MULTIDIMENSIONAL INDICES NONLINEAR SIGNED-RANK REGRESSION

BRICE M. NGUELIFACK & ASHEBER ABEBE

INTRODUCTION

Statistical modeling with multidimensional indices is an important problem in spatial ortempro-spatial process, in signal processing, and in texture modeling. Here we show the strongconsistency of the Signed-Rank estimator and also establish the asymptotic normality of the es-timator when the function f is of harmonic type.Model & Data

In the present work, we consider an extension of

yt = f (xt,) + t t = 1, 2, . . . , n (1)

with multidimensional indices,

yt = f (xt,) + t, t n (2)

where t,n Nk and {t, t Nk} is a discrete random field denotes the partial ordering m = (m1,m2, . . . ,mk) Nk and n = (n1, n2, . . . , nk) Nk,

m n if mi ni for i = 1, 2, . . . , k.Estimator

Define the Signed-Rank Dispersion:

Dn() =1

|n|tn

an(t)(|z()|(t)) (3)

Since Dn is a continuous function of , the estimator of is therefore given by:

n = Argmin

Dn() (4)

The parameter space is compact.Where |z()|(t1) |z()|(tn) are order statistics of |yti f (xti,)| : R+ R+ is continuous and strictly increasing |n| =

ni and an(t) = +(|t|/(|n| + 1)), for a bounded score function + : (0, 1) R+ that has

at most a finite number of discontinuities.with the following particular cases:1. + 1, (t) = t2 LS estimator2. + 1, (t) = t LAD estimator

Some Examples

used in texture modeling and signal processing:

y(t1, t2) =

mk=1

Ak cos(t1k + t2k) + (t1, t2) (5)

y(t1, t2) =

mk=1

[Ak cos(t1k + t2k) + Bk sin(t1k + t2k)] + (t1, t2) (6)

where Ak, Bk and k, k [0, ] are unknown parameters.

Weak Derivatives & Sobolev Spaces

D() is the space of smooth functions with compact support in .

D =||

11 ...

nn

where || =ni=1i and = (1, ..., n).

Definition: Let L1loc(). Given Nn0 , a function L

1loc() is called the

th-weak deriva-tive of if for all D()

D

udu = (1)||

du , and we put = D

.

Definition: Let m N0 and 1 p . The Sobolev space denoted by Wm,p() is defined as

Wm,p() ={

Lp() : D Lp() with || m

}.

Interlacing Sinusoidals

y(t1, t2) = 4 cos(1.886t1 + 1.1t2) + 4 sin(1.886t1 + 1.1t2)

-6

-4

-2

0

2

4

6

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Some Assumptions.

(i) |f (xt,1) f (xt,2)| at|1 2| for all 1 6= 2, where at > 0, t Nk are some constantsuch that

tn

at = 0(|n|s), for some non-negative integer s.

(ii) suptNk,

|f (xt,)| M0 for some M0 > 0

lim inf|n|

inf|0|

1|n|[f (xt,) f (xt0)] > 0

f (xt, ) is twice continuously differentiable in . Let {Mn,n Nk} be a field of k k non-singular matrices such that(i) (1/|n|)MTn

tn

[f (xt,)][f (xt,)]TMn converges to a positive definite matrix (0) uniformly

as |n| and | 0| 0.(ii) (1/|n|)MTn

tn

[f (xt,) f (xt,)]f (xt,)Mn 0 uniformly as |n| and | 0| 0.

(iii) MTn f (xt,)MnE C00 for all , for some C00 > 0. Here E is the Euclidean normon matrices.

(iv) MTn (f (xt,1) f (xt,2))MnE bt,n|1 2| for all 1 6= 2, where bt,n > 0, t Nk aresome constants such that

t,n bt,n = o(|n|r+1) for some non-negative integer r.

(v) max1tn

(1/|n|)MTn f (xt,0)E 0 as |n| .

lim|n|

max1tn

hntt = 0

STRONG CONSISTENCY

Assumptions:(A1) : P (f (xt;) = f (xt;0)) < 1 for any 6= 0,(A2) :G has a density g that is symmetric about 0 and strictly decreasing on R+.

Theorem 1. Under the above assumptions and (A1) and (A2), na.s. 0.

Lemma 1. Under (A1) and (A2), Dn()a.s. () a.e., uniformly , where : R

satisfies inf () > (0) for any a closed subset of not containing 0.

ASYMPTOTIC NORMALITY

Write (t) = [G1 (t)]. Set Sn() = D

Dn() and (t) = D

(t) for || = 1.

More Assumptions:(A3) : (t) is a map in W 3,p(B), where B is a neighborhood of 0 for every fixed t.(A4) : There exists a function W 2,p(R) such that |D(t)| (t) for every B and || 2.(A5) :A0 =

[(+)(D)

]=0

is a positive definite matrix for || = 1.

To obtain asymptotic normality of the LS estimator, Jennrich (1969) and Wu (1981) require

Hn(1,2) =1

|n|tn{f (xt,1) f (xt,2)}2

to either converge uniformly or satisfy the Lipschitz condition Hn(1,2) c1 2.Jureckova (2008) requires c112 Hn(1,2) c212 to obtain asymptotic propertiesof regression rank scores in nonlinear models.

The examples above do not satisfy these Lipschitz conditions (Kundu, 1993).

However, our approach does not require a Lipschitz condition with an absolute constant. Ourassumptions imply that Hn(1,2) cn1 2, where cn is allowed to grow to infinity.Theorem Under certain regularity assumptions,

|n|(Mn)1(n 0)D N (0, A10 0A10

),

where 0 = E[+()0()(0())

T ] and {Mn,n Nk} is a field of p p non-singular matricesand A0 is a positive definite matrix.Mn is chosen to regulate the growth of the Hessian matrix.

SIMULATION STUDY

Consider the following Model:

y(m,n) = A cos(m + n) + B sin(m + n)] + (m,n) ,

where (m,n) is generated as

(m,n) = e(m,n) + 0.25e(m 1, n) + 0.25e(m + 1, n) + 0.25e(m,n 1) + 0.25e(m,n + 1);

We used m,n = 1, . . . , 40 and the true (A,B, , ) = (4, 4, 1.886, 1.1).

1000 replications were performed.

MSEse Method A B

LS 3.803e-4 3.640e-4 2.187e-8 2.147e-8Normal(0, = .25) SR 4.092e-4 3.873e-4 2.294e-8 2.276e-8

LAD 5.792e-4 5.394e-4 3.340e-8 3.323e-8Logistic(0, = .25) LS 1.247e-3 1.153e-3 6.920e-8 6.930e-8

SR 1.122e-3 1.048e-3 6.501e-8 6.387e-8LAD 1.387e-3 1.307e-3 8.575e-8 8.864e-8

Cauchy(0, = .25) LS 65.777 47.724 0.157 0.106SR 2.245e-2 4.275e-2 3.002e-4 2.303e-7

LAD 1.660e-2 4.321e-2 2.994e-4 8.084e-7t5 LS 1.009e-2 9.315e-3 5.664e-7 5.583e-7

SR 7.464e-3 6.881e-3 4.601e-7 4.409e-7LAD 9.740e-3 9.207e-3 6.541e-7 5.620e-7

t2 LS 0.255 0.304 3.633e-3 1.211e-3SR 2.238e-2 3.367e-2 1.463e-4 7.075e-7

LAD 1.512e-2 1.385e-2 8.930e-7 8.939e-71