to be substtuted by the matrces A ( θ) B ( θ) defned n 8, Eqs. (1) (7), respectvey (see aso 6,7). Consequenty, n ths context, SB-type formuas coud be expoted to obtan A ( θ) B ( θ) needed to evauate the counterpart of the CRB n the presence of mode msspecfcaton,.e. the Msspecfed CRB (MCRB) 4,6 8,11. In partcuar, n 7 the authors derved SB-type formuas for the decouped scenaro n whch the unknown parameter vector θ can be parttoned n two sub-vectors η ν, named determnstc stochastc parameter subvectors respectvey, such that θ = η T, ν T T CN (γ(θ), (θ)) CN (γ(η), (ν)) = CN (μ, ), θ. The fndngs presented n 7 have been extended n 14 to ncude the coupng of determnstc stochastc parameters. More formay, n 14, SB-type formuas have been derved for the foowng msspecfed scenaro CN (γ(θ), (θ)) CN (γ(η, ω), (ν, ω)) = CN (μ, ), θ where the unknown parameter vector θ s parttoned as θ = η T, ν T, ω T T. The natura extenson of the works of Besson Abramovch 18, Rchmond orowtz 7 Parker Rchmond 14 woud be to derve SB-type formuas for parametrc estmaton probems nvovng CES dstrbuted data under mode msspecfcaton. Ths paper ams exacty at fng ths gap obtanng some genera msspecfed SB formuas for CES dstrbuted data. Remark : Throughout ths paper, we consder ony the case of rea parameter vectors. Ths s not a mtaton, snce we can aways maps a compex vector n a rea one smpy by stackng ts rea the magnary parts. Ceary, the proposed dervaton of the SB-type formuas coud aso be deveoped drecty n the compex fed by means of the Wrtnger cacuus as n 7,19. Notaton : Throughout ths paper, tacs ndcates scaar quanttes ( a, A ), ower case upper case bodface ndcate coumn vectors ( a ) matrces ( A ) respectvey. Each entry of a matrx A s ndcated as a, A,. ndcates the compex conugaton. The superscrpts T ndcates the transpose the ermtan operators, then A = ( A ) T. et f ( t ) be a rea scaar functon, than f ( t ) df ( t )/ dt. et A ( θ) be a matrx (or possby vector or even scaar) functon of the vector θ, then A A ( θ ) whe A A (θ) θ θ= θ A 2 A (θ) θ θ θ= θ, where the vector θ w be aways expcty defned n the paper. For two matrces A B, A B means that A B s postve sem-defnte. Fnay, for rom varabes or vectors, the notaton = d sts for has the same dstrbuton as. 2. Probem setup et x = x =1, wth x C N, be a set of ndependent compex rom vectors (or snapshots ) representng the avaabe observatons. We assume that each snapshot s samped from a CES dstrbuton 2,3,.e., x CES N ( μ,, g ), then ts pdf can be expressed as: p X (x ) p X (x ; μ, ) = c N,g g((x μ ) (x μ )) (1) where c N, g s a normazng constant, g(t) : R + R s the densty generator, μ = x s the mean vaue s a postve defnte ermtan matrx caed scatter matrx. In the rest of ths paper, we aways assume that the scatter matrx s of fu rank,.e. rank ( ) = N. From the Stochastc Representaton Theorem 2, a CES dstrbuted rom vector can be expressed as: x = d μ + R Tu, (2) where: u U(C S N ) s a N -dmensona compex rom vector unformy dstrbuted on the unt hyper-sphere wth N 1 topoogca dmenson. As reported n 2 (emma 1), u = u u = (1/N) I where I s the dentty matrx of a sutabe dmenson. R Q s a rea non-negatve rom varabe caed moduar varate, whe Q s caed second order moduar varate. Moreover, under the assumpton that rank ( ) = N, we have that: Q (x μ ) (x μ ) = d Q, N. (3) As shown n 2, the pdf of Q has a one-to-one reaton wth densty generator: p Q (t) = δ N,g t N g(t), (4) where δ N,g t N g(t) dt <. As a consequence of (3) (4), the expectaton of functons of the quadratc form Q, say

3 322 A. Mennad et a. / Sgna Processng 142 (218) h ( Q ), can be expcty derved as: E Q h (Q ) h (t ) p Q (t ) dt =δ N,g h (t ) t N g(t ) dt, N. It s cear from (5) that such expectaton does not depend on the ndex, snce the pdf n (4) of the quadratc form Q s nvarant wth respect to (w.r.t.). For ths reason, to avod confuson, n the rest of the paper we aways ndcate E Q h (Q ) smpy as E Q h (Q). T s a compex N N matrx wth rank ( T ) = N, such that = TT. If E Q Q < rank ( ) = N, then the covarance matrx M = (x μ )(x μ ) of x can be decomposed as M = σ 2, where (see Theorem 4 n 2 ): σ 2 E Q Q, (6) N σ 2 can be nterpreted as the statstca power of the CESdstrbuted vector x. From the Stochastc Representaton Theorem, t s cear that the representaton of a CES dstrbuted vector x s not unquey determned by (2). In fact, x = d μ + R Tu = d μ + (c R )(ct ) u, c >. From a dfferent, yet equvaent, stpont, ths dentfabty probem can be understood as an mpct consequence of the functona expresson of a CES dstrbuton. It s mmedate to verfy from (1) that CES N ( μ,, g ( t )) CES N ( μ, c 2, g ( t / c 2 )), c >. As ampy dscussed n 2, ths dentfabty ssue can be soved by posng a constrant on the moduar varate R ( consequenty, through (4), on the densty generator g ( t )), or on the scatter matrx. For convenence, we choose to put a constrant on R 2 = Q. Specfcay n the rest of ths paper, we aways assume that E Q Q = N consequenty, from (6), M =,.e. the scatter matrx equates the covarance matrx. Due to the ndependence assumpton, the ont pdf of the set x s gven by the product of the margna pdfs of each snapshot x gven n (1) : p X (x ) p X (x 1,..., x ; μ 1,..., μ, ) = (c N,g ) g((x μ ) (x μ )) (7) =1 As dscussed n the Introducton, the pdf n (7) s a genera powerfu mode that t s abe to statstcay characterze the mpusve, heavy-taed data behavour n a varety of appcatons aow us to overtake the Gaussanty assumpton. et us now focus on the cearng of the matched mode assumpton. Foowng the recent deveopments on ths fed, n ths paper we consder the foowng msmatched stuaton: The acqured dataset x = x (5) s characterzed by the true but =1 unknown ont pdf gven n (7). In partcuar, each data snapshot x foows a CES dstrbuton wth mean vaue μ, scatter matrx densty generator g ( t ),.e. x CES N (μ,, g), = 1,...,. In order to derve an nference agorthm, we assume that each snapshot of the dataset x s samped from a CES dstrbuton wth a densty generator w ( t ), possby dfferent from g ( t ) for a t R +, a mean vaue γ ( θ) a scatter matrx ( θ) parametrzed by a determnstc parameter vector θ R d to be estmated. In partcuar, we aow the assumed margna mode CES N ( γ ( θ), ( θ), w ) to dffer from the true one, CES N ( μ,, g ) for every θ. Ths s a recurrng scenaro n array processng appcatons, where the mean vaue /or the scatter matrx of the acqured snapshot vectors are assumed to be parametrzed by a determnstc parameter vector whose components represent the Dopper frequency, the Drecton of Arrvas (DOAs) of potenta sources so on. The assumed margna pdf of each snapshot x can then be expressed as: f X (x ; θ) f X (x ; γ (θ), (θ)) = c N,w (θ) w ((x γ (θ)) (θ) (x γ (θ))) (8), by expotng the ndependence assumpton, the ont pdf of x = x can be obtaned as: =1 f X ( x ; θ ) f X ( x 1,..., x ; γ ( θ ),..., γ ( θ ), ( θ )) = ( c N,w ) θ ( (x ) ( ) ) w γ θ θ x γ θ (9) =1 Ths scenaro ceary represents an estmaton probem n non- Gaussan data n the presence of mode msspecfcaton. et θˆ f θ ˆ f (x) be a, possby msmatched, estmator of the parameter vector θ derved under the assumed mode f X ( x ; θ) n (9) whe the data are characterzed by the true mode p X ( x ) n (7). Then, as dscussed n 6 8, under sutabe reguarty condtons, a ower bound on the error covarance matrx of any msspecfed (MS)- unbased (see 6,8) msmatched estmator θ ˆ f exsts t s gven by the MCRB defned as: C p θ ˆ f, θ ( θ ˆ f θ )( θ ˆ f θ ) T 1 A (θ ) B (θ ) A (θ ) (1) where θ s the so-caed pseudo-true parameter vector defned n 4 6 as the parameter vector that mnmze the Kuback eber dvergence (KD) between the true the assumed modes: θ arg mn θ D ( p X f θ ) = arg mn θ n f X (x ; θ), (11) where D ( p X f θ ) n ( p X ( x )/ f X ( x ; θ)). The matrces A ( θ ) B ( θ ) are defned as: A (θ ), T 2 θ θ n f X (x ; θ ) =E n f X (x ; θ) p, θ θ θ= θ (12) B (θ ), θ n f X (x ; θ ) θ T n f X (x ; θ ), n f X (x = E ; θ) p n f X (x ; θ) θ θ= θ θ θ= θ. (13) For a deep theoretca anayss of the exstence, the propertes the practca appcabty of the MCRB, we refer the reader to 6 9 to the references theren. The goa of ths paper s to provde a genera cosed-form expressons of A ( θ ) B ( θ ) n the aforementoned context. In other words, we am at dervng two SB-type formuas for the evauaton of the matrces A ( θ ) B ( θ ) that represent a generazaton of the cassca FIM for estmaton probem under mode msspecfcaton. It s mportant to note that n the foowng dervatons we aways assume the exstence the unqueness of the pseudo-true parameter vector θ defned n (11). As t s cear from (1) as t s ampy dscussed n 6 8, fndng the θ that mnmzes the KD between the true the assumed mode s a prerequste for the evauaton of the MCRB snce a the dervatves nvoved n the two matrces A ( θ ) B ( θ ) have to be evauated at θ. Fnay, t s worth notcng that, under sutabe reguarty condtons on the true the assumed pdfs, the defnton of the pseudo-true parameter vector θ

4 A. Mennad et a. / Sgna Processng 142 (218) n (11) can be expressed n an equvaent form as: D ( p X f θ ) n f X (x; θ) = E θ p =, = 1,..., d. θ θ= θ θ= θ (14) As we w see n the next secton, ths equaty w be expoted to evauate expcty the matrx B ( θ ). We note n passng the smarty between the defnton on θ gven n (14) the condton n Eq. (9) of 18. Ths fact hghghts the strong theoretca paraesm between the cassca matched theory the msspecfed framework as detaed n Sepan-Bangs formuas under msspecfed CES modes Ths Secton focuses on the dervaton of the SB formuas for A ( θ ) B ( θ ) defned n (12) (13), provded that there exsts a unque pseudo-true parameter vector θ satsfyng (11). We start by evauatng expct expressons for the foowng quanttes: V (θ ) = n f X (x; θ) n f X (x; θ),, = 1,..., d, θ θ (θ ) = 2 n f X (x ; θ) θ θ θ= θ θ= θ (15),, = 1,..., d. (16) θ= θ The entres of the matrces A ( θ ) B ( θ ) can then be obtaned by takng the expectaton operator, w.r.t. the true dstrbuton p X ( x ), of ( θ ) V ( θ ),.e. A (θ ) = (θ ) B (θ ) = V (θ ). Unfortunatey, ths expectaton can be evauated n cosed form ony n two partcuar cases, as we w deta n Sectons In a the other cases, numerca ntegraton technques, e.g. the Monte Caro ntegraton, coud be expoted Evauaton of V ( θ ) ( θ ) of ther expectaton w.r.t. the true data dstrbuton Accordng to the genera msmatched estmaton probem dscussed n the prevous secton, we consder here the genera case n whch, for each avaabe snapshot x, a parametrc CES mode f X (x ; θ) = CES N (γ (θ), (θ), w ) s assumed, whe actuay each observaton vector s dstrbuted accordng to a dfferent, possby non-parametrc CES data mode,.e. x p X (x ) = CES N (μ,, g). From (9), t s mmedate to verfy that: n f X x ; θ θ where n θ θ= θ θ= θ = n (θ) θ =1 wth φ(t) = w (t ) /w (t ) P θ= θ + φ(g (θ )) (θ) θ θ= θ = tr (P ) + φ(g ) (17) θ =1 = tr = tr (P ), (18) / 2 / 2, (19) G G (θ ) (x γ ) (x γ ), (2) where, for notatona smpcty, γ γ (θ ) ( θ ). Then, an expct expresson of V ( θ ) n (15) s gven by: V (θ ) = 2 tr (P where: θ = 2 Re tr (P + =1 m =1 ) tr (P ) tr (P ) =1 φ(g ) θ ) =1 φ(g ) φ(g m ) φ(g ) θ m, (21) θ θ (x γ ) γ (x θ γ ) S (x γ ), (22) S =. (23) The term ( θ ) n (16) can be obtaned, through drect cacuaton, from (17) as: (θ ) = tr (P P P ) + where: =1 φ (G ) θ θ + φ(g ) 2 G (24) θ θ =1 P / 2 / 2 (25) φ (t) = w (t)/w(t ) (w (t)) 2 /w 2 (t). After havng obtaned the terms V ( θ ) ( θ ), we have to evauate ther expectatons w.r.t. the true dstrbuton p X ( x ). Snce a the dervatves n (21) (24) have to be evauated n the pseudo-true parameter vector θ, we can expot the equaty estabshed n (14). Specfcay, from (17), we have that: D ( p X f θ ) = tr (P θ ) φ(g ) =, = 1,..., d, θ θ= θ consequenty, =1 φ(g ) θ =1 (26) = tr (P ), = 1,..., d. (27) Now, takng the expectaton operator w.r.t. p X ( x ) of the term V ( θ ) n (21) by expotng the equaty n (27), the matrx B ( θ ) can be expressed as: B (θ ) = 2 tr (P ) tr (P ) + =1 m =1 φ(g ) φ(g m ). θ θ (28) Smary, the matrx A ( θ ) can be obtaned by takng the expectaton operator w.r.t. p X ( x ) of the term ( θ ) n (24) as: A (θ ) = tr (P P P ) + φ (G ) θ θ =1 + φ(g ) 2 G. (29) θ θ =1

5 324 A. Mennad et a. / Sgna Processng 142 (218) As we can see from (21) (24), the expressons of V ( θ ) ( θ ) are hghy nvoved from an anaytca stpont, consequenty t s mpossbe to derve n cosed form ther expectatons n the genera case. As dscussed n Appendx A, cosed form expressons can be obtaned when the rom terms n (21) (24) satsfy certan ndependence condtons. Fortunatey, there are two scenaros of great practca nterest n whch such condtons are met consequenty a cosed form expresson for A ( θ ) B ( θ ) can be obtaned. These speca cases are: Scenaro 1: The true margna pdf s an unspecfed CES mode,.e. p X (x ) = CES N (μ,, g), whe the assumed pdf s a parametrc compex Gaussan mode,.e. f X (x ; θ) = CN (γ (θ), (θ)). Scenaro 2: The Scenaro 2 s characterzed by two assumptons: A1: The true the assumed pdfs share the same parametrc mean vaue γ ( θ) the same parametrc scatter matrx ( θ) whe the msspecfcaton s caused by a wrong assumpton on the densty generator w ( t ). More formay, post that the true margna pdf of each snapshot s gven by a parametrc CES dstrbuton such that p X (x ) p X (x ; θ) = CES N (μ,, g), where μ = γ ( θ) = ( θ) for a gven θ. The assumed mode s nstead another parametrc CES dstrbuton that share the same parameterzaton of the true one but may have a dfferent densty generator,.e. f X (x ; θ) = CES N (γ (θ), (θ), w ), θ, g(t) = w (t), t R +. Note that ths scenaro s a partcuar case of the more genera cass of msspecfed probems dscussed n 8, Secton II.D. A2: The true parameter vector θ the pseudo-true parameter vector θ are equas. In partcuar, θ s the souton of the optmzaton probem n (11). Note that these two assumpton are verfed for the scatter matrx estmaton probem dscussed n 8. The Scenaro 1 descrbes a common practce n array processng appcatons. In fact, when the true data mode s unknown, a prevaent choce s to assume a smpe Gaussan mode that guarantees an easy dervaton a consequent rea-tme mpementaton of the estmaton agorthm. The Scenaro 2 s a bt dfferent, snce t mpy the a-pror knowedge of the functona form of the true parametrc mean vaue γ ( θ) of the parametrc scatter matrx ( θ). There are, however, a varety of practca appcatons n whch ths a-pror nformaton s ndeed avaabe to the user. As an exampe, one can thnk of array sgna processng appcatons n whch the a- pror knowedge of the array geometry eads to a correct specfcaton of the parametrzed mean vaue covarance matrx of the coected snapshots, whe the uncertanty on the statstca dsturbance mode coud cause a wrong choce of the densty generator. Of course, there are cases n whch aso the knowedge of the array geometry coud be ncorrect or parta, consequenty the assumed mean vaue scatter matrx dffer from the true ones (see 7 for more detas) Scenaro 1 In ths subsecton we provde the SB formuas,.e. the cosed form expressons of the matrces A ( θ ) n (12) B ( θ ) n (13), for the Scenaro 1. We start by notcng that the assumed compex Gaussan mode beongs to the CES cass,.e. f X (x ; θ) = CES N (γ (θ), (θ), w ), where w (t) = exp ( t). Consequenty, t s mmedate to verfy that φ(t) = φ (t) =. The matrx B ( θ ) can be evauated from (28) by usng the fact that φ(g ) = : B (θ ), = 2 tr (P ) tr (P ) + =1 m =1. (3) θ θ Foowng the procedure dscussed n Appendx B, the matrx B ( θ ) can be expressed as: B (θ ), = 2 Re r + γ θ =1 ( ) γ r + E Q Q 2 + θ 1 + E Q Q 2 tr (S where: S tr (S ) tr (S ) ), (31) r μ γ. (32) Note that a the dervatves have to be evauated n the pseudotrue parameter vector θ defned n (11). et us evauate the matrx A ( θ ). From (24) φ(g ) =, φ (G ) =, through drect cacuaton (see aso Appendx B ) we obtan: (θ 2 G ) α + 2 Re θ θ r / 2 ( = 2 Re S γ θ γ θ ) γ γ 2 γ + S θ θ θ θ + r (P P + P P P r + tr (P P + P P P ) / 2 / 2. (33) ) / 2 Moreover, as n (31), we used constrant on Q. Fnay, by nsertng (33) n (24), we obtan: A (θ ), = tr (P P P ) =1 α (θ ). (34) The expressons (31) (34) represent the SB formuas for the msmatched Scenaro Scenaro 2 Ths Subsecton focuses on the Scenaro 2,.e. the case n whch the true the assumed pdfs are CES dstrbutons that share the same parametrzed mean vaue scatter matrx but are characterzed by dfferent densty generators. From the proof provded n Appendx C, the matrces B ( θ ) A ( θ ) can be expressed respectvey as: B (θ ), = B ( θ), = 2 N E Q Qφ 2 (Q) E Q Q 2 φ 2 (Q) + A (θ ), = A ( θ), = 2 N γ γ θ θ + E Q Q 2 φ (Q) =1 Re γ θ γ θ tr (P ) tr (P ) + E Q Q 2 φ 2 (Q) tr (P P ), E Q Qφ (Q) + NE Q φ(q) E Q Q 2 φ (Q) + 1 =1 Re tr (P P ) (35) tr (P ) tr (P ). (36)

6 A. Mennad et a. / Sgna Processng 142 (218) The expressons (35) (36) represent the SB formuas for the msmatched Scenaro 2. It s worth to reca that the prevous two formuas can be apped ony f, n the partcuar estmaton probem an h, the pseudo-true parameter vector equates the true parameter vector,.e. when θ = θ. The reason for ths restrcton w be carfed n Appendx C. 4. Reatonshp to prevous resuts The am of ths Secton s to show that the SB formuas derved for the Scenaros 1 2 encompass a the prevousy derved SB formuas as speca cases. In partcuar, both the SB formua for CES dstrbutons under perfect mode specfcaton, proposed n 18, the SB formuas for the scatter/covarance matrx estmaton of CES dstrbuted vectors under msspecfcaton of the densty generator, proposed n 8,11, can be obtaned as speca cases of the SB formuas shown here for the Scenaro 2. In addton, the SB formuas for msspecfed Gaussan modes, proposed n 7, are speca cases of the SB formuas dscussed here for the Scenaro 1. We note, n passng, that the SB-type formuas derved n 14 can aso be obtaned as a partcuar case of the ones proposed n ths paper n (31) (34). Ths can be easy done by parttonng the unknown parameter vector as θ = η T, ν T, ω T T by takng nto account the partcuar parameterzaton of the assumed mean vaue of the assumed scatter matrx supposed n 14,.e. γ(θ) = γ(η, ω) (θ) = (ν, ω) The SB formuas for scatter matrx estmaton under msspecfcaton of the densty generator 8,11 In 8,11, SB formuas for the scatter matrx estmaton n CES dstrbuted vectors under msspecfcaton of the densty generator have been proposed. It s easy to verfy that ths scenaro s a speca case of the more genera Scenaro 2 dscussed n Secton 3.3. In partcuar, accordng to 8,11, the dataset x s consdered to be composed of ndependent, zero mean snapshots dstrbuted as x CES N ( μ,, g ) where μ = ( θ), for a gven θ that s the true parameter vector. For each snapshot, we assume a margna pdf f X (x ; θ) = CES N (, (θ), w ),.e. we msspecfed the densty generator. Snce we am at fndng SB formuas for the estmaton of the scatter matrx tsef, we have that θ vecs( ( θ)) vecs( ), where vecs s the operator that maps a symmetrc N N matrx n a / 2 -dmensona vector whose entres are the eements of the upper (or ower) submatrx of. A smar notaton hods for the true parameter vector, n partcuar θ = vecs ( ). Note that, n the foowng, we assume that the entres of θ ( then of the scatter matrx) are rea numbers. Fnay, the resuts n 8,11 can be ready derved by (35) (36) by posng = 1, γ θ E Q Q φ(q ) = N (see Appendx B ): B (θ ), = E Q Q 2 φ 2 (Q) tr (P ) tr (P ) + tr (P P ) tr ( P ) tr ( P ), (38) 4.1. The SB formua under correcty specfed CES modes 18 The SB formua for correcty specfed CES mode has been derved n 18. Usng the formasm ntroduced n ths paper, two parametrc CES modes are sad to be correcty specfed f there exsts a vector θ, such that the assumed CES dstrbuton n (9) equates the true CES dstrbuton n (7). More formay, the CES mode f X ( x ; θ) s sad to be correcty specfed f there exsts θ, such that f X (x; θ) = p X (x), n partcuar, γ ( θ) = μ, ( θ) = g ( t ) w ( t ). As proved n 6, under correcty specfed mode, we have that θ = θ B ( θ) = A ( θ), where B ( θ) s the cassca FIM evauated at the true parameter vector θ. In the foowng, we w show that the SB formua derved n 18 can be consdered as a speca case of the one obtaned n Secton 3.3. In fact, accordng to the matched mode assumpton, we can defne the true mode as p X (x) = f X (x; θ) = CES N (γ ( θ), ( θ), w ), whe the assumed parametrc mode s f X (x; θ) = CES N (γ (θ), (θ), w ) wth θ. Wth ths n mnd, we can expot the resut n Eq. (11) of 18, that s E Q Q φ(q ) = N, where φ(t) g (t)/g(t ). Fnay, as shown n 6, we have that B (θ ) = B ( θ) = A ( θ) = FIM ( θ). Then by expotng (35), we obtan: B ( θ), = FIM ( θ), = 2 N E Q Q φ γ 2 (Q) Re θ =1 ( E Q Q 2 φ ) 2 (Q) + 1 tr ( ) tr ( ) γ θ + E Q Q 2 φ 2 (Q) tr ( ), (37) where ( θ) (θ) θ θ= θ. It s mmedate to verfy that the matrx defned n (37) s exacty the same as the FIM gven n Eq. (2) of 18. A (θ ), = E Q Q 2 φ (Q) tr (P P ) + tr (P ) tr (P ) tr (P P ), (39) where, n ths case, the matrx P (or P ) can be expressed as P = A = A where s the matrx that mnmzes the KD between the true the assumed dstrbutons, then, due to the Assumpton A2, t s equa to the true covarance matrx,.e. = = ( θ) =. A s a matrx defned as A (θ) θ 4.3. The SB formuas for msspecfed Gaussan modes 7 θ. In ths subsecton, we brefy show how to obtan the SB formuas provded n 7 by usng the genera resuts dscussed here for the Scenaro 1. The SB formuas for msspecfed Gaussan modes can, n fact, be derved as a speca case of the ones gven n (31) (34) by postng as true dstrbuton the parametrc compex Gaussan mode,.e. p X (x ) = CES N (μ,, exp ( t)) = CN (μ, ), whe the assumed margna dstrbuton s st gven by f X (x ; θ) = CES N (γ (θ), (θ), exp ( t)) = CN (γ (θ), (θ)). Note that the true the assumed densty generators are equa t s smpy gven by g(t) = w (t) = exp ( t). Ths fact can be used to evauate the term E Q Q 2 as: E Q Q 2 = exp ( t) t N+1 δ N,g = exp ( t) t N+1 δ N,g + (N + 1) Snce, exp ( t) t N+1 δ N,g E Q Q 2 = (N + 1) = exp ( t ) t N δ + t N exp ( t) δ N,g dt. (4) =, we have that t N exp ( t) δ N,g dt N,g t N exp ( t) δ N,g dt =, (41)

7 326 A. Mennad et a. / Sgna Processng 142 (218) n whch we used the fact that t N exp ( t) δ N,g dt = 1. Conse- quenty, the entres of the matrx B ( θ ) n (31) can be ready expressed as: ( γ B (θ ), = 2 Re r + =1 ( ) γ r + θ + tr (S S where r μ γ (θ ). θ ) ), (42) The so-caed Generazed Sepan formuas 7: s constant w.r.t. θ In ths partcuar case, we have that a the dervatves of w.r.t. θ are zero. In partcuar, S = S = P = P = P =. Then, by posng = 1 as n 7, the entres of the matrces B ( θ ) A ( θ ) can be obtaned from (42) (34) respectvey, as: B (θ ), = 2 Re ( γ θ ) γ A (θ ), = 2 Re θ γ θ γ 2 Re θ ( r ) (43) 2 γ, θ θ (44) where r μ γ(θ ). It s mmedate to verfy that these two expresson are exacty the same of Eqs. (38) (4) derved n The so-caed Generazed Bangs formuas 7: γ s constant w.r.t. θ ere we suppose that the mean vaue γ s ndependent of the parameter vector θ, consequenty we have that γ θ = 2 γ θ θ =. Then, f = 1, the entres of the matrx B ( θ ) n (42) can be ready expressed as: ( B (θ ), = 2 Re + tr (S ) r S ( r ) ), (45) from whch one deduces straghtforwardy Eq. (44) of 7 wth r μ γ(θ ). Fnay, et us now derve the matrx A ( θ ) for the partcuar scenaro at h. From (34), we get: / 2 A (θ ), = tr (P P P ) r (P P + P P P r + tr (P P + P P P ) / 2 / 2. (46) whch s the same as that gven n Eq. (46) of Concuson ) / 2 The am of ths paper was to provde SB formuas for CES dstrbuted data under mode msspecfcaton thus to f a theoretca practca gap n the Sgna Processng terature. Furthermore, we have shown that the proposed SB formuas encompass a the prevousy derved expressons as speca cases. Moreover, these new SB formuas nvoved reatvey sght modfcatons wth respects to the cassca counterpart obtaned for Gaussan correcty specfed data modes wth ony an expectaton of some scaar functons to derve or to evauate numercay. The practca mportance of the proposed expressons s n the fact that aow us to easy evauate Msspecfed Cramér-Rao Bounds for a ot of appcatons that are characterzed by a non-gaussan heavy-taed data behavour aong wth a mode msspecfcaton. Future works w focus exacty on the appcaton of the derved SB formuas to a pethora of engneerng probems such as the Drecton of Arrvas (DOA) estmaton n array processng the structured (.e. Toeptz) covarance/scatter matrx estmaton for adaptve detecton agorthm n non-gaussan scenaros. Acknowedgement The work of Stefano Fortunat has been supported by the Ar Force Offce of Scentfc Research under award number FA The work of M. N. E Korso A. Renaux has been supported by ANR-Astrd MARGARITA. Appendx A. Some consderatons on the expectaton of the terms V ( θ ) ( θ ) The am of ths appendx s to show under whch condtons the expectatons V ( θ ) ( θ ) coud be evauated n cosed form. Keepng n mnd the Stochastc Representaton of a CES dstrbuted rom vector x gven n (2), et us defne the vector z = Tu. Then, by recang that r (see (32)), the term θ n (22) can be rewrtten as: θ = μ γ γ θ (Q 1 / 2 z + r ) (Q 1 / 2 z + r ) (Q 1 / 2 z + r ) S (Q 1 / 2 z + r ), γ θ (A.1) where S s defned n (23). et us defne the foowng vector scaar quanttes: h = S r + γ θ A = (r ) S r + (r ) γ γ + θ θ r. Then (A.1) becomes = Q θ z S z Q 1 / 2 z h Q 1 / 2 h z A. (A.2) (A.3) (A.4) Note that, accordng to the Stochastc Representaton Theorem 2, we have that Q 1 / 2 = d R Q = d Q. Consder now the term φ(g that: ) φ(g ) θ θ. By expotng the expresson n (A.4), we have = φ(g ) Q z S z φ(g ) Q 1 / 2 z h φ(g ) Q 1 / 2 h z A φ(g ). It must be noted now that, snce G (A.5) has been defned n (2) as, G (x γ ) (x γ ) = (Q 1 / 2 z + r ) (Q 1 / 2 z + r ) (A.6) the rom varabe φ(g 1 / 2 ) the other rom quanttes Q z are mutuay dependent consequenty the cosed form evauaton of the expectaton operator n (A.5) s not feasbe. Smar consderatons hod true for a the other terms nvovng the expectaton operator w.r.t. the true pdf p X ( x ), that are φ(g ) φ(g m ) m θ θ for V ( θ ) n (21), φ (G ) θ θ

8 A. Mennad et a. / Sgna Processng 142 (218) φ(g ) 2 G θ θ for ( θ ) n (24). There are, however, two mportant cases n whch φ(g 1 / 2 ) s statstcay ndependent of Q ( consequenty of Q ) z : these are the Scenaro 1 the Scenaro 2 whch are dscussed n the foowng Appendces. Appendx B. Proof of the SB formuas for the Scenaro 1 As dscussed n Secton 3.2, n the Scenaro 1, the true margna pdf s an unspecfed CES pdf p X (x ) = CES N (μ,, g), whe the assumed one s a parametrc compex Gaussan pdf f X (x ; θ) = CN (γ (θ), (θ)). As a consequence, φ(g (θ)), θ then t trvay satsfy the ndependence condton dscussed n Appendx A. In order to obtan a cosed form expresson of the matrx B ( θ ), the second term n (3) can be rewrtten as: =1 m =1 m = θ θ =1 + θ θ =1 m =1, m =. θ θ (B.1) By usng (A.4) the Stochastc Representaton Theorem, when = m we have that: θ θ = E Q 2 Q z S z z m S z m ( A m z + E Q Q + A z m S z m S z ) + A A m, (B.2) where A s defned n (A.3) where we used the facts that z z m are ndependent z = z m =. When = m, the term n (B.2) has to be recast as: = E θ θ Q Q 2 z S z z S z ( + E Q Q A z + z h h z S z + A z S z + h z z h ) + A A, (B.3) where h s defned n (A.2) where we used the fact that the thrd-order moments of u ( then of z ) vansh (see emma 1 n 2 ). Moreover, as a consequence of the crcuarty property of z 2, the foowng equates hod: z z T = z z =, z h z h = h T z z h =, h z h z = h z z T h =. From the propertes of the trace operator, we have that: z S z = N tr (S ), z h h z = N tr (h h ), h z z h = N tr (h h ), (B.4) (B.5) (B.6) (B.7) (B.8) (B.9) whe, by expotng exacty the same procedure used n 11,2, t can be shown that: z S z z S z = tr (S ) tr (S ) + tr (S S ). (B.1) Another usefu reaton can be obtaned by expotng the equaty n (27). In fact, by combnng (A.5) wth (27), we have that: E Q Q z S z = tr (P ) A, = 1,..., d, (B.11) then, from (B.7) from the dentfabty condton for CES dstrbuton assumed n ths paper,.e. E Q Q = N, we obtan: A = tr (P ) tr (S ), = 1,..., d. (B.12) Usng (B.11) the condton E Q Q = N ( consequenty E Q 2 Q = N 2 ), the term n (B.2) can be easy evauated as: = tr (P θ θ ) tr (P ). (B.13) Smary, usng (B.12), (B.8), (B.9), (B.1) the condton E Q Q = N, the term n (B.3) can be expressed as: = tr (P θ θ ) tr (P ) + tr (h h ) + tr (h h ) =1 m =1 + ( E Q Q 2 1 ) + tr (S S ) tr (S ) tr (S ) Fnay, by combnng a the prevous resuts, we have that: = θ θ 2 tr (P ) tr (P ) + tr (h h ) + tr (h h ) =1 E Q Q tr (S + E Q Q 2 tr (S S ), ) tr (S ) (B.14) (B.15) from whch (31) foows mmedatey. Regardng the cacuaton of the matrx A ( θ ) n (34), no partcuar smpfcaton can be made. 2 G Specfcay, the term θ θ n (33) has to be obtaned through drect cacuaton from (A.1). Snce the dervaton s ong, tedous does not add any nsghtfu consderatons about the probem at h, we decded to not report t here. Appendx C. Proof of the SB formuas for the Scenaro 2 As prevousy dscussed n the paper, n the Scenaro 2 we suppose that the true margna pdf s gven by a parametrc CES dstrbuton such that p X (x ) p X (x ; θ) = CES N (μ,, g), where μ = γ ( θ) = ( θ) for a gven θ. The assumed pdf s tsef a parametrc CES dstrbuton that share the same parametrzaton of the true one but may have a dfferent densty generator,.e. f X (x ; θ) = CES N (γ (θ), (θ), w ), θ, possby g(t) = w (t), t R +. In order to guarantee the correct dentfabty of the true the assumed CES dstrbutons, as before, we may mpose a constrant of both the moduar varate, that s: (x μ ) (x μ ) (x γ ( θ) ) ( θ) (x γ ( θ)) = E Q Q = N, E f θ (x γ (θ)) (θ) (x γ (θ)) = N, θ. (C.1) (C.2) et us suppose now that, for a gven true pdf p X (x ) p X (x ; θ) for a gven assumed pdf f X ( x ; θ), the pseudo-true parameter vector θ equates the true parameter vector θ, consequenty, from (14): D ( p X f θ ) θ θ= θ = n f X (x ; θ) θ θ= θ =, = 1,..., d. (C.3)

9 328 A. Mennad et a. / Sgna Processng 142 (218) Under ths assumpton, we have that: G (x γ ) (x γ ) = (x γ ( θ)) ( θ) (x γ ( θ)) = (x μ ) (x μ ) Q = d Q = R 2, (C.4) consequenty, the Stochastc Representaton Theorem aows us to wrte the foowng equaty: x γ = x μ = d R Tu = d R 1 / 2 u. (C.5) where γ γ (θ ) ( θ ). It s worth to hghght that the equaty chans n (C.4) (C.5) hod true f ony f θ = θ. We are qute confdent that, n the context of Scenaro 2 (.e. when the msspecfcaton s ony due to a wrong assumpton on the densty generator), the equaty θ = θ aways hods true. Snce we have not a proof of ths fact yet, we consdered t as an assumpton (Assumpton A2 n Secton 3.1 ). As a consequence of (C.4), the expectaton of functons of G, say h (G ), can be expcty derved as: E Q h (G ) h (t) p Q (t) dt = δ N,g h (t) t N g(t) dt = E Q h (Q), N. (C.6) Snce such expectaton does not depend on the ndex on the unknown parameter vector θ, n the rest of ths Appendx we aways ndcate E Q h (G ) smpy as E Q h (Q). As n Appendx B for the Scenaro 1, n order to obtan a cosed form expresson of the matrx B ( θ ), the second term n (28) can be rewrtten as: φ(g ) φ(g m ) θ θ = φ 2 (G ) θ θ =1 m =1 =1 + =1 m =1, m = φ(g ) φ(g m ). (C.7) θ θ The term θ can be easy obtaned by frsty takng the derva tve of the terms n (22) then by substtutng n the obtaned expresson the Stochastc Representaton of the dfference vector x γ = Q θ 1 / 2 n (C.5), wehavethat: ( ( γ θ ) ) γ / 2 u + u / 2 Q θ u P u. et us now reca from 18 the foowng three equates: u P u u P u = tr (P ) tr (P ) + tr (P P ) u P u = tr (P u u ) = N tr (P ) (u a )(u Du ) = (C.8) (C.9) (C.1) (C.11) for some vector a hermtan matrx D. By substtutng (C.8) n (C.7) by expotng the equates (C.9) (C.11), rememberng that u u = (1/N) I (see Secton 2 ), we get: =1 m =1 φ(g ) φ(g m ) θ θ = 2 E Q Q φ 2 (Q ) N =1 Re γ θ γ θ + E Q Q 2 φ 2 (Q) tr (P ) tr (P ) + tr (P P ) E Q + ( 1) 2 Q φ(q ) tr (P N 2 ) tr (P ). (C.12) et us now mpose the equaty n (27). In partcuar, we have that: =1 φ(g ) = E θ Q Q φ(q ) =1 γ γ θ / 2 u + u / 2 θ E Q Q φ(q ) u P u =1 = N E Q Q φ(q ) tr (P ), (C.13) where we used the fact that the fact that u = whe the ast equaty foows from (C.1). Then, by expotng the equaty (27), we obtan the foowng reaton: E Q Q φ(q ) = N, (C.14) where φ(t) = w (t ) /w (t ) w ( t ) s the densty generator of the assumed CES dstrbuton f X ( x ; θ). Fnay, by substtutng (C.14) n (C.12), then repacng the obtaned term n (28), we get the cosed form expresson of the matrx B ( θ ) gven n (35). The evauaton of the matrx A ( θ ) n (36) foows drecty from the drect cacuaton of the terms φ (G ) θ θ φ(g ) 2 G θ θ. In partcuar, a the dervatves have to be evauated from (22), whe, to evauate the expectaton operator w.r.t. the true dstrbuton p X ( x ), one has to use the Stochastc Representaton n (C.5) by keepng n mnd that θ = θ. Snce ths cacuaton s tedous not nformatve, we decded to not report t here. References 1 A.M. Zoubr, V. Kovunen, Y. Chakhchoukh, M. Muma, Robust estmaton n sgna processng: a tutora-stye treatment of fundamenta concepts, IEEE Sgna Process. Mag. 29 (4) (212) E. Oa, D.E. Tyer, V. Kovunen,.V. Poor, Compex eptcay symmetrc dstrbutons: survey, new resuts appcatons, IEEE Trans. Sgna Process. 6 (11) (212) C.D. Rchmond, Adaptve Array Sgna Processng Performance Anayss n Non-Gaussan Envronments, Massachusetts Insttute of Technoogy, 1996, Ph.D. thess. Avaabe at: 4 P.J. uber, The behavor of maxmum kehood estmates under nonstard condtons, n: Proceedngs of the Ffth Berkeey Symposum n Mathematca Statstcs Probabty, Unv. of Caforna, Berkeey, CA, USA, Whte, Maxmum kehood estmaton of msspecfed modes, Econometrca (1982) Q.. Vuong, Cramér-rao Bounds for Msspecfed Modes, Dv. of the umantes Soca Sc., Caforna Inst. of Techno., Pasadena, CA, USA, Workng Paper 652, Oct. Onne. Avaabe: cramer-rao-bounds-msspec-fed-modes 7 C.D. Rchmond,.. orowtz, Parameter bounds on estmaton accuracy under mode msspecfcaton, IEEE Trans. Sgna Process. 63 (9) (215) S. Fortunat, F. Gn, M. Greco, The Msspecfed Cramér-Rao bound ts appcaton to scatter matrx estmaton n compex eptcay symmetrc dstrbutons, IEEE Trans. Sgna Process. 64 (9) (216) S. Fortunat, F. Gn, M.S. Greco, The constraned Msspecfed Cramér-Rao bound, IEEE Sgna Process. ett. 23 (5) (216) C. Ren, M.N. E Korso, J. Gay, E. Chaumette, P. arzaba, A. Renaux, Performance bounds under msspecfcaton mode for mmo radar appcaton, n: rd European Sgna Processng Conference (EUSIPCO), 215, pp M. Greco, S. Fortunat, F. Gn, Maxmum kehood covarance matrx estmaton for compex eptcay symmetrc dstrbutons under msmatched condtons, Sgna Process. 14 (214)

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