18.5 Excursion: Generating Multivariate Robust Influence Curves for Normal Scores


{A, b, V, ctrl} = 32348 ICerz (e, FI, A0, b0, N, eps, itmax, expl, fact0, aus)
generates--if possible--a (simultaneously clipped) IC to a given efficiency loss for normal scores
{A, b, V, ctrl} = 32351 ICerzsep (e, S1, S2, A0, b0, N, eps, itmax, expl, fact0, aus)
generates--if possible--a (separately clipped) IC to a given efficiency loss for normal scores, with the same output list as 32354 ICerz .
Coming from the local i.i.d. asymptotic setup, we have applied estimators to the regression model (18.9) that have proven to be optimal there; so at this point we want to give a short abridge of the theory behind it and of how these optimal IC's may be obtained, numerically.


18.5.1 Definition of IC

In the local i.i.d. asymptotic setup, we consider a parametric family $ \{P_{\theta},\quad \theta \in \Theta\}$ and want to estimate the true $ \theta_0$ based on observations $ x_1, \ldots x_n$. To do so we only allow for asymptotically linear estimators $ S_n$ for this parameter $ \theta$, i.e.,

$\displaystyle \sqrt{n} (S_n-\theta)= \frac{1}{\sqrt{n}} \sum_{i=1}^n \psi_{\theta}(x_i)+{\rm o}_{P_{\theta}}(1)$ (18.17)

for some $ L_2$-Variable $ \psi$. As is shown in Rieder (1994), to get local Fisher consistency of $ S_n$--i.e. $ S_n$ has to converge to $ \theta_0+h/\sqrt{n}$ in $ P_{\theta_0+h/\sqrt{n}}$-probability for all $ h$--we must necessarily have

$\displaystyle \mathop{\rm {{}E{}}}\nolimits _{\theta}[\psi_{\theta}]=0, \qquad ...
...p{\rm {{}E{}}}\nolimits _{\theta}[\psi_{\theta}\Lambda_{\theta}']={\rm\bf I}_n.$ (18.18)

Optimality results also to be looked up in Rieder (1994) show that for the i.i.d. setup,


18.5.2 General Algorithm

In general, for $ b$ given, $ A$ is determined by the implicite equation

$\displaystyle A^{-1} \stackrel{!}{=} \mathop{\rm {{}E{}}}\nolimits \left[\Lambda \Lambda' \min\left\{1,\frac{b}{\Vert A\Lambda\Vert}\right\}\right].$ (18.19)

18.5.2.1 Arbitrary Dimension $ n$

As clipping is done continuously, and by the integration $ \mathop{\rm {{}E{}}}\nolimits [\cdot]$, we achieve an arbitrary smoothness of (18.19) in $ A$ and $ b$.

As we know that for $ b=\infty$, $ A={\cal I}^{-1}$, for $ b$ not too small we can use $ {\cal I}^{-1}$ as a starting value for the fixed point iteration

$\displaystyle A_{i+1}^{-1} \stackrel{!}{=} \mathop{\rm {{}E{}}}\nolimits \left[[\Lambda \Lambda' \min\left\{1,\frac{b}{\Vert A_i\Lambda\Vert}\right\}\right].$ (18.20)

Proofs for local convergence may be found in Ruckdeschel (1999).

For smaller $ b$, we first solve (18.19) for a larger $ b'$ and then take $ A(b')$ as a starting value for (18.20); as a criterium whether (18.20) converges or not we take the development of the size of $ A$ which is controlled by the parameter expl, the stepsize from $ b$ to $ b'$ is controlled by fact.

Once we have determined for given $ b$ $ A(b)$, we control whether $ \mathop{\rm {{}E{}}}\nolimits \Vert\psi\Vert^2$ is smaller or larger than $ (1+\delta)\mathop{\hbox{tr}}{\cal I}^{-1}$. To determine the pair $ (b,A(b))$ for given $ \delta$, we use a bisection algorithm, as $ \mathop{\rm {{}E{}}}\nolimits \Vert\psi\Vert^2$ is strictly dicreasing in $ b$.

18.5.2.2 One Dimension

In dimension $ n=1$, the problem is only $ 2$-dimensional, so it pays off, using a simultaneous Newton procedure to determine $ (b,A(b))$. This is done by the auxiliary routines 32528 abinfonewton and 32531 absepnewton for the simultaneously and the separately clipped case, respectively.


18.5.3 Polar Representation and Explicite Calculations

In the case of $ \Lambda\sim {\cal N}_n(0,{\cal I})$, $ n>1$, we have some nice properties, which make calculations in (18.20) easier.

18.5.3.1 Polar Representation

We write $ \Lambda$ as

$\displaystyle \Lambda={\cal I}^{\frac{1}{2}} \tilde \Lambda ={\cal I}^{\frac{1}{2}}Y u$ (18.21)

with $ \tilde \Lambda \sim {N}_n(0,{\rm\bf I}_n)$, $ u=\Vert\tilde \Lambda\Vert$, $ Y=\tilde \Lambda/u$. Then $ Y\sim{\rm ufo}({\cal S}_{n-1})$, $ u^2\sim \chi^2_n$ and $ u$, $ Y$ independent.

Now we have to solve

$\displaystyle {\cal I}^{-\frac{1}{2}}A^{-1}{\cal I}^{-\frac{1}{2}}$ $\displaystyle \stackrel{!}{=}$ $\displaystyle \mathop{\rm {{}E{}}}\nolimits [YY'r(Y)]$ (18.22)
$\displaystyle (1+\delta)\mathop{\hbox{tr}}\Sigma_{t\vert t}$ $\displaystyle \stackrel{!}{=}$ $\displaystyle \mathop{\hbox{tr}}A {\cal I}^{\frac{1}{2}} \mathop{\rm {{}E{}}}\nolimits [YY's(Y)]{\cal I}^{\frac{1}{2}}A$ (18.23)

with
$\displaystyle r(Y)$ $\displaystyle =$ $\displaystyle \mathop{\rm {{}E{}}}\nolimits \left[u \min\left\{u,
\frac{b}{\Vert A{\cal I}^{\frac{1}{2}}Y\Vert}\right\}\vert Y\right]$ (18.24)
$\displaystyle s(Y)$ $\displaystyle =$ $\displaystyle \mathop{\rm {{}E{}}}\nolimits \left[\min\left\{u^2,
\frac{b^2}{\Vert A{\cal I}^{\frac{1}{2}}Y\Vert^2}\right\}\vert Y\right]$ (18.25)


18.5.3.2 Explicit Calculations for the Absolute Value

For given $ Y$, $ c={b}/{\Vert A{\cal I}^{\frac{1}{2}}Y\Vert}$ is constant, and calculation of $ r$, $ s$ using the proposition that clipped moments of $ u$ can be calculated by means of (higher) clipped moments of the standard normal--c.f. Ruckdeschel (1999).


18.5.4 Integrating along the Directions

For the remaining integration along the directions $ Y$ we do

For simultaneous clipping we additionally have (c.f. Ruckdeschel (1999)) the interesting proposition that clipping only effects the spectrum of $ {\cal I}$ but not the invariant spaces. This is used to transform integration from $ YY'r$, $ YY's$ to $ Y'Yr$, $ Y'Ys$, thus reducing the problem from $ n^2$ to $ n$ dimensions.


18.5.5 Short Description of the Auxiliary Routines Used

The further quantlets in the quantlib kalman just being auxiliary routines for the ones described up to now, we confine ourselves to shortly giving a survey of these routines in form of a table; note that in the table, $ \textrm{\tt AIh}\hat=A{\cal I}^{\frac{1}{2}}$, and $ u$, $ x$ stand for random variables, $ u^2\sim \chi_n^2$ and $ x\sim {N}(0,1)$.

quantlet input output function
32943 itera (A0,FI,b,N, (A,V,ctrl) Fixed-Point-Iteration (18.20)
  eps,itmax,   for sim. clipping; also decides
  expl)   if there was convergence or not
32946 iteras (A0,S1,S2,b, (A,V,ctrl) Fixed-Point-Iteration (18.20)
  N,eps,itmax,   for sep. clipping; also decides
  expl)   if there was convergence or not
32949 numint2 (aIh, b, N) (r,s) $ n=2$: (reduced problem)
      Romberg-integration
32952 numint2m (aIh, b, N) (r,s) $ n=2$: (full problem)
      Romberg-integration
32955 stointp (aIh, b, N) (r,s) $ n>2$: (reduced problem)
      MC-integration
32958 stointpm (aIh, b, N) (r,s) $ n>2$: (full problem)
      MC-integration
32961 ewinn (c,n) y calculates $ r(c)$
32964 ew2inn (c,n) y calculates $ s(c)$
32967 betrnormF (t,n) y calculates $ \mathop{\rm {{}E{}}}\nolimits [ (u\leq t)]$.
32974 betrnormE (t,n) y calculates $ \mathop{\rm {{}E{}}}\nolimits [ u(u\leq t)]$.
32981 betrnormV (t,n) y calculates $ \mathop{\rm {{}E{}}}\nolimits [ u^2(u\leq t)]$.
32988 nmomnorm (t,n) y calculates $ \mathop{\rm {{}E{}}}\nolimits [ x^n(x\leq t)]$.