The class of multivariate lq-spherical distributions is introduced and defined through their isodensity surfaces. We prove that, under a Jeffreys' type improper prior on the scale parameter, posterior inference on the location parameters is the same for all lq-spherical sampling models with common q. This gives us perfect inference robustness with respect to any departures from the reference case of independent sampling from the exponential power distribution.Bayesian inference; Exponential power distributions; Inference robustness; lq-norm; Symmetric multivariate distributions;

Osiewalski, Jacek

Steel, Mark F.J.

English

Research Papers in Economics

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Working Paper 92-34  Divisi6n de Economfa 
July  1992  Universidad Carlos III de Madrid 
Calle Madrid,  126 
28903 Getafe 
Fax:  (341) 624-9849 
Robust Bayesian Inference in Iq-Spherical Models 
Jacek Osiewalski and Mark F.J. Steel" 
Abstractl...  _ 
The class of multivariate Iq-spherical distributions is introduced and defined through 
their isodensity surfaces. 
We prove that, under a Jeffreys' type improper prior on the scale parameter, posterior 
inference  on  the  location  parameters  is  the  same for  all  Iq-spherical  sampling  models  with 
common  q.  This gives  us perfect inference robustness  with  respect to any departures from 
the  reference  case  of independent  sampling  from  the  exponential  power  distribution. 
Key words:
 
Bayesian  inference;  Exponential  power  distributions;  Inference  robustness;  Iq-norm;
 
Symmetric multivariate distributions.
 
·Osiewalski, Department of Econometrics, Academy of Economics, Krak6w,  Poland; Steel,
 
Departamento de Estadfstica y Econometrfa,  Universidad Carlos III de Madrid.
 J
 
) 
) 
) 
) 
,) c 
1.- Introduction 
("  A new  class  of multivariate  distributions,  which  we  name  Iq-spherical,  is  defined 
through properties of the density function.  In particular, the isodensity surfaces are spheres 
in Iq-norm for q C!: 1. Robustness results obtained in Osiewalski and Steel (1992) for the case 
q=2, are here found to extend to general qE(O,oo]. c 
Fang,  Kotz and  Ng  (1990)  mention eight different ways of constructing  symmetric 
multivariate distributions, one of which is symmetry of the density function. In their Chapter 
S they  induce  symmetry  on  the  survival  function  with  an  II-norm.  They  start  from  an 
C  exponential  distribution  (defined  on  the positive  real  line),  resulting  in  a density  function i 
proportional to  that of our Ij-spherical distribution over the positive orthant of the  sample 
space.  In their  Chapter 7  they  use  symmetry  of  the  characteristic function  to define 
I  a-symmetric distributions. For cr=2 this rorresponds to our 12-spherical distributions, usually C 
referred to in the literature as spherical and discussed  in detail in Kelker (1970), Cambanis, 
Huang and Simons (1981), Dickey and Chen (1985), and Fang, Kotz and Ng (1990, chapters 
2-4). 
,c 
In  the  same  way  as  independent  sampling  from  a  univariate  Normal  distribution 
constitutes  a  reference  case  for  the  1 2-spherical  family,  independent  sampling  from 
exponential power distributions [see Box and Tiao (1973, chapter 3)] forms a useful reference 
class for the entire Iq-spherical family. 
Within a Bayesian  framework,  we. prove that under a commonly  used  diffuse prior 
on the scale parameter posterior inference on the location vector is fully robust with respect 
to departures of any Iq-spherical sampling density from its reference case.  C 
I 
2.- Defining Iq-Spherical Distributions 
c 
In  this section we introduce the class of multivariate Iq-spherical distributions, where 
,  .. 
symmetry IS Imposed through the density function. This implies we only consider continuous 
distributions. 
'C  1 
c J
 
Let us first introduce the following notation,  for a =  (ah  ....  , aJ': 
(t  la,I')~  if  O<q<oo 
;·1
vq(a) = 
max lail  if  q=oo  . 
;·1....". 
If  we choose q in the range [1,  00] then vq =  11  a 11  q' the Iq-norm of the vector a.  In 
the case q E  (0,1) vl) does not satisfy the triangle inequality. For our purposes, however, 
the latter is not required. 
Definition:  For any scalar O<qs 00, x= (XI'  •.. , xJ' E  RD  has an Iq-spherical distribution 
with location  p.  =  (p.1'  ..., p.J'  E  RD  and scale  1'-1  E  RH denoted as x - IqD  (p.,  1'011J, if 
its density function  is given by 
and  gq  (') is a nonnegative function  such that p(x  IP.,T) is a proper density.  • 
The isodensity surfaces follow  immediately from  the Definition as 
(2) 
where ex> O.  For q ~ 1 they could be considered spheres with respect to Iq-norm centered at 
p..  This fact  motivates calling these densities  "Iq-spherical". 
Let us  now consider the conditions on gl) imposed by our Definition. For finite q, 
we define  r= v  q{ t(x-~)},  and  ~{t  1.r1~~.If.  i= 1, ... , n. Transforming x into (ZI,. .• ,Zu.h 
r),  we derive from p(x  IP.,T) a product ofan n-variate Dirichlet  density  on  z= (ZI,...,zJ' 
with parameters q.l  and 
2 
) 
) 12r (hi!
p(rlz,~,t)=p(r)=  q  r,,-tg (r),  (3) 
r (:1  i 
c 
which  is a proper density function  over R+  if and only if 
(4) 
c 
( 
Extending  the analysis  in  Dickey and  Chen  (1985),  we can represent  x in  terms  of 
three independent random quantities 
1 
t(x-J.L)=r (s  x zi), 
where  zl/q denotes a coordinatewise power,  s X  zl/q is a coordinatewise product of vectors 
and  the n elements of s independently take the value  1 or -1  with probability  1/2. 
This allows  us  to derive the following  moments for  Illt =  0,  1,  2,  ... : 
( 
o 
if  all  m j  are even, 
otherwise, 
(5) 
which exist if and only if  E(rL  III~  is finite.  In particular, if E (r) <  00, 
3 E(xl~.t)=~ 
(6) 
(7) 
The case of infinite q requires a separate treatment.  Now we define 
W'-WH and  Z·- ,  ) I r 
i= 1,  .., n where  Wo = 0 and  W j  (ic:d) is the ith ordered value of 'T I  X; - p.j I,j=I,... ,n. 
The  transformation  from  x  to  (z\> ...  ,  Zo_\>  r)  gives  us  a  Dirichlet  density  for  z  with 
, ) parameters  1 and 
p(r I  z,p.,'T)  = p(r)  =  n 2
0  ~.1  goo(r),  (8) 
.~  , 
)' 
from  which  the condition on goo(') implicit in the Definition becomes 
•

J U"-lS..(u)  du=  !  2-
11 
•  C••  (9)
 
o n 
Since Wj can  now be represented  as 
) 
I 
and  Ez.  possesses a Beta distribution with parameters i and  n-i, E(wt) exists if and only if 
Jat J 
E(fll)  is finite when i<n. Trivially, E(wo 
Dl
)  = E(r
m
)  as  Wo = r. 
Using the exchangeability property of x, we know that E[T(X,-j.l,)]",..l j:E(wl- for even 
nJa l 
m>  0,  from  which  we derive 
) 
iJ. 
4 
.,J 
--_._-------------------------,--------­t  "  m+n  E(r"'}  if  mis  event 
n(m+l)  (10) =  El t(Xi-~i)]rn 
0  if  m  is  odd . 
This immediately leads  to 
(11) E  (xl~J't')=~ 
(12) 
for  finite E(r2). 
We remark that as  q... oo  the right-hand side of (4) converges to that of (9), and the 
constant in  (5)  for  one nonzero  m j  =  m converges  to  the constant in  (10).  Clearly,  such 
convergence also  applies to  the isodensity surfaces in  (2)t which are  graphically displayed 
in Figure I for  n=2 and selected values of q. 
Fig.I.  Isodensity lines/or n=2. 
..
 
\"",  "
 
.. '  - "  , , ,  .. ' 
(values for q in the graph) 
For given a and any nt the isodensity surfaces for all qE  (Ot  00] coincide when x= 
#L±e where  e has  only  one  nonzero element  (namely  aT,I)t and  for  other values  of x the 
isodensity surfaces  move further  from  #L  as q grows. 
5 3.- Exponential Power Distributions 
An  interesting  special  class of Iq-spherical distributions is generated by independent 
sampling from exponential power distributions [Box and Tiao (1973, chapter 3)]: 
(13) 
"  ' p(x,l~ ••)=df'  e~- ~f  IXj-~jlft 
with 
I" 
df-[21'~  I'(1+~r. 
for O<q<co.  and  from 
p(x.IJ.l,t)=.! 1 (J.lj-'t-1< x,  <J.l;+'t-1)  (14) 2 
for  q = 00, where 1(') denotes  the indicator function.  Thus,  sampling  from  (13) and  (14) 
corresponds to particular choices of ~(.)  in  the Definition.  For 0 <q< 00  and  uE R+J  we  , 
'"')'
choose  8 (u)=d; exp  (_~Uf}  from which,  using (3),  we obtain q
q p(r) =2-!! tl  [r  ( ~)]"1q r"-1 exp-2 1 r  , 
Le.  the  qth power of the Iq-radius  is Gamma distributed  with parameters n/q and  1/2.  This 
implies that 
III  r (m+n) 
E (r"')= 2q  q, 
r (:) 
) 
which  can  be inserted  into  (5),  and  indicates that  all  moments of r exist for  m> -n, and, 
thus,  all positive moments of x in  (5) are finite.  For infinite q, we implicitly choose 
g""  (u)  = 2'
D I (O<u< 1), 
leading to a Beta distribution for r with parameters n and 1. Therefore, existence of moments 
is the same as in the case of finite q. 
6 Due to the independence,  the n-variate distribution considered in this Section forms 
a convenient reference case within the class of Iq-spherical distributions. For q= 1 we obtain 
the case  of independent  sampling  from  a double-exponential  or Laplace distribution  with 
variance 81.2,  for q=2 it corresponds to sampling from a Normal with variance  1,2, and for 
q=():)  we sample from  a uniform distribution with variance 1.2/3. 
Box and Tiao (1973, chapter 3) consider exponential power distributions for q~ I, 
and  parameterize instead in terms of the sampling variance,  which is given by 
2  r - jiJ
var (x.IJ1,t)= 29  t-2 
I  r  (~) 
for 0 <q < ():),  and its limit for the case q =  ():). 
4.- Posterior Inference 
We  now  focus  on conducting inference on  the location vector  p.  when  the  sampling 
distribution is Iq-spherical.  In  many  practical cases,  IJo  will be parameterized in  terms  of a 
lower-dimensional  parameter.  However,  such  a regression  context  will  not  explicitly  be 
considered here. 
Theorem: For any qE(0, ():)], if x- Iqll( 1Jo,  T,I IJ and we assume the improper prior structure 
(15) 
k>O, then 
(16) 
Proof' We need to integrate out T from the joint density p(X,J.',T), which is the product of (1) 
and (15). Using the one-to-one transformation from  (X,IJo,1) to (x,IJo,r) where r= Vq [1(X-IJo)]= 
=  1 vq(x-p.)  and  integrating  out  r through  (4)  for  finite  and  (9)  for  infinite q,  the  result 
follows.  • 
7 i 
'I' 
The prior structure in  (15) is  the product of an  improper Jeffreys'  type prior on  7, 
which is widely used  to represent a lack of prior information about the scale, and any prior 
on  p.,  either proper or improper.  Combining (15) with any  IqD  (p.,7-IIJ sampling density on 
the  observables  x results  in  the  improper  marginal  density  (16),  which  leads  to  a proper 
posterior density p(p.  Ix) provided  p(x) =fp(:qJ.)  dll<  00.  Ifpart of x is not actually observed 
r 
(e.g.  missing  data or forecasting),  then  the out-of-sample predictive density  can easily be 
obtained from p(x). 
The main  implication of the Theorem  can  be deduced  from  the functional  form  of 
(16), where  ~(.)  no longer appears.  In other words, for a given q ~ choice of  ~(.) which 
leads  to  a proper  sampling  density  in  (1)  will  produce  the ~ inference  on  p.  and  the 
unobserved  part of x.  Therefore,  the prior  in  (15) is sufficient for  complete robustness of 
posterior  (on  p.)  and  predictive inference  within  the entire class  of Iq-spherical  sampling 
distributions  with  q  fixed.  As  stressed  in  Section  3,  the  case  obtained  by  independent 
sampling from  an exponential power distribution is a particularly convenient member of the 
Iq-spherical class.  Our Theorem thus tells us that, assuming (15), the posterior results of Box 
and Tiao (1973, chapter 3), obtained under independent sampling from (13) or (14), remain 
unaffected by any  departure within  the class  of multivariate Iq-spherical  sampling densities 
with  the same q. 
Such robustness does not hold  for the scale parameter 7;  in fact,  the entire influence 
of the choice of labelling function gq(') is summarized in the conditional posterior of 7 
(17) 
which is proper by (4) and (9). Clearly,  7 given  p.  is updated by the sample information and 
its posterior does not preserve the functional form in (15). Thus, the Theorem does not hold 
for more than one independent vector observation from Iqn(p.,7-
IIJ, and achieving robustness 
seems a hopeless task in  this case. 
8￿ 
, , 
'  " ,  , 
) 
), 5.- Extensions 
We  can  easily  generalize the  Theorem  in  the previous  section  to  cases  where  we 
sample  from  y  =  Ax  with  X -lqQ(P,1'
0 
IIJ and  A  is  a  nonsingular  matrix  function  of 
parameters '1.  Under the prior structure POt,'1,1') = k 1'.lp(p,'1), k>O, we obtain 
which extends the robustness results to the posterior of '1. For q=2 the sampling distribution 
on  y is  called  elliptical  or ellipsoidal,  a case  which  was  treated  in  Osiewalski  and  Steel 
(1992) within a regression  context. 
Finally,  our Theorem  formally  suggests  a possible  extension  to  a wider  class  of 
sampling densities p(x I IJ.,1')  =  ~g  [v {1'(x-IJ.)}], where we require v(·)  ~  0, v(aa)  =  QV(a) 
for any  positive scalar a, and  we choose g(.) in the class G of nonnegative functions  such 
.. 
that  f""-18 (u)  du  cc <,..  and  p(x I IJ.,1')  is proper  (in  the case  of lq-sphericity,  the last two 
o 
conditions coincide). Then,  the prior in (15) will result in (16) with c and v(·) instead of cq 
and vq(·). If, in addition, c is the same for any member of some  G.~ G, then robustness for 
inference on the location  p.  (and on unobserved elements of x) holds within the entire class 
G•. 
Acknowledgements:  We benefitted  from  useful  discussions  with  Carmen  Fermindez,  Jean-
Pierre Florens and  Michel  Mouchart.  Of course,  the  usual  disclaimer applies.  The  second 
author gratefully acknowledges the hospitality of the Academy of Economics,  Krak6w. 
9￿ ' ..￿ 
References 
Box,  G.E.P.  &  Tiao,  G.C.  (1973).  Bayesian  Inference  in  Statistical  Analysis.  Reading, 
Mass.:  Addison-Wesley. 
Cambanis,  S.,  Huang,  S.  &  Simons,  G.  (1981).  On  the  theory  of elliptically  contoured 
distributions. J.  Multivar.  Anal.  11, 368-85. 
) 
Dickey, J.M. & Chen, C.H. (1985). Direct subjective-probability modelling using ellipsoidal 
!) 
distributions.  In  Bayesian  Statistics  2,  Bd.  J.M.  Bernardo,  M.H.  DeGroot,  D.V. 
Lindley and A.F.M. Smith, pp.  157-82, Amsterdam:  North-Holland. 
Fang,￿  K.-T.,  Kotz,  S.  &  Ng,  K.W.  (1990).  Symmetric  Multivariate  and  Related 
Distributions.  London:  Chapman and  Hall. 
Kelker,  D.  (1970).  Distribution  theory  of  spherical  distributions  and  a  location-scale 
generalization.  Sankhya A 32, 419-30. 
J'
Osiewalski,  J.  &  Steel,  M.F.J., (1992).  Robust Bayesian  inference in  elliptical regression 
models.  J.  Econometrics,  forthcoming. 
) 
10￿ 
I 

Robust Bayesian inference in Iq-Spherical models.

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Robust Bayesian inference in Iq-Spherical models.

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