Estimation accuracy of non-standard maximum likelihood estimators

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1 Estimation accuracy of non-standard maximum likelihood estimators abil Kbayer J Galy E Chaumette François Vincent Alexandre Renaux P Larzabal To cite this version: abil Kbayer J Galy E Chaumette François Vincent Alexandre Renaux et al.. Estimation accuracy of non-standard maximum likelihood estimators. International Conference on Acoustics Speech and Signal Processing ICASSP) Mar 207 ew Orleans United States. IEEE International Conference on Acoustics Speech and Signal Processing. <0.09/icassp >. <hal > HAL Id: hal Submitted on 2 May 207 HAL is a multi-disciplinary open access archive for the deposit and disseation of scientific research documents whether they are published or not. The documents may come from teaching and research institutions in France or abroad or from public or private research centers. L archive ouverte pluridisciplinaire HAL est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche publiés ou non émanant des établissements d enseignement et de recherche français ou étrangers des laboratoires publics ou privés.

2 ESTIMATIO ACCURACY OF O-STADARD MAXIMUM LIKELIHOOD ESTIMATORS. Kbayer ) J. Galy 2) E. Chaumette 3) F. Vincent 3) A. Renaux 4) and P. Larzabal 5) ) Tésa/Isae-Supaero Université de Toulouse 7 Boulevard de la Gare Toulouse France nabil.kbayer@isae.fr) 2) Université de Montpellier 2/LIRMM 6 rue Ada Montpellier France galy@lirmm.fr) 3) Isae-Supaero Université de Toulouse 0 av. Edouard Belin Toulouse France [eric.chaumette francois.vincent]@isae.fr) 4) Université Paris-Sud/LSS 3 Rue Joliot-Curie Gif-sur-Yvette France renaux@lss.supelec.fr) 5) Université Paris-Sud/SATIE 6 av. du President Wilson Cachan France pascal.larzabal@satie.ens-cachan.fr) ABSTRACT In many deteristic estimation problems the probability density function p.d.f.) parameterized by unknown deteristic parameters results from the marginalization of a joint p.d.f. depending on additional random variables. Unfortunately this marginalization is often mathematically intractable which prevents from using standard maximum likelihood estimators MLEs) or any standard lower bound on their mean squared error MSE). To circumvent this problem the use of joint MLEs of deteristic and random parameters are proposed as being a substitute. It is shown that regarding the deteristic parameters: ) the joint MLEs provide generally suboptimal estimates in any asymptotic regions of operation yielding unbiased efficient estimates 2) any representative of the two general classes of lower bounds respectively the Small-Error bounds and the Large-Error bounds has a non-standard version lower bounding the MSE of the deteristic parameters estimate. Index Terms Deteristic parameter estimation maximum likelihood estimators estimation error lower bounds. ITRODUCTIO As introduced in [ p53] a model of the general deteristic estimation problem has the following four components: ) a parameter space Θ consisting of a set of parameter vectors θ Θ R P 2) an observation space Ω consisting of a set of observation vectors x Ω C M 3) a probabilistic mapping from Θ to Ω that is the probability law that governs the effect of a vector parameters value θ on the observation x and 4) an estimation rule θ x) that is the mapping of Ω into estimates. Actually in many estimation problems the probabilistic mapping results from a two steps probabilistic mechanism leading to a probability density function p.d.f.) of the form: p x θ) = p x θ r θ) p θ r θ) dθ r θ r R Pr ) where θ r is a random vector and p x θ r θ) and p θ r θ) are known. Classical examples are the reception of M samples from a signal source either by a radar a sonar or a telecom system in the presence of Gaussian or spherically invariant thermal noise [2]: x = s θ s) a + n θ n) θ T = ) θ T s θ T n θr a. In the case of an active radar [][3] θ r a are the complex amplitudes of the received signals backscattered by P r targets which may fluctuate according to a given or measured) p.d.f. such as the Swerling laws This work has been partially supported by the icode institute research project of the IDEX Paris-Saclay by the DGA/DGCIS by the DGA/MRIS by the CES and by the AR MAGELLA. [4]. Therefore as recalled in [5] deteristic estimation problems can be divided into two subsets: the subset of standard estimation problems for which a closed-form expression of p x θ) is available and the subset of non-standard estimation problems for which only an integral form of p x θ) ) is available. Since in non-standard estimation maximum likelihood estimators MLEs) 2a) can be no longer derived the use of joint MLEs of θ and θ r 2b) are proposed as being a substitute: θ ML x) = arg max {p x θ)} 2a) θ Θ { θ x) θr x)} = arg max {p x θ r θ)} 2b) θ Θθ r Π θr where Π θr is the support of p θ r x θ). It is a sensible solution in the search for a realizable estimator of θ. Indeed the widespread use of MLEs originates from the fact that under reasonably general conditions on the observation model [7][8] the MLEs are in the limit of large sample support asymptotically unbiased Gaussian distributed and efficient. If the observation model is Gaussian some additional asymptotic regions of operation yielding unbiased Gaussian and efficient MLEs have also been identified at finite sample support [9]- [3]. Historically the open literature on the estimation accuracy of MLEs in terms of mean squared error MSE) including the associated lower bounds LBs) has remained focused on standard deteristic estimation 2a) [7][4]-[33]. It is the reason why θ x) and θ r x) 2b) are referred to as non-standard MLEs SMLEs). Interestingly enough despite its frequent occurrence in practical estimation problems the study of SMLEs has received little attention and the contributions have been limited to the derivation of two representatives of the Small-Error bounds namely the Cramér-Rao bound CRB) [6][34] and the Battacharayya bound BaB) [35]. The aim of the present communication is to complete this initial characterization of estimation accuracy of SMLEs. First the intuitive idea [6][34][35] that SMLEs are generally suboptimal estimates in any asymptotic region of operation yielding unbiased efficient estimates) is rigorously established. Therefore it is of interest to investigate the suboptimality of the SMLEs which can be in some extent quantified by lower bounds LBs) derivation and comparison. Thus as a second contribution we show that any representative of the two general classes of LBs on the MSE respectively the Small-Error bounds and the Large-Error bounds has a non-standard version lower bounding the MSE of SMLEs. Small-Error bounds are not able to handle the threshold phenomena whereas it is revealed by Large-Error bounds that can be used to predict the threshold value. Last some of the results introduced are exemplified by a new look at the well known Gaussian complex observation models.

3 We focus on the scalar case i.e. θ θ although the results are easily extended to the estimation of a vector of parameters [30][3]. 2. RELATIO TO PRIOR WORK Despite its frequent occurrence in estimation problems the study of SMLEs 2b) has received little attention and the contributions have been limited to the derivation of the appropriate CRB [6][34] and BaB [35]. Our contribution is two-fold: we show that ) SM- LEs are generally suboptimal estimates in any asymptotic regions of operation yielding unbiased efficient estimates 2) any standard Small-Error or Large-Error bound on the MSE has a non-standard version lower bounding the MSE of SMLEs. 3. O THE SUBOPTIMALITY OF SMLES In the present communication the discussion is restricted to the case where the supports of p x θ r θ) and p θ r x θ) are independent of θ: θ) = { x θ r) C M R Pr p x θ r θ) > 0 } and Π θr x θ) = { θ r R Pr p x θ r θ) > 0 } Π θr x. Let L 2 Ω) respectively L 2 ) be the real Euclidean space of square integrable real-valued functions over the domain Ω respectively. Let us denote φ = ) θ θ T T r Θ R P r p x φ) p x θ r θ) and E x φ [ ] E x θrθ [ ]. Then any estimator φ ) T = θ θt r φ x θ r) L 2 ) of a selected vector value φ uniformly unbiased for p x φ) must comply with: [ φ] φ Θ R Pr : E x φ = φ 3) which implies that: ] θ Θ : E T xθ [ φ = E T [ r θ θ r φ ] = θ θ E T θ r θ [ θ r ] ) 4) that is φ is an uniformly unbiased estimate of g θ) T = θ E T θ r θ [θ r] ) for p x θ r θ). As the reciprocal is not true: θ Θ : E xθ r θ [ φ ] φ = 0 φ Θ R Pr : E x φ [ φ ] φ = 0 } then U ) = { φ L 2 ) verifying 3) V ) = { φ L 2 ) verifying 4)}. Let U Ω) and V Ω) be the restriction to L 2 Ω) of U ) and V ). As φ L 2 ): [ ) ) ] T E xθr θ = [ [ φ]) ]) ] T E xθr θ φ Ex φ φ Ex φ [ φ + [ ] ) ] ) ] T E x φ [ φ g θ) E x φ [ φ g θ) 5a) therefore if φ U ): [ ) ) ] T E xθr θ = [ ) ) ] T φ φ φ φ + C θ φ) 5b) E xθr θ In most cases the inclusion is strict leading to strict inequalities 6a-6c) [ ] 0 0 where C θ φ) = and C 0 C θ θ θ θ r) is the covariance r) matrix of θ r. Also as U Ω) V Ω) and U Ω) U ) finally: φ VΩ) φ UΩ) { E xθr θ φ φ φ φ + C θ φ) 6a) and in particular as θ = e T φ where e = ) T : [ ) ]} 2 θ θ θ VΩ) θ UΩ) [ θ θ ) 2 ]}. 6b) In any asymptotic regions of operation yielding unbiased efficient estimates θ ML V Ω) θ U Ω) and both reach the imum MSE. Thus according to 6b) the SMLEs of θ is generally an asymptotically suboptimal estimator of θ in the MSE sense) in comparison with the MLE of θ. Therefore from a theoretical as well as a practical viewpoint it is of interest to investigate the suboptimality of the SMLEs which can be in some extent quantified by LBs derivation and comparison. 4. O-STADARD LOWER BOUDS It is worth noticing that an equivalent form of 6a) is: C θ φ) φ VΩ) φ UΩ) { [ ) ) ]]} T [E x φ φ φ φ φ 6c) since the latter form 6c) is the corner-stone to derive LBs on the MSE of SMLEs. Indeed since U Ω) U ) any LB on the MSE over U ) is a LB on the MSE over U Ω). In its seal work in standard estimation [9] Barankin has shown that the locally-best at θ uniformly unbiased estimator is the solution of a norm imization under linear constraints [5 Section 2]: { } 2 θ x) θ θ L 2 Ω) where υ θ x; θ ) = px θ ) px θ) under θ θ x) θ υθ x; θ ) θ = θ θ θ Θ denotes the likelihood ratio LR) and: ] MSE θ [ θ = θ x) θ g x) h x) θ = E x θ [g x) h x. 2 θ Unfortunately if Θ contains a continuous subset of R then the norm imization 7) leads to an integral equation with no analytical solution in general. As a consequence many studies quoted in [28]- [32] have been dedicated to the derivation of computable LBs approximating the MSE of the locally-best uniformly unbiased estimator aka the Barankin bound BB). All these approximations derive from sets of discrete or integral linear transform of the Barankin constraint 7): E x θ [ θ x) θ ) υ θ x; θ = θ θ θ Θ. These results are readily generalizable to the parameters vector case [30][3] that is any Barankin bound approximation BBA) 7)

4 on {E x φ φ φ φ φ can be derived from φ U ) discrete or integral linear transforms of the set of constraints: ) ] n [ ] E x φ [ φ φ x; φ n ) = φ n φ 8a) where x; φ n ) = px φn ) that is as the solution of: px φ) {E x φ φ φ φ φ φ U ) under ) E x φ [ φ φ υ T φ Φ = Ξ Φ ) where Φ = [ φ... φ ] Ξ Φ ) = [ φ φ... φ φ ] 8b) υ ) φ Φ υ ) φ x; Φ = υ ) φ x; φ... υ )) φ x; φ T which defines the following BBA [30 Lemma ]: ) C φ φbba = Ξ Φ ) R ) Φ Ξ Φ ) T φ BBA = Ξ Φ ) R ) Φ υ ) 8c) φ x; Φ where R ) [ υφ Φ = E ) x φ υφ Φ υ T φ Φ and ) [ ) ] T C φ φbba = E x φ φbba φ) φbba φ is the covariance matrix of φ BBA. Even if in general φ BBA φ BBA x; φ) 8c) is a clairvoyant estimator and does not belong to U Ω) as: φbba { 9) E xθr θ φ φ φ φ φ UΩ) U Ω) containing asymptotically the SMLEs it seems sensible to call φbba a non-standard LB SLB) and to denote SLB φbba to make the difference with the{ modified [ LBs MLB) which are LBs on ) ) ]} T E xθr θ [5]. In the same φ V ) vein φbba can also be regarded as a non-standard BBA SBBA). ote that in general the SLBs cannot be arranged in closed form due to the presence of the statistical expectation. They however can be evaluated by numerical integration or Monte Carlo simulation. Last since U ) V Ω) and V Ω) U ) no general result can be drawn from 6c) on the ordering between φbba C θ φ) and φ VΩ) BBA computed on V Ω). 4.. Lower bound examples or any A typical example is the CRB obtained for = 2 where φ = ) θ θ T T r and φ 2 = ) θ + dθ θ T T r leading to the following subset of constraints: ) [ 0 ) = E x φ θ x θr) θ dθ υ ) φ x; φ 2 0a) which is equivalent to [33 Lemma 3]: ) [ 0 ) = E x φ θ x θr) θ υ φx;φ 2 ) 0b) dθ and can be reduced to [33 Lemma 2]: [ ) ] p x θr θ + dθ) p x θ r θ) = E x φ θ x θr) θ dθp x θ r θ) [ 0c) since E ) x φ υφ x; φ 2 = 0. Then by letting dθ be infinitesimally small 9) becomes: [ [ ) ] 2 ] ln p x φ) SCRB E x φ ) θ that is the Miller and Chang bound [6 7. Following the rationale introduced in [22] a straightforward extension of ) is obtained for Φ = [ φ... φ ] φ n = ) θ + n ) dθ θ T T r n. Indeed the set of associated constraints: ) dθ 0... ) T = E x φ [ θ x θr) θ Φ 2a) by letting dθ be infinitesimally small becomes equivalent to [22]: ) ] ) T = E x φ [ θ x θr) θ b x; φ) 2b) ) T where b x; φ) = p x φ) px φ)... px φ) px φ) θ θ. [ ] As E x φ [b x; φ) b n x; φ = E n px φ) x φ = 0 2 n n θ 2b) is actually equivalent to [33 Lemma 2]: ) ] e = E x φ [ θ x θr) θ b x; φ) 2c) ) T where b x; φ) = px φ)... px φ) px φ) θ θ e = ) T and 9) leads to the BaBs [8] of order [35]: [ ] ] SBaB [e T E x φ b x; φ) b T x; φ) e. 3) Last a key representative of the Large Error bounds is the Mcaulay- Seidman bound MSB) [24] which is the practical form of the BB. Its non-standard form is : SMSB = φbba Tighter non-standard lower bounds Interestingly enough it is quite simple to introduce tighter SLBs. It suffices to note that the addition of any subset of K constraints: k [ + + K] φ k φ = + ) E x φ [ φ x θr) φ x; φ k φ k = ) T T θ k θr) k to 8b) restricts the class of viable estimators φ x θ r) and therefore increases the associated SBBA 8c) leading to: [ ) Ξ Φ R Φ ) ) ] Ξ Φ T [ Ξ Φ +K) R Φ +K) Ξ Φ +K) ] T 4a) and regarding the estimation of θ to: [ ) ξ θ T R Φ ) ξ θ [ ξ θ +K) T R Φ +K) ξ θ +K 4b)

5 where ξ θ L) = θ θ... θ L θ ) T Φ L = [ φ... φ L] Ξ Φ L) = [ φ φ... φ L φ ] for L { + K}. A typical example is given by the SCRB ). Indeed by adding to 0a) the following K = P r constraints: ) 0 = E x φ [ θ x θr) θ x; Φ K x; Φ K) = x; φ )... x; φ K)) T where φ k = θ θ r + u k hr) k T ) T and uk is the kth column of the identity matrix I Pr one obtains the following equivalent set of constraints [33 Lemma 3+Lemma 2]: ) e = E x φ [ θ x θr) θ c x; Φ K+ c x; Φ K+) ) = px θrθ+dθ) dθ px θ rθ) ) )) px θr+u h θ) h px θ rθ)... px θr+uk h K θ) h K px θ rθ). By letting dθ h... h Pr ) be infinitesimally small then c x; Φ K+) ln px φ) and 4b) becomes [34 24: φ [ [ ) ] 2 ] ln p x φ) E x φ θ [ [ ] ] ln p x φ) e T ln p x φ) E x φ e φ φ T. 5) The above example illustrate that the tightest form of any SLB is obtained when the set of unbiasedness constraints are expressed for φ as in 4b) however at an additional cost in numerical integration or Monte Carlo simulation to evaluate the additional statistical expectations Further considerations Since any existing standard Small-Error [7][5]-[8] or Large-Error bound [9]-[26][28]-[3] on φ can be obtained from 8c) it has a non-standard form obtained from φbba [5]. Let us recall that in general φ BBA φ BBA x; φ) U ) therefore the associated SLB can not be compared a priori neither with the MSE of θ ML V Ω) nor with any of its LBs computed with p x θ)). In particular SLBs are not in general neither upper bounds on the MSE of θ ML nor on any of its LBs; comparisons are possible only on a case-by-case basis. However if p θ r θ) does not depend on θ then one can derive inequalities between modified [5] and nonstandard LBs proofs are given in [36]). In the general case where p θ r θ) does depend on θ no general inequalities between modified and non-standard LBs can any longer be exhibited; comparisons are possible only on a case-by-case basis. 5. A EW LOOK AT GAUSSIA OBSERVATIO MODELS In many practical problems of interest radar sonar communication...) the complex M-dimensional observation vector x consists of a bandpass signal which is the output of an Hilbert filtering leading to a complex Gaussian circular vector x C m x C x) [][37 3][38]. Two particular signal models are mostly considered: the deteristic conditional) signal model and the stochastic unconditional) signal model [39]. In the deteristic case the unknown parameters are connected with the expectation value whereas they are connected with the covariance matrix in the stochastic one. A simple and well known instantiation is: x t = s τ) a t + n t t T 6) where a... a t are the complex amplitudes of the signal s ) is a vector of M parametric functions depending on a single deteristic parameter τ n t C ) 0 σ 2 ni M t T are independent and identically distributed i.i.d.) Gaussian complex circular noises independent of the signal of interest. Additionally if a = a... a T ) T C ) 0 σ 2 ai T then 6) is an unconditional signal model parameterized by θ = τ σ 2 a σn) 2 T and the MLE 2a) of τ aka the unconditional MLE UMLE) is obtained by{ imization of the concentrated criterion []: τ = arg σ2 a s τ) s H τ) + σ } 2 τ ni M. The associated CRB aka the unconditional CRB UCRB) is [ 4.64[40]: ) UCRB τ = σ 2 n 2h τ) T σ 2 SR a 7) SR + SR = σ2 a s τ) 2 h τ) = σ 2 n H s τ) Π sτ) τ s τ) τ where Π a = I M aa H a 2 and SR is the signal-to-noise ratio computed at the output of the single source matched filter []. The SMLE 2b) of τ is actually the conditional MLE CMLE) obtained{ by imization of the concentrated criterion []: τ = T } arg τ t= xh t Π sτ)x t and the associated SCRB is: SCRB τ = E a σ 2 a [CCRB τ a CCRB τ a) = σ 2 n 2h τ) a 2 where CCRB τ denotes the conditional CRB associated to the CMLE [ First it has been shown [ 4.74 in the case of a vector of unknown parameters τ that asymptotically where T : C θ τ ) C θ τ ) = UCRB τ CCRB τ 8) which illustrates that the act of resorting to the SMLE here the CMLE) is in general an asymptotic suboptimal choice in the MSE sense. However in the case of single unknown parameter τ 8) becomes: C θ τ) = C θ τ) = UCRB τ which highlights that in some particular cases the SMLE may be asymptotically equivalent to the MLE in the MSE sense. Second since a 2 χ 2 σ 2 2T i.e. a chi-squared random variable with a 2T degrees of freedom then [37]: SCRB τ = Therefore if T 2: SCRB τ UCRB τ σ 2 n T 2T σ 2 a hτ) T = SCRBτ C θ τ) if T 2 if T =. 9) = T SR T SR + 20) which illustrates the facts that SLB are not in general neither upper bounds on the MSE of MLEs nor on any of its LBs.

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