B Chouvion

S Mcwilliam

A A Popov

C H J Fox

Review and comparison of different support loss models for MEMS resonators undergoing in-plane vibration

Keywords: Support loss, anchor loss, Q-factor, perfectly matched layer, infinite elements

Several approaches for calculating support loss in MEMS resonators undergoing in-plane vibration are reviewed. In each of them, the support is approximated as a semi-infinite domain. The first approach is analytical and models the support as a semi-infinite thin plate. This is compared with two different finite element approaches that introduce artificial boundaries to their finite domain. In order to absorb outgoing waves and model the infinite support, a perfectly matched layer method and the use of infinite elements are considered. Simple test cases are studied and the results for the support losses predicted by the different methods are compared. It is shown that each of the methods yield similar trends. Using the developed analytical model, a parametric study is performed on the support losses of a ring-based resonator. General strategies for improving the quality factor by reducing support losses are provided.

Introduction

Damping is an important consideration in the design of resonant MEMS devices as their performance can be highly sensitive to the presence of damping. A variety of loss mechanisms contribute to the damping in MEMS resonators, and the required level of damping is application dependent. For example, accelerometers require significant levels of damping to ensure the vibrations are critically damped, whilst it is advantageous for vibrating rate sensors to have low levels of damping. The latter forms the motivation for the study performed here. By removing the air that surrounds the resonator, the effects of fluid damping can be eliminated. Thermoelastic damping, caused by the interaction between elastic strain and thermal effects, can be reduced by using an appropriate geometric design for the resonator [START_REF] Lifshitz | Thermoelastic damping in microand nanomechanical systems[END_REF][START_REF] Yi | Geometric effects on thermoelastic damping in MEMS resonators[END_REF].

Internal friction and surface loss are ever-present sources of energy loss in micro-resonators [START_REF] Yasumura | Quality factors in micron-and submicron-thick cantilevers[END_REF][4][START_REF] Tabrizian | Effect of phonon interactions on limiting the f.Q product of micromechanical resonators[END_REF] and can be reduced by surface treatment [START_REF] Yang | Investigating surface stress: surface loss in ultrathin single-crystal silicon cantilevers[END_REF][START_REF] Henry | Controlling energy dissipation and stability of micromechanical silicon resonators with selfassembled monolayers[END_REF]. Support losses account for the loss of energy from the resonator through the supporting structure, and have been less widely studied compared to the other loss mechanisms. This fundamental loss mechanism forms the main focus of this study.

Practical MEMS resonators incorporate a resonator that is connected to a supporting structure. Vibration of the resonator causes the supporting structure to deform slightly, allowing some of the vibrational energy to flow from the resonator to the surrounding material. This flow arises because elastic deformations are created at the attachment point to the support, which generate stress waves. Careful design of the resonator and its at-tachment to the support can reduce support loss by reducing motion of the attachment point, thereby reducing the coupling between the resonator and the surrounding material [START_REF] Langdon | Resonator sensors -review[END_REF]. Such systems may involve a mounting mechanism which isolates shear and moment reactions from the support, or the introduction of blocking masses to reduce the vibration transmitted to the boundaries [9].

To predict support loss, the coupled resonator-support system must be considered in order to understand the interaction and energy transmission between them. It is generally straightforward to model the resonator in isolation, without having to make too many assumptions, using classical finite element analysis and/or modal analysis techniques. The admittance that links velocity and applied forces at the attachment point provides a way to characterise the influence of the resonator on the support when they are modelled independently [START_REF] Judge | Attachment loss of micromechanical and nanomechanical resonators in the limits of thick and thin support structures[END_REF]. However, the disadvantage of this approach is that the physical flexibility of the attachment area is not taken into account properly in the resonator vibration. A simple and precise modelling of the interaction between resonator and support can be difficult to obtain. A more complete model that considers resonator and support together, such as for instance the one presented by Park and Park [START_REF] Park | High-fidelity modeling of MEMS resonators -Part I: Anchor loss mechanisms through substrate[END_REF][START_REF] Park | High-fidelity modeling of MEMS resonators -Part II: Coupled beam-substrate dynamics and validation[END_REF], is desirable, but the associated complexity also greatly increases, and makes it difficult to perform detailed modelling of the entire resonator-support system; so simplifications are often made. For example, the support is often assumed to be semiinfinite when calculating support losses [START_REF] Judge | Attachment loss of micromechanical and nanomechanical resonators in the limits of thick and thin support structures[END_REF][START_REF] Park | High-fidelity modeling of MEMS resonators -Part I: Anchor loss mechanisms through substrate[END_REF][START_REF] Park | High-fidelity modeling of MEMS resonators -Part II: Coupled beam-substrate dynamics and validation[END_REF][START_REF] Hao | An analytical model for support loss in micromachined beam resonators with in-plane flexural vibrations[END_REF], and this assumption will be used in this paper.

The objective of the present paper is to review and compare different support loss models that assume the support is semi-infinite. One of these models is used to perform a parametric study so as to gain improved understanding of the factors that influence support loss and develop strategies to minimise support loss.

In Section 2, several support loss models available in the literature are presented. For each model, convergence issues are analysed. A comparison between the different support models is given in Section 3. In Section 4, one of the models is used to calculate support loss for a ring-based resonator supported on folded-beam legs, and the influence of the leg design is considered in order to gain insights into support loss characteristics.

2 Review of different support loss models

Analytical model

Based on the fundamental work of Miller and Pursey [START_REF] Miller | The field and radiation impedance of mechanical radiators on the free surface of a semi-infinite isotropic solid[END_REF], who modelled the radiation impedance of resonators on semi-infinite solids, closed-form expressions have been derived for support losses in beam resonators [START_REF] Hao | An analytical model for support loss in micromachined beam resonators with in-plane flexural vibrations[END_REF].

These expressions follow on from work undertaken by Jimbo and Itao [START_REF] Itao | Energy loss of a cantilever vibrator[END_REF].

The physical arrangement of a MEMS resonator and its support can take different forms. Typically the resonator and support are etched from the same silicon wafer and it is reasonable to assume they lie in the same plane and have the same thickness (denoted h in Figure 1) [START_REF] Hao | An analytical model for support loss in micromachined beam resonators with in-plane flexural vibrations[END_REF], particularly when only in-plane vibrations are considered.

It has been proved theoretically [START_REF] Cross | Elastic wave transmission at an abrupt junction in a thin plate with application to heat transport and vibrations in mesoscopic systems[END_REF] that, when the wavelength of the propagating elastic wave in the support is much larger than the size of the attachment region, the coupling between the vibration modes of a MEMS resonator and the modes in its support is very weak. On this basis, a MEMS resonator and its support can reasonably be analysed separately and the stresses at the attachment region from the resonator can be considered as excitation sources for the elastic wave propagation in the support. Shear stress only was considered in [START_REF] Hao | An analytical model for support loss in micromachined beam resonators with in-plane flexural vibrations[END_REF]. However, in more complex resonators such as the ring-based resonator studied in Section 4, the supporting legs produce both shear and normal stresses simultaneously. Support excitation from normal stress was studied in [START_REF] Hao | Vibration displacement on substrate due to timeharmonic stress sources from a micromechanical resonator[END_REF][START_REF] Chouvion | Vibration transmission and support loss in MEMS sensors[END_REF]. Over the source region (the width of the clamped resonator), the following boundary conditions for the support are used.

• For flexural vibrations: the shear stress is assumed uniform and the normal stress is linearly dependant of the position along the width (due to the bending moment). However, the energy loss due to the bending moment has been proved to be negligible compared to that due to shear stress [START_REF] Chouvion | Vibration transmission and support loss in MEMS sensors[END_REF].

• For longitudinal vibrations, the shear stress is zero and the normal stress is uniform.

The support is modelled using the classical two-dimensional dynamic equations for a thin plate [START_REF] Graff | Wave Motion in Elastic Solids[END_REF] subjected to the previous stress excitation sources.

The attachment of the resonator to the support is assumed to be rigid initially. In other words, the deformation of the support is not considered when calculating the reactions between the resonator and support.

An analytical expression for the displacement at any point in the support is obtained by solving the governing equations of the thin-plate in the wavenumber domain. For each different boundary condition, the mean displacements (normal and tangential) along the source region, responsible for the transmitted power, are derived in order to calculate the amount of energy lost. The velocities induced by these displacements are used to determine the mean power over a single cycle, denoted Π, transmitted from the resonator to the support in each situation. This power is obtained by integrating the product of resultant forces at the attachment with the associated velocity, over one cycle of vibration, such that:

Π = 1 2 Re (Force • Velocity * ) , (2.1)
where Re denotes the real part, and * the complex conjugate. The term 'Force' represents either the normal force or the shear force, and the term 'Velocity' is the associated velocity (normal or tangential direction) at the attachment area induced by the application of the corresponding force. The amount of energy lost per cycle ∆W from the support can be calculated explicitly as:

∆W = 2π ω Π, (2.2)
where ω is the frequency. The total energy lost is the sum of the energy lost from shear force (∆W shear ) and normal force (∆W normal ) sources. ∆W shear and ∆W normal are dependent on the Poisson's ratio (ν), Young's modulus (E), thickness (h) of the support material, and the coupling shear force S and tensile force T at the attachment point. From equations (2.1) and (2.2), these energy losses can be expressed as [START_REF] Hao | Vibration displacement on substrate due to timeharmonic stress sources from a micromechanical resonator[END_REF][START_REF] Chouvion | Vibration transmission and support loss in MEMS sensors[END_REF]:

∆W shear = α |S| 2 Ψ s , (2.3)
∆W normal = α |T | 2 Ψ n , (2.4)
where α = 4(1 + ν)/(Eh(1 -ν)), and Ψ n and Ψ s are the imaginary parts of integrals, function of ν only, which arise from the transformation of the displacement expressions from the wavenumber domain back to the frequency domain. For a Poisson's ratio of 0.28, Ψ n = 0.22153 and Ψ s = 0.33503 [START_REF] Chouvion | Vibration transmission and support loss in MEMS sensors[END_REF].

The ratio of the maximum stored energy during a single cycle, denoted W , to the energy lost per cycle gives the Q-factor for the device, such that:

Q = 2π W ∆W shear + ∆W normal . (2.5)
The analytical approach has its limitations. The wavelength of propagating waves must be much larger than the thickness of the vibrating structure for the two-dimensional theory to be valid. It is also unable to capture the effects of wave reflections at discontinuities in the support, since the support is modelled as a perfect semi-infinite thin plate. Nevertheless, because the support model is analytical and different stress sources can be considered, calculating Q-factors for different resonators is straightforward because it only requires the resonator model (and not the support model) to be modified. However, the main drawback is that the model is only valid under the assumptions made and results are only available for a restricted range of geometries.

In [START_REF] Hao | An analytical model for support loss in micromachined beam resonators with in-plane flexural vibrations[END_REF], the response characteristics of the resonant structure, needed to calculate the Q-factor (total energy and reaction forces appearing in (2.3) and (2.5)), were determined from analytical modal expansions. If the resonator geometry is complex such that modal expressions are not available, a ray tracing method [START_REF] Chouvion | In-plane free vibration analysis of combined ring-beam structural systems by wave propagation[END_REF][START_REF] Chouvion | Vibration modelling of complex waveguide structures[END_REF] can be used to obtain detailed insight into the resonator motion and energy distribution, and this approach is employed in Sections 3 and 4.

In order to simulate the response of a semi-infinite domain, computational methods are also available, such as the use of boundary dampers, infinite elements, boundary integrals, or exact Dirichlet-to-Neumann boundary conditions (see [START_REF] Engquist | Absorbing boundary conditions for the numerical simulation of waves[END_REF][START_REF] Givoli | Discrete Dirichletto-Neumann maps for unbounded domains[END_REF][START_REF] Astley | Infinite elements for wave problems: a review of current formulations and an assessment of accuracy[END_REF][START_REF] Cheng | Heritage and early history of the boundary element method[END_REF]). The Perfectly Matched Layer method and the use of infinite elements are presented in the next sections.

Perfectly matched layer

The Perfectly Matched Layer (PML) technique was first introduced by Bérenger [START_REF] Bérenger | A perfectly matched layer for the absorption of electromagnetic waves[END_REF] for problems in electromagnetic wave propagation to model infinite boundaries. The general feature of the PML is the construction of a layer exterior to the domain of interest that damps all the waves that enter it from the interior domain, independent of the angle of incidence and the frequency. Waves passing through the layer are also further damped when they are reflected back, so that most of the vibration and energy entering the layer are absorbed. This layer is represented by a system of equations that are not necessarily physically realisable. Their only purpose is to prevent reflections from the outer artificial boundary. Simple "sponge layers" [START_REF] Kosloff | Absorbing boundaries for wave propagation problems[END_REF] that add dissipative terms to damp the outgoing waves were first used. However, these methods were only moderately successful because reflections were also created by gradients within the layer. To be effective, the layer must be designed so that there is no impedance mismatch to reflect waves back from the interface between the layer and the rest of the domain. The PML "perfectly matches" the impedance of the interior and exterior domains, and there are therefore no spurious reflections at the interface.

Bindel and Govindjee [START_REF] Bindel | Elastic PMLs for resonator anchor loss simulation[END_REF] considered the PML method for support loss calculation in MEMS resonators, and implemented the approach in a finite element code called HiQLab [29]. When a PML is used to model infinite boundaries in a finite element code, complex mass and stiffness matrices are generated. As a consequence, complex eigenvalues ω are obtained for the governing equations and the Q-factor can be calculated directly using [START_REF] Bindel | Elastic PMLs for resonator anchor loss simulation[END_REF]:

Q = |ω| 2 Im(ω) (2.6)
where ω is the natural frequency and Im denotes the imaginary part.

Bindel and Govindjee interpreted the PML as a complex-valued change of coordinates which could be applied to any linear wave equation. The elements use complex stretching of the coordinate axes to introduce artificial damping to the wave. It is worth noting that the complex coordinate stretching has as a prerequisite the analytic continuation of the equations and their solutions. The coordinate transformation for a 1D case can be expressed in the form:

x = x - i k x 0 σ(s) ds, (2.7)
where x is the new coordinate, x is the "real" axial coordinate, i = √ -1, and k is the wavenumber. σ is a PML attenuation function that is used to reduce the amplitude of the waves when they travel in both directions: on the way to the outer boundary, and on their way back to the resonator. As a general idea, when σ is chosen to be zero at the boundary with the finite medium, the solution is the same for both media and hence the wave enters the PML without any reflection at the boundary. When σ > 0, the wave decays in the direction of travel. Figure 2 illustrates the case of a linear attenuation function σ with end magnitude β(L p -L), where L and L p are respectively the length of the finite region and the PML domain. Since waves decay so rapidly in the PML region, a good approximation to the infinite domain is obtained even if the condition of zero-displacement at the outer boundary (in L p ) is enforced [START_REF] Bindel | Elastic PMLs for resonator anchor loss simulation[END_REF].

In order to correctly model a semi-infinite domain, the main objective of the PML is to minimise the waves that are reflected back in the finite domain. It should ideally absorb completely the outgoing waves and no reflection should occur. However, due to the way in which the problem is discretised, some energy is always reflected back. A precise analysis of these reflections was presented by Bindel and Govindjee [START_REF] Bindel | Elastic PMLs for resonator anchor loss simulation[END_REF], and Koyama [START_REF] Koyama | Efficient evaluation of damping in resonant MEMS[END_REF],

where the total reflection due to the PML is separated into: the end termination reflection, introduced by a finite termination of the PML; the interface reflection, introduced by the discretised PML at the PML interface; and the computed reflection, due to round-off errors in computing calculations.

The different parameters that have an effect on the Q-factor are:

• Geometrical parameters: the dimension of the PML region (L p -L), and the dimension of the "finite" support region (L) meshed using standard finite elements;

• Meshing parameters: the number of elements in the PML region, and the interpolation order of the elements in the PML (linear, quadratic, cubic);

• PML function properties: the end value parameter β of the PML absorbing function σ, and the polynomial order of the PML absorbing function σ.

The work reported in [START_REF] Bindel | Elastic PMLs for resonator anchor loss simulation[END_REF] and [START_REF] Koyama | Efficient evaluation of damping in resonant MEMS[END_REF] provides guidelines for selecting PML parameters in a rational way and these are used as the starting point for the results presented later. Different parameter values are investigated to ensure that converged results are obtained. All of the results presented in Section 3 are obtained after the convergence study has been completed.

These results indicate that the PML method is time-consuming and that the results obtained are not "exact" Q-values. For this reason, an alternative finite element method with infinite elements was also investigated, and is discussed next.

Infinite elements

Infinite boundaries can be modelled using the finite element method using specific "infinite elements". First-and second-order infinite elements based on the work of Zienkiewicz et al. [START_REF] Zienkiewicz | A novel boundary infinite element[END_REF] for static response and of Lysmer and Kuhlemeyer [START_REF] Lysmer | Finite dynamic model for infinite media[END_REF] for dynamic response are implemented in Abaqus TM to model infinite boundaries. The energy is artificially attenuated and lost in these elements. They are used to mesh the far-field region, in conjunction with standard finite elements, which model the area around the resonator.

The infinite elements provide a theoretically non-reflective boundary to the model by using special shape functions that eliminate artificial wave reflections from the truncated domain of the model. Regular shape functions are used to model the displacement variables u while growing shape functions are used for position variables r [START_REF] Systèmes | Abaqus Online documentation[END_REF].

A one-dimensional case is presented next and illustrated in Figure 3.

The infinite element contains three nodes, whose nodes are identified by s i = -1, 0 and 1 (i = 1, 2, 3). r i and u i respectively represent the distance from the "pole" that corresponds to the source of propagating waves, and the displacement of node i. The particular chosen shape function such that:

r = -2s 1 -s r 1 + 1 + s 1 -s r 2 , (2.8)
and the following standard quadratic interpolation for the displacement:

u = 1 2 s(s -1)u 1 + (1 -s 2 )u 2 (2.9)
give at s = -1 : r = r 1 and u = u 1 , at s = 0 : r = r 2 and u = u 2 , at s = 1 : r → ∞ and u → 0.

Hence the position coordinate approaches infinity, and the displacement of the propagating wave decreases to zero at the outer edge of the infinite element.

The infinite element is shown to work well for static and dynamic analyses [START_REF] Systèmes | Abaqus Online documentation[END_REF]. In any case, it is assumed that the response adjacent to the boundary of the infinite elements is of sufficiently small amplitude that the medium responds in a linear elastic fashion. The infinite elements have to be placed sufficiently "far" from the pole. They also have to be arranged such that the dominant direction of wave propagation is orthogonal to the boundary, which means that the outer boundary must be semi-circular.

The fundamental in-plane mode of a simple cantilever beam is studied in Abaqus TM to check convergence using infinite elements. The beam is meshed using classical two-dimensional quadratic plane stress elements. The support is divided in two different sections: a bounded domain of length R, meshed with standard finite elements, and an infinite domain, meshed using the infinite elements described previously. The beam width is meshed using 4 elements, and the element size in the support gradually increases to fill the entire bounded domain. The outer ends of the infinite elements are placed at an approximate distance 2R from the pole (origin of the propagating waves -the beam attachment point).

The support is supposed to be infinite and a change in the bounded domain length R should theoretically not influence the results for the natural frequency and Q-factor. For different values of R, a steady state analysis is performed in the vicinity of the expected first natural frequency. The Qfactor is then calculated from the resulting amplitude-frequency curve and defined as: where ω 0 is the natural frequency of interest and ∆ω is the half-power bandwidth at -3 dB of the maximum amplitude. This is the common method used to measure Q-factors.

Q = ω 0 ∆ω , (2.10)
It is found that the calculated Q-factor is actually a function of R, and it is expected that this dependency is also a function of the transverse wavelength λ of the propagating shear waves in the support. The results are illustrated in Figure 4, which shows the calculated Q-factor for different mesh refinements and different values of R. With a very coarse mesh, the Q-value quickly diverges as R increases. This is probably due to the difference in size between the bounded domain and the resonator itself. It is important to notice that, for example, when R/λ ≈ 10, the bounded domain is actually 3000 times larger than the beam width. It is understandable that a coarse mesh struggles to correctly model the energy propagation and dissipation arising from a relatively small resonator. With a fine mesh, the Q-value first shows a maximum around R/λ ≈ 1.75, then seems to stabilise between 3 < R/λ < 5, and finally oscillates for R/λ > 6. This simple convergence test demonstrates the complexity of calculating Q. Here Q is highly dependent on the mesh density and on the support dimensions. However this study shows that a range of approximate values of Q can still be obtained using the infinite elements. From the analysis of Figure 4 and especially the results obtained with the finest mesh, it seems sensible to use as a maximum Q-value, the value obtained when R/λ ≈ 1.75 and as an appropriate minimum Q-value, the value obtained when R/λ ≈ 4. These two particular values of R/λ have been used in the studies presented in Section 3.

Comparison of the models

The objective is to compare and validate the different support models for any resonator design. The following applications consider a wide range of possible resonator-support characteristics, in which support losses occur either from shear stress excitation, normal stress excitation, or a combination of those. Due to the complexity of creating meshes in the HiQLab program, the difficulty to obtain a rapid converged result, and the computing cost of using finite elements methods, the methods presented in Section 2 are applied to simple test cases only.

A cantilever beam vibrating in its fundamental mode is studied first.

Resonator and support are made in silicon, from the same wafer, and have identical material properties. For a fixed beam width, the calculated Q- factor is plotted as a function of the length/width ratio of the beam in ). Even though the infinite element method gives a "range" of correct Q-values, it is shown that they are all in reasonable agreement. A possible explanation of the discrepancies is the different way the attachment is modelled. A rigid attachment assumption is considered in the analytical model, whereas the coupled resonator-support system, with high stress concentration in the attachment area, is modelled in the finite element approaches.

It is important to note that for more complex-shaped resonators, the attachment resonator-support may undergo flexural and longitudinal vibra- For a fixed beams width, the length of beam '2' is varied. The ray tracing method [START_REF] Chouvion | In-plane free vibration analysis of combined ring-beam structural systems by wave propagation[END_REF] is used to calculate the natural frequency of the first mode, its corresponding strain energy, and shear and normal forces at the clamped boundary. The energy loss caused by the shear force and that caused by the normal force are calculated separately using (2.3) and (2.4), and then added up to obtain the overall energy loss. The objective was to take account of a wide range of resonator-support configurations that includes several complex-shaped resonator characteristics. For instance, in Figure 6: Two-beam case. Q-factor calculated using the different methods as a function of the length/width ratio of beam '2'. The system studied is shown in the inset. Schematic representations of the mode shapes, for the shortest and longest beam '2' considered, are also illustrated.

the nominal design of the ring-based resonator studied in Section 4, the energy lost due to shear force is approximately 4 times larger than the one due to normal force, for the modes of interest. From the range of length/width ratio considered in the present test case, the ratio of energy lost due to shear force over the energy lost due to normal force varies between 7000 for short beam '2' and 0.3 for long beam '2' [START_REF] Chouvion | Vibration transmission and support loss in MEMS sensors[END_REF]; showing that this test case can represent a wide range of features.

The Q-factor calculated with the different methods, when the length of beam '2' is varied, is presented in Figure 6. For the infinite element method, the maximum Q-value, obtained when R/λ ≈ 1.75, has been used. Again, all the curves show a similar trend. However, it appears in this case, which is not true for the simple cantilever beam study, that the relative Q-difference increases when the length/width ratio of beam '2' is higher. In the high length/width range, normal forces are predominant at the attachment point.

Thus, a possible explanation is that the analytical support model does not properly quantify the loss occurring when normal stresses are predominant.

An alternative explanation could be that the ray tracing method struggles to correctly model the beam corner when it is subjected to high stresses.

The thin beam theory used in the ray tracing does not consider high stress concentrations at the corner, as only the centreline is modelled, and the complexity of a real corner could not be properly modelled.

However, a simple study has ruled out the hypothesis of inaccurate modelling by the ray tracing method. The same system has been modelled in Abaqus TM using quadratic elements. A modal analysis in Abaqus TM gives the modal strain energy of the entire system and the forces at the clamped boundary for the first mode of vibration. These clamped forces and strain energy are inputted into the analytical model of the support presented in Section 2.1, and the Q-factor is calculated. These results are illustrated in Figure 6. This study represents a combination between the analytical model of the support and a finite element analysis of the vibrating structure. The excellent agreement achieved indicates that the ray tracing method correctly models the vibrations, and that the difference between analytical and PML methods is only due to the support model.

In conclusion, Figures 5 and6 indicate that the three different support models give the same range of Q-values and same trends when the 4 Influence of leg design on the Q-factor of a ringbased resonator

Nominal study

The method referred to as the Analytical Model in Section 2 is used here to calculate the Q-factors of a ring-based resonator developed by Atlantic Inertial Systems (AIS). Figure 7 shows a photograph of this complex-shaped resonator on its silicon wafer and a schematic representation of the nominal device. The resonator is composed of a thin ring and eight identical uniformly-spaced legs in the form of folded beams. Table 1: Q-factors predictions and measurements for the 2θ and 3θ modes.

The process employed to calculate Q can be summarised as follows.

Firstly, assume that the resonator is clamped and use the ray tracing approach [START_REF] Chouvion | In-plane free vibration analysis of combined ring-beam structural systems by wave propagation[END_REF][START_REF] Chouvion | Vibration modelling of complex waveguide structures[END_REF] to calculate its natural frequencies and associated mode shapes. Then, from the mode shape, calculate the total energy stored W and the forces T and S at the clamped end. Use (2.3) and (2.4) to obtain the energy loss due to the forces S and T , and then use (2.5) to calculate the Q-factor.

The nominal dimensions and material properties of the ring-based resonator are identical to those given in [START_REF] Chouvion | In-plane free vibration analysis of combined ring-beam structural systems by wave propagation[END_REF][START_REF] Chouvion | Vibration modelling of complex waveguide structures[END_REF]. The Q-factors for the so-called 2θ and 3θ modes, whose mode shapes are illustrated in Figure 8, are calculated using the analytical method and are given in Table 1.

These results are compared with Wong's experimental Q-factor measurements and theoretical predictions [START_REF] Wong | Thermoelastic damping in MEMS ring resonators[END_REF] for thermoelastic damping (denoted "Thermo." in Table 1) in similar resonators. The predicted total Q-factor is also shown for comparison in Table 1 and is defined as the inverse of the total energy loss (Q

Total = 1/Q i -1
). The measured overall Q-factors consider all damping mechanisms. The comparison of the simulated and measured Q-factors gives confidence in the support loss model developed and used here. Indeed, it is interesting to note that the total energy loss predicted (inverse of the Q-factor) considering support loss and thermoelastic damping is only slightly smaller than the energy loss measured experimentally. It may be possible to obtain improved agreement by including damping contributions arising from surface losses and air damping in the computational model.

In applications, the excited and sensed modes are usually the 2θ mode and its orthogonal companion. Only results relating to the 2θ modes are presented in the following sections as they are of primary interest.

Change of leg geometry

In this section, the geometry of the leg is modified. The beam lengths and angles between the beams that make up the leg are changed. The beams are numbered as follows: beam '1' (length L 1 ) is attached to the ring, beam '2' connects beams '1' and '3', and beam '3' (length L 3 ) is clamped to the support. The beam numbering is illustrated in Figure 7. In the parametric study performed, the geometry is constrained such that:

• Beam '1' and beam '3' are oriented radially (i.e. their centrelines pass through the centre of the ring).

• The total angle subtended by the leg, angle between beam '1' and beam '3', is equal to 40 • . This ensures that the ring contains eight legs with a 5 • angular separation between each.

• The point of attachment to the support is a fixed point. This means that the distance between the clamped end and the centre of the ring is constant. The reason for this is that the modifications to the leg designs are not intended to change the dimension of the silicon wafer on which the resonators are etched nor their position on the wafer.

L 1 and L 3 are varied independently over a range defining the shortest and longest beam configurations in the allocated space (between the ring and the support). When one of these lengths varies, the other parameters (e.g. angles between them) vary accordingly by following the constraints mentioned above. For each value of the couple (L 1 , L 3 ), the Q-factor is calculated following the procedure described in Section 4.1.

Results for the Q-factors are presented in Figure 9, where it can be seen that leg designs with a longer beam '1' generally have a larger Q-factor.

If at the same time, beam '3' is shortened, then the largest Q is obtained, resulting in a significant improvement in performance. For this particular design, the Q-factor is approximately 3 times larger than the value for the nominal design. The associated mode shape is presented in Figure 10(a).

It can be seen clearly from the mode shape that the ring contributes in an important way to the overall energy of the resonator, maximising the of approximately 350 Hz is found, which is small compared to the nominal frequency of operation (≈ 14 kHz). It is important to control the associated change in frequency because a large variation will modify the thermoelastic loss [START_REF] Wong | Thermoelastic damping in MEMS ring resonators[END_REF].

From Figure 9, it can be seen that the design with a short beam '1' and a short beam '3', whose mode shape is illustrated in Figure 10(b), is clearly inefficient compared to the nominal design. The Q-factor in this case is approximately 100 times smaller than the nominal one and the 2θ frequency is greatly increased. In this case the resonator frequency almost matches one of the leg natural frequencies considered separately. For this particular design, the legs are as straight as possible within the applied constraints, and have a large influence on the ring vibration. The leg flexibility is greatly reduced and the contribution of the legs to the overall resonator deformation is increased. From the mode shape, it is clear that the ring deformation is less important compared to the legs in beam portion '2', because the leg vibrates at a frequency close to one of its own natural frequencies. This indicates that most of the energy is contained within the legs, increasing the support losses and reducing the Q-factor.

Looking directly at the energetic transmission and reflection coefficients at the different corners of the legs could give some general ideas and guidelines about which angles are to be used or avoided to minimise the energy propagating away through the legs. Wu and Lundberg [START_REF] Wu | Reflection and transmission of the energy of harmonic elastic waves in a bent bar[END_REF] presented quantitative results on these transmission coefficients and they generally confirmed the fact that large angles facilitate the direct transmission of waves of the same type, whereas acute angles would tend to increase the reflection of waves. Even though their work neither takes account of the transmission and creation of evanescent terms after the discontinuity, nor takes account of the infinite reflection-transmission process that occurs between the corners, it still indicates that the angles present in the leg when beam '1' and beam '3' are short are inefficient for energy transmission. Another general guideline to minimise support losses would be to avoid designs where the natural frequency of the resonator coincides (or is in the near vicinity of) one of the leg natural frequencies, as this facilitates vibrational energy transmission.

Change of leg width

In this section, the geometry of the leg is maintained at the nominal design.

However, the in-plane width of the leg is modified. The nominal leg and ring widths are equal to 61 µm and 121 µm respectively. Here, the leg width takes values between 30 µm and 105 µm.

Results for the natural frequencies and the associated Q-factors as a function of leg width are shown in Figure 11. From this graph, it can be seen that the 2θ frequency increases when the leg width increases, whilst the Q-factor decreases when the leg width increases. This is mainly because thin legs are more flexible in bending and constrain the ring less. This ensures that the vibration of the ring is closer to that of a ring without support, maximising the energy stored. In contrast, wide legs have a strong influence on the ring vibration and restrain its amplitude. From Figure 11, the Q-factor trend as a function of the leg width b can be approximated to Q ≈ 1/b 4 . This means that when the width of the leg is half the nominal value, the Q-factor is approximately 16 times larger than the nominal Qfactor. This relationship between Q and b can be compared to that obtained for other designs: Q ∝ 1/b 3 for a simple cantilever beam resonator [START_REF] Hao | An analytical model for support loss in micromachined beam resonators with in-plane flexural vibrations[END_REF] and Q ∝ 1/b 6 for a disk resonator supported on a single straight beam [START_REF] Hao | Support loss in micromechanical disk resonators[END_REF].

The Q-factor here, where the legs stay relatively flexible for any width, is therefore more sensitive to a change of width than for a cantilever beam, in which the internal stored energy in the resonator increases at the same time as the energy propagating away; but less than for a disk resonator, in which a larger supporting beam clearly increases its stiffness, and has more influence on the disk vibration.

In order to better understand the effect of the width change on the resonator vibration characteristics, it is interesting to consider the energy distribution within the resonator as a function of the leg width. shows how the kinetic and strain energies are distributed (as percentages)

between the different parts of the structure as the leg width is changed. The legs are divided into three beam portions, numbered from the ring to the support, as explained in Section 4.2. The energy corresponding to a specific portion of a leg is combined with the energies for the same portion of all other legs.

From Figure 12(a), it is clear that the ring moves less when the leg is 29 wider. One can see that the kinetic energy, which is mainly present within the ring for very thin legs is shifted to the legs themselves when the leg width is increased. The kinetic energy percentage of beam '3' remains close to zero, as expected because beam '3' is "clamped" at its end. The fact that the percentage of kinetic energy contained within the ring decreases when the leg width increases shows that the total energy stored decreases.

It can be seen in Figure 12(b), that the strain energy of the ring decreases when the leg width increases. The ring does not deform as much as for the case when the leg width is very thin. The strain energies in the leg portions increase, which can be explained because they have a larger width and also show a greater deformation. It is interesting to look at the energy of the third beam portion. As this increases, it is expected that the forces at the clamped end will also increase, meaning an increase in the amount of energy lost. This explains the Q-factor decrease shown in Figure 11.

Conclusion and discussion

Different models for the analysis of support loss, which all assume the support to be semi-infinite, have been compared. It has been found that they

give Q-values in the same range of magnitude and exhibit similar trends when a parameter is varied. This has been shown for simple and for more complex resonator vibrations. However, the results obtained for the Q-factor

show that the analytical model tends to overestimate Q, or underestimate the quantity of energy lost. The most plausible reason for this is the rigid attachment assumption. This assumption over-evaluates the stored energy (and therefore the associated Q-factor) as the vibrational energy is constrained to stay within the resonator and is thus artificially increased. In contrast, the PML and the infinite element methods model the resonator and support together. The support therefore presents a non-zero flexibility when analysing the resonator vibration. Consequently the energy stored within the resonator is smaller. It is also very difficult to predict and to model the real effect of the attachment on the resonator. Even a finite element analysis makes approximations in this region of high stress concentration.

It is worth noting that the semi-infinite support assumption is usually not realistic for practical MEMS devices. This is because the supporting structure to which the resonator is attached, often has dimensions of a similar order to the dimensions of the resonator itself. Furthermore, the supporting structure is often bonded rigidly to other materials (e.g. glass), which may also be attached to a metal base. Hence the support structure is finite in extent, with external boundaries that would in practice reflect some energy back into the resonator, reducing the likelihood that all energy propagating away from the resonator is lost. As the support typically has finite extent, its own end boundaries may also have an important effect on the motion of the resonator-support attachment point. In addition, internal structural damping will be present and attenuate waves propagating into the support. These factors are not taken into account in the semi-infinite model. In general, MEMS resonators have Q-factors that have high values which are sensitive to external factors. This characteristic helps to explain why the models presented yielded numerical values significantly different from experimental measurements. Despite this, it is expected that the value of the theoretical models is to predict the influence of different parameters on the Q-factor, so that resonators can be designed to have lower levels of damping.

The presented analytical method was applied to a practical ring-based resonator. In comparison to thermoelastic damping, support loss of the nominal design was found to be negligible. However, recent research on thermoelastic damping [START_REF] Duwel | Engineering MEMS resonators with low thermoelastic damping[END_REF] indicates the possibility of designing resonators with reduced levels of thermoelastic damping, increasing the importance of support losses in resonator applications. In addition, other resonator designs, such as bulk-mode resonators [START_REF] Hao | VHF single-crystal silicon elliptic bulk-mode capacitive disk resonators-part I: design and modeling[END_REF][START_REF] Abdelmoneum | Stemless wine-glass-mode disk micromechanical resonators[END_REF], have been shown to have very high Q-factors that are support loss limited.

A parametric study was performed to gain a better understanding of the factors that influence support loss for a ring-based resonator supported on legs. The main findings relating to resonator and support design were that thin, flexible legs produced a significant increase in Q-factor for the operating modes, whilst the Q-factor decreased if the leg frequencies coincided with the ring frequencies. In order to minimise the support loss, it is necessary to consider both the clamping force and energy contained in the resonator simultaneously.

Essential future work includes validation of the presented models for simple and complex-shaped resonators with experimental results. To increase the Q-factors, the use of blocking masses that reduce the transmission of vibration to the attachment points (suggested in [9]) is an interesting area of possible future research. Support loss models for out-of-plane vibrations are also essential for the next generation of multi-axis resonators [START_REF] Fox | Development of multi-axis rate sensors based on silicon ring structures[END_REF].

Figure 1 :

 1 Figure 1: Semi-infinite thin plate with excitation sources.

Figure 2 :

 2 Figure 2: Linear attenuation function used in the PML method for a onedimensional wave.

Figure 3 :

 3 Figure 3: Nodal variable position for a one-dimensional infinite element in Abaqus TM .

Figure 4 :

 4 Figure 4: Convergence analysis. Q-factor function of R/λ calculated for different mesh densities.

Figure 5 :

 5 Figure 5: Cantilever beam case. Q-factor calculated using the three different methods as a function of the length/width ratio.

Figure 5 .

 5 Figure 5. This graph compares the results obtained using the PML method in HiQLab, infinite elements in Abaqus TM , and the analytical model. In the analytical model, only shear stresses are considered at the resonatorsupport interface (normal stresses are neglected -see Section 2.1). The energy stored and clamped forces in the resonator are obtained using modal expressions. All curves show the same trend, and the results display a cubic character when the beam is made longer and/or thinner (as explained by Hao et al. in [13]). Even though the infinite element method gives

  tions simultaneously, opposed to the simple flexural vibration of a beam resonator. For instance in a xylophone bar resonator, complex relations between the geometry and Q, arising from modal tuning of the component, have been found[START_REF] Grigg | Towards a parametrically pumped xylophone microbar magnetometer: design optimisation of xylophone bar resonator[END_REF] using a PML method. In the analytical model presented in Section 2.1, the coupled flexural-longitudinal vibration provides a complication as the support model uses two different expressions for shear and normal stress sources. To validate the model when flexural and longitudinal vibrations occur simultaneously, another test case is studied. It consists of a two-beam system, where the beams are at right angles to each other. The first beam is attached to the infinite support, whereas the second beam (called beam '2') is free. The Q-factor corresponding to the first mode of vibration is studied.

Figure 7 :

 7 Figure 7: Photograph of a ring-based rate sensor on its silicon wafer (courtesy AIS), and schematic representation of the resonator.

Figure 8 :

 8 Figure 8: 2θ and 3θ mode shapes of the ring-based resonator with nominal dimensions. Damping predictions Measurements [35] Support loss Thermo. [35] Total 2θ mode 147 600 11 731 10 867 9 455 3θ mode 80 660 17 783 14 571 12 434

Figure 9 :

 9 Figure 9: Q-factors of the 2θ mode for different values of L 1 and L 3 .

Figure 10 :

 10 Figure 10: Deformed shape of the 2θ mode for a long beam '1' and a short beam '3' (a), and for a short beam '1' and a short beam '3' (b)

Figure 11 :

 11 Figure 11: 2θ frequencies (Hz) and the associated Q-factors function of the leg width.

Figure 12 :

 12 Figure 12: Kinetic energy (a) and strain energy (b) percentages in the different portions of the ring-based resonator, for the 2θ mode, as a function of the leg width.

Funding acknowledgements

This work was supported by Atlantic Inertial Systems.


Piotr Graczyk Larema

Fisher Information and Exponential Families Parametrized by a Segment of Means

Keywords: AMS Subject Classification: 62H12, 62H10 Fisher information, efficient estimator, exponential family, multivariate Gaussian distribution, Wishart distribution, parsimony

We consider natural and general exponential families (Q m ) m∈M on R d parametrized by the means. We study the submodels (Q θm 1 +(1-θ)m 2 ) θ∈[0,1] parametrized by a segment in the means domain, mainly from the point of view of the Fisher information. Such a parametrization allows for a parsimonious model and is particularly useful in practical situations when hesitating between two parameters m 1 and m 2 . The most interesting examples are obtained when R d is a linear space of matrices, in particular for Gaussian and Wishart models.

Introduction

Fisher information is a key concept in mathematical statistics. Its importance stems from the Cramér-Rao inequality which says that the variance of any unbiased estimator T (X 1 , . . . X n ) of an unknown parameter θ, is bounded by the inverse of the Fisher information: Var θ (T ) -(I(θ)) -1 is semi-positive definite. Fisher information is therefore mainly used as a measure of how well a parameter can be a estimated. This justifies the use of Fisher information in experimental design for predicting the maximum precision an experiment can provide on model parameters. This also justifies the important role Fisher information plays in estimation theory where it provides bounds for confidence regions and also in Bayesian analysis where it provides a basis for noninformative priors. Fisher information can be used to investigate the trade-off between parsimony of parameters and precision of the estimation of the parameters [START_REF] Andersson | Cramer-Rao bound and the parsimony principle[END_REF].

Besides its importance in statistical theory, Fisher information has different interpretations that lead to some practical applications. For example, the interpretation of Fisher information 1 as a measure of the state of disorder of a dynamic system leads to the use of Fisher information in stochastic optimal control as a tuning tool to stabilise the performance of a dynamic system [START_REF] Ramirez | Fisher information as a novel tool for process control applications[END_REF]. Viewing Fisher information as a measure of information, leads to the statement of a "minimum information principle" akin to the well-known maximum entropy principle for determining the "maximally unpresumptive distribution" satisfying some predefined constraints [START_REF] Bercher | On minimum Fisher information distributions with restricted support and fixed variance[END_REF]. [START_REF] Gupta | On the comparison of Fisher information of the Weibull and GE distributions[END_REF] describe the use of Fisher information in model selection as a tool to discriminate between two models with otherwise very similar fit to some data. The use of Fisher information however goes far beyond statistics; [START_REF] Frieden | Science from Fisher information: a unification[END_REF] shows that Fisher information is in fact a key concept in the unification of science in general, as it allows a systematic approach to deriving Lagrangians.

The objective of this work is the study of the Fisher information for exponential models (P m ) m∈M parametrized by a segment of means [m 1 , m 2 ]. Exponential families of distributions have been extensively studied [START_REF] Brown | Fundamentals of Statistical Exponential Families with Applications in Statistical Decision Theory[END_REF][START_REF] Barndorff-Nielsen | Information And Exponential Families In Statistical Theory[END_REF][START_REF] Letac | Lectures on natural exponential families and their variance functions[END_REF][START_REF] Letac | Natural exponential families[END_REF]. A parametrization of the family by a segment instead of the whole means domain allows to obtain a parsimonious model when the mean domain is high-dimensional. The parametrization of the mean parameter by a segment is particularly useful in practical situations when hesitating between two equally convenient mean values m 1 and m 2 . Such parametrization will also serve in sequential data collection, when an updated estimate of a parameter largely differs from the previous estimate. An important practical example is a Gaussian model N(u, θC + D) in R d with the mean vector u known and the covariance matrix in a segment IC + D, where

θ ∈ I = [a, b] ⊂ R.
From the Fisher information point of view, exponential families constitute an interesting and important class of models. Their Fisher information coincides with the second derivative of the cumulant generating function of the measure generating the family and they are the only models for which the Cramér-Rao bound can always be attained [START_REF] Brown | Fundamentals of Statistical Exponential Families with Applications in Statistical Decision Theory[END_REF][START_REF] Letac | Lectures on natural exponential families and their variance functions[END_REF].

The paper is organised as follows. In Section 2, basic definitions and results on Fisher information and exponential families are recalled and extended to matrix-parametrized models. Section 3 contains new results on the Fisher information of exponential families parametrized by the domain of the means and the sub-families parametrized by a segment of means [m 1 , m 2 ]. In Section 4, these results are applied to Gaussian and Wishart families of distributions. When m 1 and m 2 are colinear, we construct efficient estimators for the segment parameter θ.

Preliminaries

In most expositions of the theory of exponential families and of the concept of Fisher information, the parameter is considered to be a vector whereas cases abound in multivariate analysis where the canonical parameter is a matrix. In this preliminary section we adapt the presentation of the usual objects of exponential families (mean function, variance function and Fisher information) to the case where the canonical parameter is a matrix.

We denote by R k×m the space of real matrices with k rows and m columns and by A ⊗ B the Kronecker product of two matrices. We use the usual notations A T for the transpose matrix and A, B = Tr(A T B) for the scalar product of two matrices. The operator Vec converts a k × m matrix A into a vector Vec(A) ∈ R km by stacking the columns one underneath the other. The Vec operator is commonly used in applications of the matrix differential calculus in statistics, cf. [START_REF] Magnus | Matrix Differential Calculus with Applications in Statistics and Econometrics[END_REF][START_REF] Muirhead | Aspects of multivariate statistical theory[END_REF].

The following properties of the Kronecker product are used in this work [Magnus and Neudecker, 2007, p.32,35]. For non-singular squared matrices A, B we have (A ⊗ B) -1 = A -1 ⊗ B -1 . For all matrices A, B and C such that the product ABC is well defined

Vec(A B C) = (C T ⊗ A) Vec(B).
(1)

In this paper we use the following convention of the matrix differential calculus: if a function f : R k×p → R n×m is differentiable then its derivative is a matrix

f ′ (x) ∈ R nm×kp such that Vec(df (x)(u)) = f ′ (x) Vec(u), u ∈ R k×p . (2)
The only exception we will make is the derivative of a function K : R k×m → R, for which the following convention is used: the derivative of K is not a row vector but the matrix K ′ (x) ∈ R k×m , related to the differential of K by dK(x)(u) = K ′ (x), u = Tr(K ′ (x) T u), for all u ∈ R k×m . This convention is needed to give sense to formula (5) for the mean of an exponential family.

In this section we consider probability models (P s (dω)) s∈S , S ⊂ R k×m , on a measurable space (Ω, A) such that there exists a σ-finite positive measure ν on (Ω, A) and a real function (ω, s) → l ω (s) such that P s (dω) = e lω(s) ν(dω)

and e lω(s) ν(dω) = 1. These models encompass but are not reduced to exponential families of distributions. We suppose that S is open and that the function s → l ω (s) is twice differentiable. We impose on l ω (s) classical regularity conditions, allowing double differentiation under the integral sign in e lω(s) ν(dω).

The score function l ′ ω (s) is the derivative with respect to s of the log-likelihood function l ω (s). It is a k × m matrix with zero mean. The extension of the definition of Fisher information matrix from vector-parametrized models to matrix-parametrized models is straightforward.

Definition 2.1. The Fisher information matrix of the model (P s (dω)) s∈S , S ⊂ R k×m on a measurable space (Ω, A) is the km × km symmetric matrix

I(s) = Cov(l ′ ω (s)) = Ω Vec(l ′ ω (s)) Vec(l ′ ω (s)) T P s (dω).
Similarly as for vector parametrized models, the Fisher information can be written as the negative of the mean of the second derivative of the log-likelihood function.

Proposition 2.1. The Fisher information of the model (P s (dω)) s∈S , S ⊂ R k×m equals

I(s) = - Ω l ′′ ω (s)P s (dω).
Proof. As a probability distribution, P s (dω) satifies

Ω P s (dω) = 1 =⇒ d ds Ω P s (dω) = Ω d ds P s (dω) = Ω e lω(s) l ′ ω (s)ν(dω) = 0 R k×m =⇒ d ds Ω e lω(s) l ′ ω (s)ν(dω) = Ω d ds e lω(s) l ′ ω (s) ν(dω) = 0 R km×km =⇒ Ω e lω(s) Vec(l ′ ω (s)) Vec(l ′ ω (s)) T + e lω(s) l ′′ ω (s) ν(dω) = 0 R km×km
and the Proposition follows.

Consider a σ-finite measure space (Ω, A, ν), a finite-dimensional normed vector space E and a measurable map T : Ω → E. Let µ be the image of the measure ν by T and let S µ be the interior of the domain of the moment generating function of µ, i.e. the set {s ∈ E : E e s,x dµ(x) < ∞}. We suppose that µ is a σ-finite measure on E and that S µ = ∅. The cumulant generating function of µ is defined as the logarithm of the moment generating function of µ:

k µ (s) = log E e s,x µ(dx) = log Ω e s,T (ω) ν(dω), s ∈ S µ .
Definition 2.2. The general exponential family generated by the measure ν and the map T is the family of probability distributions

{P (s, T, ν)(dω) = e s , T (ω) -kµ(s) ν(dω) : s ∈ S µ }. (3)
The natural exponential family associated with the above general exponential family is the family of probability distributions defined on the space E by

P (s, µ)(dx) = e s , x -kµ(s) µ(dx), s ∈ S µ . (4)
Natural exponential families may be viewed as a special case of general exponential families with Ω = E, T (ω) = ω and ν = µ. The following result is well-known for vector-valued and matrix-valued exponential families [START_REF] Letac | Natural exponential families[END_REF].

Proposition 2.2.

1. The set S µ is convex. If µ is not concentrated on some affine hyperplane of E, then k µ is a strictly convex function on S µ . The name "domain of the means" for the set M is justified by formula (5) of the following Proposition, well-known in the vector case.

The map s

→ k ′ µ (s) is an analytic diffeomorphism from S µ to its image M = k ′ µ (S µ ) ⊂ R k×m called the
Proposition 2.3. The mean and covariance of a random matrix X following the distribution P (s, µ) belonging to the natural exponential family generated by a measure µ are given by

m(s) = E s (X) = k ′ µ (s) (5) v(s) = Cov s (Vec(X)) = k ′′ µ (s). ( 6
)
Proof. Formula (5) follows from

k ′ µ (s) = Sµ xe s,x µ(dx) Sµ e s,x µ(dx) = Sµ xe s,x -kµ(s) µ(dx) = E s (X).
Next, using (5), we obtain

k ′′ µ (s) = Sµ d ds xe s,x -kµ(s) µ(dx) = Sµ Vec(x) Vec(x -k ′ µ (s)) T e s,x -kµ(s) µ(dx) = Sµ Vec(x -m(s)) Vec(x -m(s)) T e s,x -kµ(s) µ(dx) = Cov s (Vec(X)).
Remark 2.1. If W is a random matrix with a law P (s, T, ν) from the general exponential family, then T (W ) = X in law and m(s) and v(s) are the mean and the covariance of T (W ).

Now we compute the Fisher information of the exponential families parametrized by the canonical parameter s ∈ S µ .

Proposition 2.4. The Fisher information for the parameter s of exponential families (3) and ( 4) is given by

I(s) = k ′′ µ (s) = v(s). (7)
Proof. The log-likelihood is equal to

l ω (s) = s , T (ω) -k µ (s), so l ′′ ω (s) = -k ′′ µ (s)
does not depend on ω. Formula (7) follows by Proposition 2.1.

Definition 2.3. Denote by ψ : M → S µ , m → ψ(m) = k ′ µ -1 (m) the inverse of the diffeomorphism k ′ µ .
The general exponential family, parametrized by the domain of the means M is given by the family of distributions

Q(m, T, ν)(dω) = e ψ(m) , T (ω) -kµ(ψ(m)) ν(dω), m ∈ M. ( 8
)
The natural exponential family, parametrized by the domain of the means M is the family of probability distributions defined on the space E by

Q(m, µ)(dx) = e ψ(m) , x -kµ(ψ(m)) µ(dx), m ∈ M. ( 9
)
The mean of the families ( 8) and ( 9) is equal to m. We denote the covariance of the families (8) and ( 9) by V (m) and we have by ( 6)

V (m) = v(ψ(m)) = k ′′ (ψ(m)). ( 10
)
The function V : m ∈ M → V (m) is called the variance function of the exponential family.

We will compute the Fisher information of the exponential families ( 8) and ( 9) parametrized by the mean m ∈ M in the next section. We will need the following formula giving the Fisher information for a reparametrized model.

Theorem 2.1. Consider a model (P s (dω)) s∈S and a reparametrization f : S ⊂ R k×p → S ⊂ R n×m , where f is a differentiable map. Let I(s) be the information matrix of (P s (dω)) s∈S . The Fisher information matrix of the model

(Q t (dω)) t∈ S = P f (t) (dω) t∈ S is Ĩ(t) = f ′ (t) T I(f (t))f ′ (t).
(11)

Proof. Let us denote h ω (t) = l ω (f (t)). We have Q t (dω) = e hω(t) ν(dω). For all t ∈ S and u ∈ R k×p ,

dh ω (t)(u) = dl ω (f (t))(df (t)(u)) = l ′ ω (f (t)), df (t)(u) = = Vec (l ′ ω (f (t))) T Vec (df (t)(u)) = Vec (l ′ ω (f (t))) T f ′ (t) Vec (u) .
Thus, using the convention introduced after (2), Vec (h

′ ω (t)) T = Vec (l ′ ω (f (t))) T f ′ (t) and Vec (h ′ ω (t)) Vec (h ′ ω (t)) T = f ′ (t) T Vec (l ′ ω (f (t))) Vec (l ′ ω (f (t))) T f ′ (t).
Therefore, by Definition 2.1 we get Ĩ

(t) = f ′ (t) T I(f (t))f ′ (t).
3 Fisher information of exponential families parametrized by the mean

In this section we first compute the Fisher information of the exponential families ( 8) and ( 9) parametrized by the mean. Next we consider the same problem for a submodel parametrized by a segment of means. In order to avoid confusion, when the parameter of an exponential family is the mean m we will denote the Fisher information by J(m).

Theorem 3.1. The Fisher information of the exponential families ( 8) and ( 9) parametrized by the mean m ∈ M equals

J(m) = V (m) -1 = ψ ′ (m), ( 12
)
where V (m) is the variance function of the exponential family, given by (10).

Proof. We use Theorem 2.1 with

f = ψ : M → S µ . Since ψ(m) = k ′ µ -1 (m), we have ψ ′ (m) = [k ′′ µ (ψ(m))] -1 . Thus J(m) = [k ′′ µ (ψ(m))] -1 k ′′ µ (ψ(m))[k ′′ µ (ψ(m))] -1 = [k ′′ µ (ψ(m))] -1 = V (m) -1 . Remark 3.1.
Note a striking contrast in the formulas ( 7) and ( 12) for the Fisher information of an exponential family parametrized either by the canonical parameter s ∈ S µ or by the mean m ∈ M; in the first case we have I(ψ(m)) = V (m), in the second J(m) = V (m) -1 .

Fisher information of exponential families parametrized by a segment of means

In statistical practice, the following situation will be concerned by such models. Let m 1 ∈ M and m 2 ∈ M be two different estimations of the true mean m of an exponential family (8) or (9) . When one hesitates between them as estimators, and when M is convex, it is natural to consider the model

{Q(θm 1 + (1 -θ)m 2 , T, ν) : θ ∈ [0, 1]}.
Writing

θm 1 + (1 -θ)m 2 = θ(m 1 -m 2 )
+ m 2 we see that this is a special case of the model ( 13).

The following theorem gives the Fisher information of a general exponential family parametrized by a segment of means. By analogy to the notation J(m), we denote this information by J(θ).

Theorem 3.2. The Fisher information of the model {Q(θA + B, T, ν) : θ ∈ I} equals

J(θ) = Vec(A) T V (θA + B) -1 Vec(A). ( 14
)
Remark 3.2. This and the following results are also true for submodels {Q(θA + B, µ) : θ ∈ I} of natural exponential families.

Proof. By Theorem 3.1, the Fisher information of the model {Q(m, T, ν) : m ∈ M} is J(m) = V -1 (m). We apply Theorem 2.1 to the reparametrization f : I → M, f (θ) = θA + B. We have f ′ (θ) = Vec(A). Then formula ( 11) gives ( 14).

The following Lemma is useful for the derivation of an alternative formula for the Fisher information of an exponential family parametrized by a segment of means and verifying an additional condition (15). We will see in Section 4 that this condition holds for Gaussian and Wishart models.

J(θ) = - d 2 dθ 2 [k µ (ψ(θA + B))] . ( 17
) Proof. Let h(θ) = k µ (ψ(θA + B)) and f (θ) = θA + B. We want to compute h ′′ (θ). If θ, u ∈ R, dh(θ)(u) = dk µ (ψ(f (θ))) dψ(f (θ))(df (θ)(u)) = k ′ µ (ψ(f (θ))) , dψ(f (θ))(df (θ)(u)) = f (θ) , dψ(f (θ))(df (θ)(u)) = -df (θ)(u) , ψ(f (θ)) = -u A , ψ(f (θ)) ,
where we used successively: the convention on k ′ µ introduced after (2), the equality k ′ µ •ψ(m) = m, Lemma 3.1 and the formula df (θ)(u) = uA. Thus we have h ′ (θ) = -A , ψ(f (θ)) . Now, starting as in the computation of h ′ (θ) and using (2), we get

h ′′ (θ) = -A , dψ(f (θ))(A) = -Vec(A) T Vec(dψ(f (θ))(A)) = -Vec(A) T ψ ′ (θA + B) Vec(A).
We conclude using (12) and Theorem 3.2.

Applications

In this section, we apply the results from preceding sections to the study of some important exponential families parametrized by a segment of means.

We denote by S d the vector space of d × d symmetric matrices and by S + d the open cone of positive definite matrices.

Exponential families of Gaussian distributions

Let us recall the construction of the multivariate Gaussian model {N(u, Σ); Σ ∈ S + d } as a general exponential family. We consider Ω = R d equiped with a normalised Lebesgue measure ν(dω) = dω/(2π) d/2 , the space E = S d and the map

T : R d → S d , T (ω) = - 1 2 (ω -u)(ω -u) T .
The image measure µ on E is concentrated on the opposite of the cone of semi-positive definite matrices of rank one. For s ∈ S d , the moment generating function of µ equals

Ω e s,T (ω) ν(dω) = 1 (2π) d/2 R d e -1 2 Tr(s(ω-u)(ω-u) T ) dω = (det s) -1/2
when s ∈ S + d and it is infinite otherwise. Thus S µ = S + d and the cumulant function is

k µ (s) = - 1 2 log det(s), s ∈ S µ = S + d .
The general exponential family is therefore

P (s, T, ν)(dω) = 1 (2π) d/2 e s , -1 2 (ω-u)(ω-u) T + 1 2 log det(s) dω = (det s) 1/2 (2π) d/2 e -1 2 (ω-u) T s(ω-u) dω, (18)
which is the family of Gaussian distributions N(u, s -1 ) on R d with a fixed mean u ∈ R d , parametrized by s = Σ -1 , the inverse of the covariance matrix Σ supposed to be invertible. The derivative of the function X ∈ R d×d → det X is the cofactor matrix X ♯ which equals (det X)(X -1 ) T when X is inversible. It follows that

m(s) = k ′ µ (s) = - 1 2 s -1 , s ∈ S + d .
This can be also deduced from Remark 2.1; if W is a random vector with law N(u, s -1 ), then

m(s) = k ′ µ (s) = E T (W ) = E(- 1 2 (W -u)(W -u) T ) = - 1 2 Cov W ∈ -S + d .
The means domain is M = -S + d and the inverse mean map is ψ(m) = -1 2 m -1 . The Gaussian general exponential family parametrized by m in the means domain

M = -S + d is therefore the family Q(m, T, ν) = N(u, -2m). ( 19
)
Up to a trivial affine change of parameter Σ = -2m, this parametrization by the covariance parameter is more natural than the parametrization of the family (N(u, s -1 )) s∈S + d by the canonical parameter s.

In order to compute the variance function, recall that

XX -1 = I d implies that dX -1 = -X -1 dX X -1 and (X -1 ) ′ = -X -1 ⊗ X -1 . Thus k ′′ µ (s) = 1 2 s -1 ⊗ s -1 and formula (10) implies that V (m) = 2m ⊗ m. (20)
By Proposition 2.4, the Fisher information of the family (N(u,

s -1 )) s∈S + d is I(s) = 1 2 s -1 ⊗ s -1
. By Theorem 3.1 and formula (20), the Fisher information of the model (N(u, -2m)) m∈-S + d equals J(m) = 1 2 m -1 ⊗ m -1 . Corollary 4.1. The Fisher information matrix of the Gaussian model (N(u, Σ))

Σ∈S + d is J(Σ) = 1 2 Σ -1 ⊗ Σ -1 .
Proof. Using Theorem 2.1 and a reparametrization Σ = -2m we obtain J

(Σ) = 1 2 Σ -1 ⊗ Σ -1 = J(Σ).
Let us now consider Gaussian models parametrized by a segment of covariances.

Corollary 4.2. Let C and D be two symmetric matrices and let I ⊂ R be a non-empty segment such that

I ⊂ Θ = {θ ∈ R : θC + D ∈ S + d }. The Fisher information of the Gaussian model {N(u, θC + D), θ ∈ I} is J(θ) = 1 2 Tr C(θC + D) -1 C(θC + D) -1 .
The last formula allows to compute easily d 2 dθ 2 (log det(θC + D)). First we see that

d dθ (log det(θC + D)) = d dθ det(θC + D) det(θC + D) = d j=1
a j θa j + 1 .

One more derivation and formula (21) lead to (22).

We finish by computing the Fisher information of two Gaussian models in R d , parametrized by an explicitely given segment of covariances. First, let A be a circulant matrix with the first row e 2 + e d = (0, 1, 0, . . . , 0, 1). Then for a segment I ⊂ R containing 0 and θ ∈ I θA

+ I d =          1 θ 0 . . . 0 θ θ 1 θ 0 . . . 0 0 θ 1 θ 0 . . . . . . . . . . . . 0 . . . 0 θ 1 θ θ 0 . . . 0 θ 1          ∈ S + d . (23)
Corollary 4.4. The Fisher information of the model (N(0, θA + I d )) θ∈I is given by

J(θ) = 1 2 d-1 j=0 2 cos( 2πj d ) 1 + 2θ cos( 2πj d ) 2 . (24)
Proof. Let A be a circulant matrix with the first row (r 0 , r 1 , . . . , r d-1 ). It is well known (see e.g. [START_REF] Gray | Toeplitz and Circulant Matrices: A review[END_REF]) and easy to check that if ǫ is a d-th root of unity, ǫ d = 1, then a = d-1 l=0 r l ǫ l is an eigenvalue of A with an eigenvector (1, ǫ, ǫ 2 , . . . , ǫ d-1 ).

Therefore if ǫ j = e 2πji d , j = 0, . . . , d -1 are the d distinct d-th roots of unity, then the matrix A has d distinct eigenvalues a j = d-1 l=0 r l ǫ l j . In our particular case,

a j = e 2πji d + e 2(d-1)πji d = 2 cos 2πj d .
Formula (24) follows from Theorem 4.1. Now, let us consider a tridiagonal matrix C such that

θC + I d =          1 θ 0 0 0 . . . θ 1 θ 0 0 . . . 0 θ 1 θ 0 . . . . . . . . . . . . . . . 0 . . . 0 θ 1 θ 0 . . . 0 0 θ 1          . ( 25
)
As in the preceding case, there exists a segment I ⊂ R such that θC + I d ∈ S + d for θ ∈ I.

Corollary 4.5. The Fisher information of the model (N(0, θC + I d )) θ∈I is given by

J(θ) = 1 2 d j=1 2 cos j d+1 π 1 + 2θ cos j d+1 π 2 . ( 26
)
Proof. We will apply Theorem 4.1 with C and D = I d . Expanding ψ d (λ) = det (C -λI d ) along the first row, we get ψ d (λ) = -λψ d-1 (λ) -M 1,2 . Expanding the minor M 1,2 along its first column gives M 1,2 = ψ d-2 (λ) and

ψ d (λ) = -λψ d-1 (λ) -ψ d-2 (λ), d ≥ 3.
We set ϕ d (λ) = (-1) d ψ d (2λ) and we obtain

ϕ d (λ) = 2λϕ d-1 (λ) -ϕ d-2 (λ), d ≥ 3
with initial conditions ϕ 1 (λ) = 2λ, ϕ 2 (λ) = 4λ 2 -1. Therefore ϕ d is a Tchebyshev polynomial of the second kind [START_REF] Mason | Chebyshev polynomials[END_REF] and it satisfies ϕ d (cos x) = sin(d+1)x sin x , d ≥ 1. We have, for all λ ∈ [-2, 2],

ψ d (λ) = 0 ⇐⇒ ϕ d λ 2 = 0 =⇒ sin(d + 1)x sin x = 0, x = arccos λ 2 .
Therefore λ j = 2 cos j d+1 π , 1 ≤ j ≤ d, are d distinct eigenvalues of the matrix C.

Exponential families of Wishart distributions

Let E = S d be the space of symmetric real matrices of order d. Wishart distributions are multivariate analogs of the gamma distributions λ p Γ(p) -1 e -λx x p-1 dx on R + ( p > 0, λ > 0), considered with a canonical parameter s = -λ < 0. Similarly as in dimension 1, the Wishart distributions are often parametrized by a scale parameter σ = (-s) -1 ∈ S + d and then the notation γ(p; σ) = W (p; (-σ) -1 ) is used, cf. [START_REF] Letac | The noncentral Wishart as an exponential family, and its moments[END_REF]

(p) -1 (det x) p-d+1 2 , x ∈ S + d , Γ d (p) = Γ(p)Γ(p -1 2 ) . . . Γ(p -d-1 2 ). Otherwise, when p ∈ { 1 2 , . . . , d-1 2 },
Q(m, µ p ) = W (p; -pm -1 ) = γ(p; 1 p m), m ∈ S + d . ( 27
) As v(s) = k ′′ µp (s) = p(s -1 ⊗ s -1
), it follows that the variance function is

V (m) = 1 p (m ⊗ m). (28)
By Proposition 2.4, the Fisher information of the model {W (p; s)

: s ∈ -S + d } is I(s) = ps -1 ⊗ s -1 . By Theorem 3.1 the Fisher information of the model {Q(m, µ p ), m ∈ M} is J(m) = pm -1 ⊗ m -1 .
Consequently, using Theorem 2. (29)

J(θ) = -p d 2 dθ 2 (log det(θC + D)) J(θ) = p d j=1 a j 1 + a j θ 2 (30)
where a 1 , . . . , a d are the eigenvalues of the matrix D -1/2 CD -1/2 . Proof. The proofs are similar to the proofs of the analogous results for exponential Gaussian families in the previous subsection. The condition ( 15

θ = θσ 1 + (1 -θ)σ 2 ∈ S + d . The Fisher information of the model {γ(p; σ θ ) : θ ∈ I} is equal to J(θ) = p Tr (σ 1 -σ 2 ) σ -1 t 2 .
Proof. We write θσ 1 + (1θ)σ 2 = θ(σ 1σ 2 ) + σ 2 and we apply formula (29).

Using (30) we obtain the following corollary, analogous to Corollaries 4.4 and 4.5. Remark 4.1. Let P (s, µ) be the natural exponential family corresponding to the Gaussian general exponential family (18). If W has the law N(u, s -1 ) given by ( 18), then T (W ) has the law P (s, µ). On the other hand it is well known that -T (W ) = 1 2 (Wu)(Wu) T has the Wishart law γ( 1 2 ; 2s -1 ). This explains why the formulas for the Fisher information are the same for the Gaussian family and for the Wishart family with p = 1 2 .

Exponential families of noncentral Wishart distributions. Let us finish the section on the Wishart models by considering the non-central case. The main reference is [START_REF] Letac | The noncentral Wishart as an exponential family, and its moments[END_REF]. As before, we denote σ = (-s) -1 . We see that the mean equals m(s) = k ′ µ (s) = p(-s) -1 + (-s) -1 a(-s) -1 = pσ + σaσ (31)

  domain of the means of the family. In particular M is open.

  Consider a general exponential family {Q(m, T, ν)(dω) : m ∈ M} parametrized by the domain of the means M. Let A = 0, B ∈ R k×m be two matrices. Define Θ = {θ ∈ R : θA + B ∈ M}. The set Θ ⊂ R is open because M is open. Suppose that Θ = ∅. The parametrization by a segment of means I ⊂ Θ consists in considering the submodel {Q(θA + B, T, ν) : θ ∈ I}.

Lemma 3. 1 .

 1 Assume that for all m ∈ M, m , ψ(m) = C, (15) for some constant C ∈ R. Then, for all u ∈ M, m , dψ(m)(u) =u , ψ(m) . (16) Proof. By (15) the differential of the function g : M → R, m → m , ψ(m) is zero. Therefore, for all m, u ∈ M dg(m)(u) = m , dψ(m)(u) + u , ψ(m) = 0 and (16) follows. Corollary 3.1. Let {Q(θA + B, T, ν)(dω) : θ ∈ I} be an exponential model parametrized by a segment of means. If the condition (15) holds then the Fisher information of the model equals

  the measures µ p are singular and concentrated on semipositive symmetric matrices of rank 2p. The family of Wishart distributions W (p; s) on S + d is defined as the natural exponential family generated by the Riesz measure µ p . According to (4), it means that p ∈ Λ d , s ∈ S µp = -S + d and W (p; s)(dx) = e s , x L µ p (-s) µ p (dx) = e s , x (det(-s)) p µ p (dx) = e s , x -kµ p (s) µ p (dx), with k µp (s) = -p log det(-s). It follows that L W (p; s)(t) = det(I d +(-s) -1 t) -p and that µ p (dx) = e Tr x W (p; -I d ).

  . The study of Wishart distributions is motivated by their importance as estimators of the covariance matrix of a Gaussian model in R d . Let us apply our results on the Fisher information to a natural exponential family of Wishart distributions {W (p; s) : s ∈ -S + d }. The mean equals m(s) = k ′ µp (s) = p(-s) -1 ∈ M = S + d and the inverse mean map ψ : S + d → -S + d is ψ(m) = -pm -1 . Thus the Wishart family Q(m, µ p ) parametrized by the domain of means is, up to a trivial reparametrization m → 1 p m, the family parametrized by its scale parameter:

  1 and a reparametrization m → 1 p m = σ we see that the the Fisher information matrix of the Wishart model {γ(p; σ) : σ ∈ S + d } parametrized by a scale parameter σ equals J(σ) = pσ -1 ⊗ σ -1 . Theorem 4.2. Let I = (a, b) ⊂ R and C, D ∈ S d such that IC +D ⊂ S + d . The Fisher information J(θ) of the Wishart model {γ(p; θC + D) : θ ∈ I} verifies the formulas J(θ) = p Tr C(θC + D) -1 2

  ) holds true: m , ψ(m) = -pd, the model {γ(p; θC + D) : θ ∈ I} is equal to the model {Q(θpC + pD, µ p ) : θ ∈ I} parametrized by the means and we have k µ (ψ(θpC + pD)) = p log det(θC + D).

Corollary 4. 6 .

 6 Let σ 1 , σ 2 ∈ S + d and let I be the open interval containing θ such that σ

  Corollary 4.7. 1. Consider the model {γ(p; θA + I d ) : θ ∈ I} with θA + I d as in (23). Then its Fisher information equals J(θ) = p dthe model {γ(p; θC +I d ) : θ ∈ I} with θC +I d as in (25). Then its Fisher information equals J(θ)

  Let p ∈ Λ d , a ∈ S + d and σ ∈ S + d . The noncentral Wishart distribution γ(p, a; σ) is defined by its Laplace transform L γ(p, a; σ)(t) = S + d e -Tr(tx) γ(p, a; σ)(dx) = det(I d + σt) -p e -Tr(t(I d +σ t) -1 σaσ) , for all t ∈ S + d . When p ≥ d-1 2 , then non-central Wishart laws exist for all a ∈ S + d ; when p ∈ { 1 2 , . . . , d-2 2 } then a must be of rank at most 2p [Letac and Massam, 2011]. When p = n 2 , n ∈ N, the non-central Wishart distributions are constructed in the following way from n independent d-dimensional Gaussian vectors Y 1 , . . . , Y n . Let Y j ∼ N(m j , Σ) and let M be the d × n matrix [m 1 , . . . , m n ]. Then, the d × d matrix W = Y 1 Y T 1 + . . . + Y n Y T n has the noncentral Wishart distribution γ(p, a; σ) with p = n2 , σ = 2Σ and σaσ = MM T . Such Wishart distributions are studied in[START_REF] Muirhead | Aspects of multivariate statistical theory[END_REF].The non-central Wishart distributions may be constructed as a natural exponential family {W (p, a; s) : s ∈ -S + d } generated by the positive measure µ = µ a,p (dx) = e Tr(a+x) γ(p, a; I d )(dx). Its moment generating function is given for s ∈ -S + d by S + d e Tr(sx) µ a,p (dx) = det(-s) -p e Tr(a(-s) -1 ) .We have W (p, a; s) = γ(p, a; (-s) -1 ). Like for central Wishart families, S µ = -S + d . The cumulant generating function is k µ (s) = -p log det(-s) + Tr(a(-s) -1 ).

  The Riesz measures µ p on the cone S + d are unbounded positive measures such that their Laplace transform equals for t ∈ S +

			d
	L µ p (t) =	d S +	e -t,x dµ p (x) = (det t) -p .
	By the celebrated Gindikin theorem, such measures exist if and only if p belongs to the Gindikin
	set Λ d = { 1 2 , . . . , d-1 2 } ∪ d-1 2 , ∞ . They are supported by the cone S + d if and only if p > d-1 2
	and they are absolutely continuous in that case, with a density Γ d

Acknowledgements. The authors would like to thank Gérard Letac for his insightful suggestions. The second author acknowledges financial support(co-tutelle fellowship) from the French Ministry of Foreign Affairs and the Embassy of France in South Africa. The authors are greatly indebted to l'Agence Nationale de la Recherche for the research grant ANR-09-Blan-0084-01.

Supported by l'Agence Nationale de la Recherche ANR-09-BLAN-0084-01.

Proof. We use Corollary 4.1 and Theorem 2.1 with f (θ) = θC + D. It follows that J(θ) = Vec(C) T J(θC + D) Vec(C) = 1 2 Vec(C) T (θC + D) -1 ⊗ (θC + D) -1 Vec(C).

Applying (1) we get

On the other hand we have the following alternative formula for the information J(θ).

Corollary 4.3. The Fisher information of the Gaussian model {N(u, θC + D), θ ∈ I} is

Proof. Observe that the condition (15) holds for the Gaussian exponential families Q(m, t, ν):

The model

We apply Corollary 3.1 and the fact that

Formula (21) follows.

Now we characterize the information J(θ) in terms of the eigenvalues of the matrix D -1/2 CD -1/2 . Theorem 4.1. Let C and D be two symmetric matrices and let I ⊂ R be a segment such that IC + D ⊂ S + d . Let a 1 , . . . , a d be the eigenvalues of the matrix D -1/2 CD -1/2 . The Fisher information of the Gaussian model {N(u, θC + D), θ ∈ I} equals

Proof. The idea of the proof is to use formula (21). Let P (λ) be the characteristic polynomial of the matrix D -1/2 CD -1/2 . We have

On the other hand P (λ) = n j=1 (a jλ). It follows that

(θa j + 1).

and the covariance

When the matrix a is non-singular, the inverse mean map ψ(m) = s is such that

For other cases see [Letac and Massam, 2008, Prop.4.5]. In order to write the variance function

we compose the last expression from (32) and the formula (33).

For a model {W (p, a; ψ(θA + B)) : θ ∈ I} parametrized by a segment of means, the Fisher information J(θ) is obtained from the expression of V (m) and Theorem 3.2. Example. Suppose that a = I d , A = αI d and B = βI d , α, β > 0. The Fisher information on θ is

.

Estimation of the mean in exponential families parametrized by a segment of means

Consider a sample X 1 , . . . , X n of a random variable X from a natural exponential family Q(m, µ) parametrized by the domain of means M, where the parameter m = E X is unknown and M is open. The following qualities of the sample mean Xn as an estimator of m seem to be known; for the sake of completeness we provide a short proof of properties which are less evident.

Proposition 4.1. The sample mean Xn is an unbiased, consistent and efficient estimator of the parameter m. It is also a maximum likelihood estimator of m.

Proof. By Theorem 3.1 we have Cov X = V (m) = J(m) -1 , so the Cramér-Rao bound is attained by X. Consequently, the sample mean Xn is an efficient estimator of m. It follows from ( 9) that the sample mean Xn is a maximum likelihood estimator of m. One can also first show by (4) that the maximum likelihood estimator of s is ŝ = k ′-1 µ (X) = ψ(X) and next use the functional invariance of the maximum likelihood estimator [Casella and Berger, 2002, Theorem 7.2.10].

Remark 4.2. For general exponential families Q(m, T, ν) parametrized by an open domain of means M, all these properties remain valid for m = T (X) n as an estimator of m = E T (X).

Consider an exponential family Q(θA+B, µ) parametrized by a segment of means IA+B ⊂ M with A = 0, B ∈ E and θ ∈ I, a segment in R. We will now discuss estimators of the real parameter θ when we know that the mean E X = m ∈ IA + B.

The segment IA + B ⊂ M is of dimension one and has an empty interior. That's why the efficiency and maximum likelihood properties of the estimator m = Xn are not automatically inherited by natural estimators of the real parameter θ. Determining a maximum likelihood estimator for θ seems impossible explicitly. This is the "price to pay" for the parsimony of the segment model parametrized by m ∈ IA + B. On the other hand, the efficiency of estimators of θ may be studied thanks to Theorem 3.2 and its corollaries.

Knowing that

for a value θ ∈ I, we have many possibilities of writing down a solution θ of equation ( 34). If A = 0 then the solution θ is unique (Aθ + B = Aθ ′ + B implies θ = θ ′ when A = 0.) For any C such that A , C = 0 we have

We define an estimator θC of the parameter θ by θC = Xn -B , C

A , C

All the estimators θC are unbiased and consistent. The natural question is whether they are efficient. The variance of θC may be computed using the variance function V (m) of the exponential family:

On the other hand, the Cramér-Rao bound is equal by Theorem 3.2 to

When the space E is a squared matrix space R d×d and the matrix A is invertible, we can take C = A -1 and consider the estimator

The following theorem shows that for Gaussian and central Wishart exponential families and for linearly dependent A and B the estimator θA -1 is efficient as an estimator of the mean m (with

In conclusion, we obtain efficient estimators for Gaussian models parametrized by a covariance segment parameter and for Wishart models parametrized by a scale segment parameter. Proof. For the first part of the Theorem, we give the proof in the Wishart case. The proof in the Gaussian case is identical, with p = 1 2 , cf. Remark 4.1. By formulas ( 35) and ( 28)

On the other hand, by ( 36) and ( 28)

Thus Var θ = 1 nJ(θ) and the estimator θA -1 is efficient. The second part of the Theorem follows by necessary reparametrizations. For (2a), using ( 19