Robustness issues in Kalman filtering have long been a research topic, with first (non-verified) hits on a quick search for "robust Kalman filter" on scholar. google.com as early as 1962 and 1967, i.e.; the former even before the seminal Huber (1964) paper, often referred to as birthday of Robust Statistics.
In the meantime there is an ever growing amount of literature on this topic - Kassam and Poor (1985) have already compiled as many as 209 references to that subject in 1985. Excellent surveys are given in, e.g. Kassam and Poor (1985) , Stockinger and Dutter (1987) , Schick and Mitter (1994) , Künsch (2001) .
In these references you find many different notions of robustness, all somewhat related to stability but measuring this stability w.r.t. deviations of various "input parameters"; in this paper we are concerned with (distributional) minimax robustness; i.e.; we work with suitable distributional neighborhoods about an ideal model, already used by Birmiwal and Shen (1993) and Birmiwal and Papantoni-Kazakos (1994) , and then solve the problem to find the procedure minimizing the maximal predictive inaccuracy on these neighborhoodsmeasured in terms of mean squared error (MSE)-in quite generality, compare Theorems 3.2, 3.10, 4.1. In the particular situation that the ideal conditional mean is linear in the observation innovation (for a definition see subsection 2.3.2), the minimax filter is a straight-forward Huberization, the rLS filter, which is extremely easy to compute. For this linearity we obtain a surprising characterization in Propositions 3.4 and 3.6. This motivates a corresponding optimal test for linearity, Proposition 3.8. Even in situations where no or only partial knowledge of the size of the contamination is available we can distinguish an optimal procedure, compare Lemma 3.1.
In this section, we start with some definitions and assumptions. We are working in the context of state space models (SSM's) as to be found in many textbooks, cf. e.g. Anderson and Moore (1979) , Harvey (1991) , and Durbin and Koopman (2001) .
The most prominent setting in this context is the linear, time-discrete, Euclidean setup, which will serve as reference setting in this paper: An unobservable p-dimensional state X t evolves according to a possibly time-inhomogeneous vector autoregressive model of order 1 (VAR(1)) with innovations v t and transition matrices F t , i.e., X t = F t X t−1 + v t (2.1)
The statistician observes a q-dimensional linear transformation Y t of X t and in this makes an additive observation error ε t ,
In the ideal model we work in a Gaussian context, that is we assume
3) X 0 , v s , ε t , s, t ∈ N stochastically independent (2.4)
As usual, normality assumptions may be relaxed to working only with specified first and second moments, if we restrict ourselves to linear unbiased procedures as in the Gauss-Markov setting. For this paper, we assume the hyper-parameters F t , Z t , Q t , V t , a 0 to be known.
Parts of our results (more specifically, all of sections 3.2, 3.4) also cover much more general SSMs; in this paragraph we sketch some of these. To begin with, as long as MSE makes sense for the range of the states, these results cover general Hidden Markov Models for arbitrary observation space as given by
In this setting, we assume known (and existing) [regular conditional] densities p X0 0 , p · | · t , q · | · t w.r.t. known measures ν t , µ t on B q and B p , respectively. Dynamic (generalized) linear models as discussed in West et al. (1985) and West and Harrison (1989) are covered as well -under corresponding assumptions as to (conditional) densities and range of the states. In applications of Mathematical Finance we also need to cover continuous time settings, i.e.; there is an unobservable state evolving according to an SDE
where for X 0 we assume (2.5), while W t , is a Wiener process, and f and q are suitably measurable, known functions, and observations Y t are either formulated as a time-continuous observation process (as in Tang (1998) ) or-more oftenat discrete, but not necessarily equally spaced times, compare, e.g. Nielsen et al. (2000) and Singer (2002) . In this context, but also for corresponding nonlinear time-discrete SSMs, a straightforward approach linearizes the corresponding transition and observation functions to give the (continuous-discrete) Extended Kalman Filter (EKF) After this linearization we are again in the context of a (time-inhomogeneous) linear SSM, hence the methodology we develop in the sequel applies to this setting as well. So far we do not cover approaches to improve on this simple linearization, notably the second order nonlinear filter (SNF) introduced in Jazwinski (1970), also cf. Singer (2002, sec. 4.3.1) . the unscented Kalman filter (UKF) (Julier et al., 2000) and Hermite expansions as in Aït-Sahalia (2002) , see also Singer (2002, sec. 4.3) .
Going one more step ahead, to cover applications such as portfolio optimization, we may allow for controls U t to be set or determined by the statistician, and which are fed back in the state equations. In the context of the continuous time model, this is also known as SDEX, cf. Nielsen et al. (2000) , and for the application of stochastic control to portfolio optimization, cf. Korn (1997) . In this setting, controls U t are usually assumed measurable w.r.t. σ(Y t− ); to integrate them into our setting, we simply have to integrate them in the corresponding condition vectors.
Finally, the question of specifying the order of conditioning left aside, we do not make use of the linearity of time, so our minimax results also cover suitable formulations of indirectly observed random fields.
As usual with Robust Statistics, the ideal model assumptions we have specified so far are extended by allowing (small) deviations, most prominently generated by outliers. In our notation, suffix "id" indicates the ideal setting, "di" the distorting (contaminating) situation, "re" the realistic, contaminated situation.
In SSM context (and contrary to the independent setting), outliers may or may not propagate. Following the terminology of Fox (1972) , we distinguish innovation outliers (or IO's) and additive outliers (or AO's). Historically, AO's denote gross errors affecting the observation errors, i.e., AO ::
where L(ε di t ) is arbitrary, unknown and uncontrollable (a.u.u.) and 0 ≤ r AO ≤ 1 is the AO-contamination radius, i.e.; the probability for an AO. IO's on the other hand are usually defined as outliers which affect the innovations,
where again L(v di t ) is a.u.u. and 0 ≤ r IO ≤ 1 is the corresponding radius. We stick to this distinction for consistency with literature, although we rather use these terms in a wider sense, unless explicitly otherwise stated: IO's denote endogenous outliers affecting the state equation in general, hence distortion propagates into subsequent states. This also covers level shifts or linear trends; which if |F t | < 1 are not included in (2.10), as IO's would then decay geometrically in t. We also extend the meaning of AO's to denote general exogenous outliers which enter the observation equation only and thus do not propagate, like substitutive outliers or SO's defined as SO ::
where again L(Y di t ) is a.u.u. and 0 ≤ r SO ≤ 1 is the corresponding radius. Apparently, the SO-ball of radius r consisting of all L(Y re t ) according to (2.11) contains the corresponding AO-ball of the same radius when Y re t = Z t X t + ε re t . However, for technical reasons, we make the additional assumption that
and then this relation no longer holds.
In the presence of AO's we would like to attenuate their effect, while when there are IO's, the usual goal in online applications would be tracking, i.e.; detect structural changes as fast as possible and/or react on the changed situation. A situation where both AO's and IO's may occur poses an identification problem: Immediately after a suspicious observation we cannot tell IO type from AO type. Hence a simultaneous treatment of both types will only be possible with a certain delay-see Ruckdeschel (2010) .
The most important problem in SSM formulation is to reconstruct the unobservable states X t based on the observations Y t . For abbreviation let us denote
Then using MSE risk, the optimal reconstruction is distinguished as
Depending on s this is a prediction (s < t), a filtering (s = t) and a smoothing problem (s > t). In the sequel we will confine ourselves to the filtering problem.
It is well-known that the general solution to (2.14) is the corresponding conditional expectation E[X t |Y 1:s ]. Except for the Gaussian case, this exact conditional expectation may be computational too expensive. Hence similar to the Gauss-Markov setting, it is common to restrict oneself to linear filters. In this context, the seminal work of Kalman (1960) (discrete-time setting) and Kalman and Bucy (1961) (continuous-time setting) introduced effective schemes to compute this optimal linear filter X t|t . In discrete time, we reproduce it here for later reference:
Init.:
Corr.:
and where ∆X t is the prediction error, ∆Y t the observation innovation, and Σ t|t = Cov(∆X t ), Σ t|t−1 = Cov(X t − X t|t−1 ), ∆ t = Cov(∆Y t ); M 0 t is the socalled Kalman gain, and ∆ − t stands for the Moore-Penrose inverse of ∆ t .
Realizing that M 0 t ∆Y t is an orthogonal projection, it is not hard to see that the (classical) Kalman filter solves problem (2.14) (for s = t) among all linear filters. Using orthogonality of {∆Y t } t once again, we may setup similar recursions for the corresponding best linear smoother; see, e.g. Anderson and Moore (1979) , Durbin and Koopman (2001) . Under normality, i.e.; assuming (2.3), we even have X t|t[−1] = E[X t |Y 1:t[−1] ], i.e.; the Kalman filter is optimal among all Y 1:t[−1]measurable filters. It also is the posterior mode of L(X t |Y 1:t ) and X t|t can also be seen to be the ML estimator for a regression model with random parameter; for the last property, compare Duncan and Horn (1972) .
The Kalman filter stands out for its clear and understandable structure: it comes in three steps, all of which are linear, hence cheap to evaluate and easy to interpret. Due to the Markovian structure of the state equation, all information from the past useful for the future may be captured in the value of X t|t−1 , so only very limited memory is needed. From a (distributional) Robustness point of view, this linearity at the same time is a weakness of this filter-y enters unbounded into the correction step which hence is prone to outliers. A good robustification of this approach would try to retain as much as possible from these positive properties of the Kalman filter while revising the unboundedness in the correction step.
In a first step we limit ourselves to AO's. Notationally, where clear from the context, we suppress the time index t. As no (new) observations enter the initialization and prediction steps, these steps may be left unchanged. In the correction step, we will have to modify the orthogonal projection present in (2.17). Suggested by H. Rieder and worked out in Ruckdeschel (2001, ch. 2), the following robustification of the correction step is straightforward: Instead of M 0 ∆Y , we use a Huberization of this correction
for some suitably chosen clipping height b. Apparently, this proposal removes the unboundedness problem of the classical Kalman filter while still remaining reasonably simple, in particular this modification is non-iterative, hence especially useful for online-purposes.
For the choice of the clipping height b, we have two proposals. Both are based on the simplifying assumption that E id [∆X|∆Y ] is linear, which will turn out to only be approximately right. The first one, an Anscombe criterion,
δ may be interpreted as "insurance premium" to be paid in terms of loss of efficiency in the ideal model compared to the optimal procedure in this (ideal) setting, i.e.; the classical Kalman filter. The second criterion for a given radius r
Assuming linear ideal conditional expectations, this will produce the minimax-MSE procedure for U SO (r) according to Theorem 3.2 below. One might object that (3.3) assumes r to be known, which in practice hardly ever is true. If r is unknown however, we translate an idea worked out in Rieder et al. (2008) : Assume we have limited knowledge about r, say r ∈ [r l , r u ], 0 ≤ r l < r u ≤ 1. Then we distinguish a least favorable radius r 0 defined in the following expressions
and use the corresponding b(r 0 ). Procedure rLS(b(r 0 )) then minimizes the maximal inefficiency ρ 0 (s) among all procedures rLS(b(r)), i.e.; each rLS for some clipping height b(r) = b(r 0 ) has an inefficiency no smaller than ρ 0 (r 0 ) for some r ∈ [r l , r u ]. Radius r 0 can be computed quite effectively by a bisection method: Let
Then the following analogue to Kohl (2005, Lemma 2.2.3) holds:
and there exists somer 0 ∈ [r l , r u ] such that Ar 0 /A r l = Br 0 /B ru . Thisr 0 is least favorable, i.e., min r∈[r l ,ru] ρ 0 (r) = ρ 0 (r 0 ). Moreover, if r u = 1, r 0 = r u .
In particular, the last equality shows that one should restrict r u to be strictly smaller than 1 to get a sensible procedure.
The (so-far) ad-hoc robustification proposed in the rLS filter has some remarkable optimality properties: Let us first forget about the time structure and instead consider the following simplified, but general "Bayesian" model:
We have an unobservable but interesting signal X ∼ P X (dx), where for technical reasons we assume that in the ideal model E |X| 2 < ∞. Instead of X we rather observe a random variable Y taking values in an arbitrary space of which we know the ideal transition probabilities; more specifically, we assume that these ideal transition probabilities for almost all x have densities w.r.t. some measure µ,
Our approach uses MSE as accuracy criterion for the reconstruction, so is limited to ranges of X where this makes sense. On the other hand it is this reduction to the "Bayesian" model which makes the generalizations sketched in section 2.1 possible. As (wide-sense) AO model, we consider an SO outlier model, i.e.;
for U independent of (X, Y id ) and (X, Y di ) and some distorting random variable Y di for which, in a slight variation of condition (2.12) we assume Y di , X stochastically independent (3.11) and the law of which is arbitrary, unknown and uncontrollable. As a first step consider the set ∂U SO (r) defined as ∂U SO (r) = L(X, Y re ) | Y re acc. to (3.10) and (3.11) (3.12)
Because of condition (3.11), in the sequel we refer to the random variables Y re and Y di instead of their respective (marginal) distributions only, while in the common gross error model as present in (2.9) or (2.10), reference to the respective distributions would suffice. Condition (3.11) also entails that in general, contrary to the usual setting, L(X, Y id ) is not element of ∂U SO (r), i.e.; not representable itself as some L(X, Y re ) in this neighborhood. As corresponding (convex) neighborhood we define
In the sequel where clear from the context we drop the superscript SO and the argument r.
With this setting we may formulate two typical robust optimization problems:
Minimax-SO problem Minimize the maximal MSE on an SO-neighborhood, i.e.; find a measurable reconstruction f 0 for X s.t.
Lemma5-SO problem As an analogue to Hampel (1968, Lemma 5) , minimize the MSE in the ideal model but subject to bound on the bias to be fulfilled on the whole neighborhood, i.e.; find a measurable reconstruction f 0 for X s.t.
The solution to both problems can be summarized as
The value of the minimax risk of Problem (3.14) is
(2) f 0 from (3.16) also is the solution to Problem (3.15) for b = ρ/r.
or in SSM formulation: M 0 is just the classical Kalman gain and f 0 the (one-step) rLS.
Identifying X in model (3.9) with ∆X t and π(y, x) µ(dy) with L(Z t ∆X t + ε t )(dy), our "Bayesian" Model (3.9) covers the SSM context. Hence, if ∆X t is normal, (3) applies and rLS is SO-minimax.
To illustrate the result of Theorem 3.2, we have plotted the ideal density of P Y id , the (least favorable) contaminated density of P Y re 0 , and the (least favorable) contaminating density of P Y di 0 in Figure 1 .
for P X = P ε = N (0, 1), r = 0.1; note the "thin" tails.
Remark 3.3. (a) Without using this name, SO neighborhoods have already been used by Birmiwal and Shen (1993) and Birmiwal and Papantoni-Kazakos (1994) , although only in a one-dim. model. (b) Explicit solutions to robust optimization problems in a finite sample setting are rare, which is why one usually appeals to asymptotics instead. Important exceptions are Huber (1968) , Huber and Strassen (1973) , and even there, in the former case one is limited to a special loss function and to one dimension. Our results however are valid in a finite sample context and in whole generality. (c) Although the structure of our model resembles a location model-interpreting X as a random location parameter-our saddle-point differs from the one obtained in Huber (1964) . To see this, let us look at the tails of the least favorable P Y re 0 assuming a Gaussian model for simplicity: while in Huber's setting the tails decay as ce −k|x| for some c, k > 0, in our setting they decay as c |x|e −x 2 /2 so appear even "less harmful" than in the location case. (d) Attempts to solve corresponding optimization problems in a (narrow-sense) AO neighborhood are much more difficult and only partial results in this context have been obtained in Donoho (1978) , Bickel (1981) , and Bickel and Collins (1983) ; in particular one knows, that in the setup of our example the least favorableP ε = P ε di 0 must be discrete with only possible accumulation points ±∞. In addition, existence of a saddlepoint follows from abstract compactness and continuity arguments, but in order to obtain specific solutions one has to recur to numeric approximation techniques as e.g. worked out in Ruckdeschel (2001, sec. 8.3) ; in particular, one obtains redescending optimal filters. (e) Redescenders are also used in the ACM filter by Masreliez and Martin (1977) which formally translates the Huber (1964) minimax variance result to this dynamic setting (formally, because of the randomness of the "location parameter" ∆X). It should be noted though that the least-favorable SO-situation for the ACM then is not in the tails but rather where the corresponding ψ function takes its maximum in absolute value. An SO outlier could easily place contaminating mass on this maximum, while this is much harder if not impossible to achieve in a (narrow-sense) AO situation. Hence in simulations where we produce "large" outliers, the ACM filter tends to outperform the rLS filter, as these "large" outliers are least favorable for the rLS but not for the ACM. The "inliers" producing the least favorable situation for the ACM on the other hand will be much harder to detect on naïve data inspection than "large" outliers, in particular in higher dimensions.
So far, in this section, we have ignored the fact that our X in model (3.9) resp. ∆X t in the SSM context will stem from a past which has already used our robustified version of the Kalman filter. In particular, the law of ∆X t (even in the ideal model) is not straightforward and hence (ideal) conditional expectation appearing in the optimal solution f 0 in Theorem 3.2 in practice are not so easily computable.
The issue to assess the law ∆X t from a non-linear filter past is common to other robustifications, and hence there already exist a couple of approaches to deal with it: Masreliez and Martin (1977) and Martin (1979) assume L(∆X t ) normal and propose using robust location estimators (with redescending ψ-function) as alternatives to the linear correction step. Contradicting this assumption in the rLS case, we have the following proposition Proposition 3.4. Whenever in one correction step in the ∆X t past one has used the rLS-filter, then {∆X t } (as a process) cannot be normally distributed; this assertion cannot even hold asymptotically, as long as
Similar assertions can also be proven for particular ψ-functions used in the ACM filter of Masreliez and Martin (1977) and Martin (1979) . Schick (1989) and Schick and Mitter (1994) use Taylor-expansions for nonnormal L(∆X t ); doing so they end up with stochastic error terms but do not give an indication as to uniform integrability. Hence it is not clear whether the approximation stays valid after integration. More importantly, at time instance t, they come up with a bank of (at least t) Kalman-filters which is not operational. Birmiwal and Shen (1993) work with the exact L(∆X t ) and hence have to split up the integration according to the the history of outlier occurrences which yields 2 t different terms-which is not operational either.
Remark 3.5. One of the features of the ideal Gaussian model is that Eid[∆Xt|Y1:t] is Markovian in the sense that Eid[∆Xt|Y1:t] = Eid[∆Xt|∆Yt] hence only depends on the one value of ∆Yt. When using bounded correction steps, however, this property gets lost, hence the restriction to strictly recursive procedures as is the rLS filter is a real restriction.
Theorem 3.2 does not make any normality assumptions, but in assertion (3), we have seen that the rLS would result optimal once we can show that E id [∆X t |∆Y t ] for ∆X stemming from an rLS past is linear. This leads to the question: When is E id [∆X|∆Y ] linear? Omitting time indices t, the answer is Proposition 3.6. Assume rk(I p − M Z) = p, p = q and rk Z = p, and that where Π is the projector onto ker Z andΠ = Ip − Π. In fact we prove Proposition 3.6 in this more general case. Assumption (3.25) is needed, as Π∆X is invisible for ∆Y . (c) Equivalence (3.23) together with Proposition 3.4 shows that, stemming from an rLS-past, rLS can only be SO-optimal in the very first time step.
(d) Simulations however show that rLS gives very reasonable results. So in fact we could/should be close to an ideal linear conditional expectation. "Closeness" to linearity could be quantified by the second derivative ∂ 2 /∂y 2 Eid[∆X|∆Y = y], which in fact leads us to expression (3.24). (e) Equivalence (3.24), i.e.; conditional unskewedness of ∆X, is somewhat surprising, as it seems much weaker than normality of the prediction error. 
In particle filter context where you simulate many stochastically independent filter realizations in parallel, Proposition 3.6 suggests the following test for linearity/normality: Proposition 3.8. Let ∆X i , i = 1, . . . , n be an i.i.d. sample from L(∆X t ), the law of the prediction errors of some filter at time t; let Σ = Cov(∆X t ), σ 2 its maximal eigenvalue and e a corresponding eigenvector (of norm 1); let Σ n ,σ 2 n , andê n the corresponding empirical counter parts (all assumed consistent). Define the test statistic T n = 1 n n i=1 (ê τ n ∆X i ) 3 . Then under normality of L(∆X t ), √ n T n −→ w N (0, 15σ 6 ) (3.26) and the test I(|T n | > 15/nσ 3 n u α/2 ) (3.27)
for u α the upper α-quantile of N (0, 1) is asymptotically most powerful among all unbiased level-α-tests for testing
One explanation for the good empirical findings for the rLS is given by a further extension of the original SO-neighborhoods-the extended SO or eSO-model :
In this model, we also allow for model deviations in X, i.e.; we assume a realistic (X re , Y re ) according to
U and (X id , Y id ) independent as well as (mutually) U, X di , Y di (3.30) and the joint law P X id ,Y id and the radius r = r eSO are known, while P X di , P Y di are arbitrary, unknown and uncontrollable; however, we assume that
for some known 0 < E id |X id | 2 ≤ G < ∞, and accordingly define Remark 3.9. At first glance, moment condition (3.31) seems to violate (distributional) robustness; however, this condition has not been introduced to induce a higher degree of robustness, but rather to extend the applicability of Theorem 3.2.
Theorem 3.10 (minimax-eSO). The pair (f 0 , P Y di 0 ), optimal in the Minimax-SO-problem to radius r SO = r from Theorem 3.2, extended to f 0 , P Y di 0 ⊗ P X di 0 for any P X di 0 such that E di |X di | 2 = G, remains a saddle-point in the corresponding Minimax-Problem on the eSO-neighborhood U eSO to the same radius r-no matter what bound G in equation (3.31) holds. The value of the minimax risk is
As an application of Theorem 3.10, we now invoke a coupling idea: In the Gaussian setup, i.e.; we assume (2.3), we no longer regard the (SO-) saddle-point solution to an U(r)-neighborhood around L(∆X) stemming from an rLS-past, but use Theorem 3.10 as follows:
Proposition 3.11. Assume that for each time t there is a (fictive) random variable ∆X N ∼ N p (0, Σ) such that ∆X rLS t stemming from an rLS-past can be considered an X di in the corresponding eSO-neighborhood around ∆X N with radius r. Then, rLS is exactly minimax for each time t.
Remark 3.12. (a) Existence of ∆X N ∼ Np(0, Σ) in a general setting is not yet proved. To this end one has to show moment condition (3.31) and that
where p ∆X N t , p ∆X t are the corresponding Lebesgue densities and sup λ is the corresponding essential supremum w.r.t. Lebesgue measure in the respective dimension. Clearly condition (3.34) is the difficulty, while condition (3.31) is not hard to fulfillwe only need to check that Eid ∆Xt = 0, which for the rLS follows from symmetry of the distributions in the ideal model, and that the second moment is bounded-which also clearly holds. (b) As to the choice of covariance Σ for ∆X N t , we have two candidates: Σ = Cov ∆X rLS t and Σ = Σ t|t−1 from the classical Kalman filter. While the former takes up the actual error covariances, the latter is much easier to compute. In our numerical examples in Ruckdeschel (2001), we could not find any significant advantages for the former in terms of precision and hence propose the latter for computational reasons.
(c) For p = 1, (3.34) could be checked numerically in a number of models, cf. Ruckdeschel (2001 , Table 8 .1). For p > 1, particle filter techniques should be helpful.
In this section, we translate the preceding optimality results to the IO situation.
We have already noted that in this case, instead of attenuating (the influence of) a dubious observation we would rather want to follow an IO outlier as fast as possible. It is well-known that the Kalman filter tends to be too inert for this task and faster tracking filters are needed. To do so, let us go back to our "Bayesian" model (3.9) but now we specify the transition densities π(y, x) to come from an observation Y which is built up additively as
Equation (4.1) reveals a remarkable symmetry of X and ε which we are going to exploit now: Apparently
This is helpful if we are now assuming that ε will be ideally distributed, and instead the states X get corrupted. To this end, we retain the SO-model from the preceding sections, i.e., Y id will be replaced from time to time by Y di . Contrary to the AO formulation however, we now assume that this replacement by Y di reflects a corresponding change in X, as we now want to track the distorted signal. As a consequence this gives the following IO-version of the minimax problem (where the only visible difference is the superscript "re" for X).
But, using X re = Y re − ε, and settingf (y) = y − f (y) we obtain the equivalent formulation
and we are back in the situation of subsection (3.2) with the respective rôles of X and ε interchanged. That is; the corresponding theorems translate word by word. Skipping the Lemma 5 solution we obtain (1)' In this situation, there is a saddle-point Note that contrary to Theorem 3.2 where E X need not be 0, here E ε = 0, which simplifies the definition ofD in (4.7). Details on how to use this for a corresponding IO-robust variant of rLS are given in Ruckdeschel (2010) .
In the extremely flexible class of dynamic models consisting in SSMs we were able to obtain optimality results for filtering. In this generality this is a novelty. We stress the fact that our filters are non-iterative, recursive, hence fast, and valid for higher dimensions.
So far, we have not said much about the implementation of these filters. rLS.AO was originally implemented to XploRe, compare Ruckdeschel (2000) . In an ongoing project with Bernhard Spangl, BOKU, Vienna, and Irina Ursachi (ITWM), we are about to implement the rLS filter to R, (R Development Core Team (2010)), more specifically to an R-package robKalman, the development of which is done under r-forge project https://r-forge.r-project. org/projects/robkalman/, (R-Forge Administration and Development Team (2008)). Under this address you will also find a preliminary version available for download.
In an extra paper, which for the moment is available as technical report, Ruckdeschel (2010) , we also check the properties of our filters at simulations and discuss the extension of these optimally-robust filters to a filter that combines the two types (for system-endogenous and -exogenous outlier situation). This hybrid filter is capable to treat (wide-sense) IO's and AO's simultaneouslyalbeit with minor delay.
Proof to Lemma 3.1 We use the fact that for 0 ≤ a, b, c, d,
Hence A r is increasing in r, and B r decreasing, B r from ∞ to 0. By dominated convergence b(r), and hence A r and B r are continuous in r. Thus existence of r 0 follows. For r u = 1, one argues letting r n ∈ [0, 1) tend to 1. To show equality in (6.1), we parallel Kohl (2005, Lemma 2.2.3), and first show that for r ≥ s, s fixed, ρ(r, s) is increasing and correspondingly, for r ≤ s, s fixed, decreasing, which entails (3.8): Let 0 ≤ s < r 1 < r 2 ≤ 1. Then by monotony of A r , B r , (A s B −1 s + r 1 ) −1 ≥ (A r1 B −1 r1 + r 1 ) −1 ; multiplying this inequality with (r 2 − r 1 ), we get (r 2 − r 1 )B s (A s + r 1 B s ) −1 ≥ (r 2 − r 1 )B r1 (A r1 + r 1 B r1 ) −1 . Now, due to optimality of A r + rB r for radius r,
Multiplying with (A s +r 1 B s )/(A r2 +r 2 B r2 ), we obtain indeed ρ(r 2 , s) ≥ ρ(r 1 , s), and similarly for 0 ≥ s > r 1 > r 2 ≥ 1. Next, forr 0 least favorable, we show that for r fixed, and s ≥ r, ρ(r, s) is increasing and correspondingly, for s ≤ r, decreasing: Let 0 ≤ r < r 1 < r 2 ≤ 1. Then, due to optimality of A r1 − r 1 B r1 ,
and similarly for 0 ≥ r > r 1 > r 2 ≥ 1. For the last assertion, note that by (3.3), b(1) = 0, hence B 1 = 0. Hence max A s /A r l , B s /B 1 = ∞ for s < 1, while for s = 1, we get ρ 0 (1) = max{A 1 /A r l , 1} = 1.
Proof to Theorem 3.2 (1) Let us solve max ∂U min f [. . .] first, which amounts to min ∂U E re [ E re [X|Y re ] 2 ]. For fixed element P Y di assume w.l.o.g. that µ P Y di for µ from (3.9)-otherwise we replace µ by µ + P Y di ; this gives us a µ-density q(y) of P Y di . Determining the joint (real) law P X,Y re (dx, dy) as
we deduce that µ(dy)-a.e.
a 1 q(y)+a 2 (y) a 3 q(y)+a 4 (y) (6.3)
Hence we have to minimize F (q) := |a 1 q(y) + a 2 (y)| 2 a 3 q(y) + a 4 (y) µ(dy)
To this end, we note that F is convex on the non-void, convex cone M = {q ∈ L 1 (µ) | q ≥ 0} so, for someρ ≥ 0, we may consider the Lagrangian Lρ(q) := F (q) +ρ q dµ . So by continuity, there is some ρ ∈ (0, ∞) with H(ρ) = 1. On M 0 , q dµ = 1, butq ρ = q s=ρ ∈ M 0 and is optimal on M ⊃ M 0 hence it also minimizes F on M 0 . In particular, we get representation (3.17) and note that, independently from the choice of µ, the least favorable P Y di 0 is dominated according to P Y di 0 P Y id , i.e.; nondominated P Y di are even easier to deal with.
As next step we show that
To this end we first verify (3.16) determining f 0 (y) as f 0 (y) = E re;P [X|Y re = y]. Writing a sub/superscript "re; P " for evaluation under the situation generated by P = P Y di andP for P Y di 0 , we obtain the the risk for general P as
This is maximal for any P that is concentrated on the set |D(Y di;,q )| > ρ , which is true forP . Hence (6.5) follows, as for any contaminating P
Finally, we pass over from ∂U to U: Let f r ,P r denote the components of the saddle-point for ∂U(r), as well as ρ(r) the corresponding Lagrange multiplier and w r the corresponding weight, i.e., w r = w r (y) = min(1, ρ(r) / |D(y)|). Let R(f, P, r) be the MSE of procedure f at the SO model ∂U(r) with contaminating P Y di = P . As can be seen from (3.17), ρ(r) is antitone in r; in particular, asP r is
Note that R(f s , P, 0) = R(f s , Q, 0) for all P, Q-hence passage toR(f s , P, r) = R(f s , P, r) − R(f s , P, 0) is helpful-and that
Hence the saddle-point extends to U(r); in particular the maximal risk is never attained in the interior U(r) \ ∂U(r). (3.19) follows by plugging in the results.
(2) Letf (Y ) = f (Y ) − E X, and X 0 = X − E X; then (3.15) becomes
The assertion follows upon noting that sup U | E ref | = sup |f | (to be shown just as in Rieder (1994, chap. 5) ) and writing
is linear in Y , the corresponding optimal matrix M 0 is just the respective Fourier coefficient, i.e.; Cov(X, Y ) Var Y − . We have already recalled that the classical Kalman filter is optimal among all linear filters; hence the corresponding Kalman gain M 0 is then the optimal linear transformation in the SSM context. Remark 6.1. (a) Birmiwal and Shen (1993) proceed similarly for their result. However, they invoke a minimax result by Ferguson (1967) which in our infinite dimensional setting is not applicable. Also their setting is restricted to one dimension, and they assume Lebesgue densities right away-also in the contaminated situation. In particular, they do not realize the connection to the exact conditional mean present in equation (3.18).
(b) For an alternative proof, see Ruckdeschel (2001, pp.156-163) : It uses Rieder (1994, App. B) , showing existence of Lagrange multipliers in (1) by abstract compactness and continuity arguments. (c) The fact that the solutions to Problems (3.14) and (3.15) coincide parallels the situation in the estimation problem for a one-dimensional location parameter.
Proof to Proposition 3.4 Recall that by the Cramér-Lévy Theorem (cf. Feller (1971, Thm. 1, p. 525) ) the sum of two independent random variables has Gaussian distribution iff each summand is Gaussian. This can easily be translated into a corresponding asymptotic statement, cf. Ruckdeschel (2001, Prop. A.2.4) , i.e.; the sum of two independent random variables converges weakly to a Gaussian distribution iff each summand converges weakly to a Gaussian distribution. We first consider (for fixed t, omitted from notation where clear) the filter error,
where we assume ∆X, ε, and v normal. Then for the conditional law of ∆X given ∆Y is N p (g, (I p − M 0 Z)Σ) for Σ = Cov ∆X and g :
which by Cramér-Lévy cannot be normal, as g is obviously not normal. Consequently ∆X t+1 = F t+1 ∆X t + v t+1 cannot be normal either. Hence starting with normal ∆X t and ε t , ∆X t+1 cannot be normal. The same assertion clearly holds if v t is not normal. As by (3.21), g t does neither converge to 0 nor to M 0 ∆Y , the asymptotic version of Cramér-Lévy also excludes asymptotic normality.
Remark 6.2. A similar assertion for the case that vt is normal but not both ∆Xt and εt are, seems plausible and we conjecture that this is true; it may also be proven in particular cases, but in general, it is hard to obtain due to the lack of independence of ∆X − g and ∆Y .
Proof to Proposition 3.6 For the second equivalence in Proposition 3.6 we use the following lemma and a corollary of it:
Lemma 6.3. Let ε ∼ N q (0, V ), X ∼ P X and for some measurable function h : range(X) → R q let Y = h(x) + ε. Let g ∈ L l 1 (P X ), i.e., g : range(X) → R l measurable and E P X |g(X)| < ∞. Then
Proof. For simplicity, we only consider rk V = q; otherwise we may pass to ε = Aε for someε ∼ Nq(0,Ṽ ) with rkṼ =q and use the generalized inverse V − instead of V −1 everywhere in the proof.
Let p ε be the Lebesgue density of ε and denote Λ ε (ε) := ∂ ∂ε log p ε (ε). Then, no matter whether ε is Gaussian, it holds that
As ε is normal, we may interchange differentiation and integration and obtain that ∂ ∂y
But as ε ∼ N q (0, V ), it holds that Λ ε (ε) = −V −1 ε, which entails (6.9) as
Corollary 6.4. In our linear time discrete, Euclidean SSM, ommiting indices t, assume that rk V = q and let
Proof. During the proof we will omit ∆ in notation. Equation (6.11) is just plugging in Lemma 6.3. We note that equivalently to (6.9) we could have written
Hence applying Lemma 6.3 for g(X) = X i U j and g(X) = U j to the last two terms we obtain
Proof to Proposition 3.6 Equivalence (3.23):
If L(∆X) is normal, the uncorrelated random variables Π∆X andΠ∆X are independent and again normal, while the random variables ∆X, ∆Y are jointly normal, hence linearity of conditional expectation is a well-known fact.
If E id [∆X|∆Y ] is linear, after subtracting E M Z∆X from both sides, the defining equation for the conditional expectation P Y (dy)-a.e. reads M (y − Zx)p ε (y − Zx) P X (dx) = (I p − M Z) xp ε (y − Zx) P X (dx) (6.13) Let us introduce q ε (y) = yp ε (y) and the signed measure Q X (dx) = x P (dx); if we denote the mapping h : R q → R, y → h(y) = f (y − Zx) G(dx) by f * Z G, (6.13) becomes M q ε * Z P X = (I p − M Z)p ε * Z Q X (6.14)
We pass over to the Fourier transforms (denoted with· ) for s ∈ R p , t ∈ R q q X (s) = e is τ x Q X (dx),p X (s) = e is τ x P X (dx), q ε (t) = e it τ x q ε (y) dy,p ε (t) = e it τ x p ε (y) dy,
As usual, convolution translates into products in Fourier space, in our case
and hence (6.14) in Fourier space is Mq εpX (Z τ · ) = (I p − M Z)p εqX (Z τ · ). For the derivatives (p X ) (s), (p ε ) (t) for s ∈ R p and t ∈ R q , we obtain (p X ) (s) = iq X (s), (p ε ) (t) = iq ε (t) (6.15) By assumption, I p − M Z is invertible and ε ∼ N q (0, V ), hencep ε (t) = exp(−t τ V t/2) > 0 and together with (6.15), this gives the linear differential equation This is the characteristic function of a normal distribution, so Z∆X, hence alsō Π∆X are normal, and together with (3.25) the assertion follows. On the other hand, Cov Z∆X = ZΣZ τ , so we have also shown that Z(I p − M Z) −1 M V = ZΣZ τ , which otherwise is tricky unless assuming Σ and ∆ invertible.
If E id [∆X|∆Y ] is linear, by equivalence (3.23) ∆X and ∆Y are jointly normal with expectation 0, so the conditional law of ∆X given ∆Y is again normal with expectation 0, hence in particular symmetric so the assertion follows. Now assume E e τ (∆X − E[∆X|∆Y ]) 3 ∆Y = 0 ∀ e ∈ R p (6.17)
Apparently, E id [∆X|∆Y ] is linear iff ∂ 2 /∂y∂y τ E id [∆X|∆Y ] = 0. But Corollary 6.4 gives (in the notation of (6.10)) ∂ 2 ∂y j ∂y k E[∆X i |∆Y = y] = E(∆X 0 i U 0 j U 0 k |∆Y = y) (6.18) By complete polarization (compare Weyl (1997, Chap. I.1)), (6.17) also entails that the symmetric multilinear form given by E[∆X 0 i ∆X 0 j ∆X 0 k |Y = y] i,j,k∈{1,...,p} is identically 0. So the assertion follows, as withZ = ZV −1 , the RHS of (6.18) is just p h,l=1Z j,hZk,l E(∆X 0 i ∆X 0 h ∆X 0 l |∆Y = y)
Proof to Theorem 3.10 We proceed as in Theorem 3.2, but note that in the eSO context (6.2) becomes P (X ∈ A, Y re ∈ B) = (1 − r) I A (x) I B (y)π(y, x) P X id (dx) µ(dy)
+r I A (x) I B (y)q(y) P X di (dx) µ(dy) and hence (6.3) becomes
But by (3.31), the RHS of (6.19) is exactly F (q) from (6.3). Thus, we may jump to the proof of Theorem 3.2 from this point on, replacing tr Cov X bỹ G := tr Cov P X di 0 X di = G − | E id X id | 2 in equation (6.6). For passing from ∂U eSO to U eSO , let f r ,P r ⊗Q r be the components of the saddle-point at ∂U eSO (r) and R(f, P ⊗ Q, r) be the MSE of procedure f at ∂U eSO (r) with contaminating P Y di ⊗ P X di = P ⊗ Q. Instead of equation (6.7), we use ∆G :=G − tr Cov id X id = G − E id |X id | 2 ≥ 0 and abbreviating R(f, P ⊗ Q, r) − R(f, P ⊗ Q, 0) byR(f, P ⊗ Q, r) we obtaiñ R(f s , P ⊗ Q, r) = r tr Cov Q X di − Cov id X id + E P [min(|D(Y di )|, ρ(s)) 2 ] ≤ ≤ r ∆G + E id |D(Y id )| 2w s (Y id ) + ρ(s) 2 = =R(f s ,P r ⊗Q r , r) <R(f s ,P r ⊗Q r , s) =R(f s ,P s ⊗Q s , s)
Hence the saddle-point extends to U eSO (r). (3.33) follows by plugging in the results. ∼ N (0, σ 2 ). Thus by the Lindeberg-Lévy CLT,
But the sixth moment of N (0, σ 2 ) is just 15σ 6 . Hence by the assumed consistency ofê n for e, Slutsky's Lemma yields (3.26). Asymptotically, the testing problem is a test for a normal mean µ to be 0 or not, which yields the corresponding optimality for the Gauss test given in (3.27).
Proof to Proposition 3.11 Let us identify X ∆X N , Y ∆Y N := Z∆X N + ε, and set P ε = N q (0, V ), P X = N p (0, Σ), and let p ε the corresponding Lebesgue density, then π(y, x) = p ε (y − Zx). Assertions (1') and (3') of Theorem 3.10 show that the eSO-optimal f 0 in our "Bayesian" model of subsection 3.2 is just f 0 (y) = M 0 (y) min{1, ρ/ M 0 y } with ρ according to (3.17) such that dP Y di 0 = 1 and M 0 = ΣZ τ (ZΣZ τ + V ) −1 . By assumption, ∆X rLS lies in the corresponding eSO-neighborhood U(r) about ∆X N so the value of the saddle-point from equation (3.19) is also a bound for the MSE of X rLS t|t on U(r). Remark 6.5. One should mention, however, that due to assumption (2.12) resp.
(3.11), members of an SO-neighborhood U (r ) about L(∆X rLS , ∆Y rLS ) need not lie in an eSO neighborhood U(r + r ) about L(∆X N , ∆Y N ).