On the role played by the fixed bandwidth in the Bickel-Rosenblatt goodness-of-fit test

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On the role played by the fixed bandwidth in the Bickel-Rosenblatt goodness-of-fit test
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  Statistics & Operations Research TransactionsSORT 29 (2) July-December 2005, 201-216 Statistics &Operations ResearchTransactions On the role played by the fixed bandwidth in theBickel-Rosenblatt goodness-of-fit test c   Institut d’Estad´ıstica de Catalunyasort@idescat.esISSN: 1696-2281www.idescat.net  /  sort Carlos Tenreiro Universidade de Coimbra Abstract For the Bickel-Rosenblatt goodness-of-fit test with fixed bandwidth studied by Fan (1998) we deriveits Bahadur exact slopes in a neighbourhood of a simple hypothesis  f  = f  0  and we use them to get abetter understanding on the role played by the smoothing parameter in the detection of departuresfrom the null hypothesis. When  f  0  is a univariate normal distribution and we take for kernel the standardnormal density function, we compute these slopes for a set of Edgeworth alternatives which give us adescription of the test properties in terms of the bandwidth  h  . A simulation study is presented whichindicates that finite sample properties are in good accordance with the theoretical properties based onBahadur local efficiency. Comparisons with the quadratic classical EDF tests lead us to recommenda test based on a combination of bandwidths in alternative to Anderson-Darling or Cram´er-von Misestests. MSC:  62G10, 62G20 Keywords:  goodness-of-fit test, kernel density estimator, Bahadur efficiency. 1 Introduction Let  X  1 ,  X  2 ,...,  X  n ,...  be a sequence of independent and identically distributed  d  -dimensional random vectors with unknown density function  f  . As it has been shownby Bickel and Rosenblatt (1973), a test of the simple hypothesis  H  0  :  f   =  f  0  against thealternative  H  a  :  f     f  0 , where  f  0  is a fixed density function on R d  , can be based on the  L 2  distance between the kernel density estimator of   f   introduced by Rosenblatt (1956)  Address for correspondence:  Carlos Tenreiro. Departamento de Matem´atica, Universidade de Coimbra,Apartado 3008, 3001-454 Coimbra, Portugal. Phone: (351) 239 791 155. Fax: (351) 239 832 568. E-mail:tenreiro@mat.uc.ptReceived: July 2004Accepted: April 2005  202  On the role played by the fixed bandwidth in the Bickel-Rosenblatt goodness-of-fit test  and Parzen (1962), and its mathematical expectation under the null hypothesis (see alsoFan (1994) and Gouri´eroux and Tenreiro (2001)):  I  2 n ( h n )  =  n     {  f  n (  x  ) − E 0  f  n (  x  ) } 2 dx  ,  (1)where, for  x   ∈ R d  ,  f  n (  x  )  =  1 n n  i = 1 K  h n (  x  −  X  i ) , K  h n  =  K  ( · / h n ) / h d n  with  K   a kernel, that is, a bounded and integrable function on R d  , and( h n ) is a sequence of strictly positive real numbers converging to zero, when  n  goes toinfinity (bandwidth). The Bickel-Rosenblatt test is asymptotically consistent and has anormal asymptotic distribution under the null hypothesis.Following an idea of Anderson, Hall and Titterington (1994) that have used kerneldensity estimators with fixed bandwidth for testing the equality of two multivariateprobability density functions, Fan (1998) uses the statistic (1) with a constant bandwidthfor testing the composite hypothesis that  f   is a member of a general parametricfamily of density functions. He provides an alternative asymptotic approximation forthe finite-sample properties of the Bickel-Rosenblatt test by showing that, for a fixed h , the asymptotic distribution of   I  2 n ( h ) is an infinite sum of weighted  χ 2 randomvariables. Moreover, Fan (1998) proves that  I  2 n ( h ) can be interpreted as a  L 2  weighteddistance between the empirical characteristic function and the parametric estimate of the characteristic function implied by the null model with weight function  t  →| φ K  ( th ) | 2 .In the important case of testing univariate or multivariate normality, and taking for  K  the standard normal density function, the role played by  h  in the power performance of the test is assessed in simulation studies by Epps and Pulley (1983), Henze and Zirkler(1990) and Henze and Wagner (1997).Restricting our attention to the test of a simple hypothesis, the main purpose of thispaper is to derive the Bahadur local exact slopes of goodness-of-fit tests based on  I  2 n ( h ),for a fixed  h  >  0, and use them to get a better understanding of the role played bythe smoothing parameter in the detection of departures from the null hypothesis. Forcompleteness reasons we give in Section 2 the asymptotic null distribution and theconsistency of the test based on kernel density estimators with a fixed bandwidth. Usingthe integral and quadratic form of   I  2 n ( h ), we derive in Section 3 its Bahadur local exactslopes. They naturally depend on the smoothing parameter, on the kernel, on the nulldensity  f  0  and, finally, on the considered departure direction from the null hypothesis. InSection 4, in the particular case of a test for a simple univariate hypothesis of normalityand taking for  K   the standard normal density function, the Bahadur local slopes arenumerically evaluated for di ff  erent values of   h  for a set of Edgeworth alternatives. Thesealternatives express departures from the null hypothesis in terms of each one of the first  Carlos Tenreiro   203 four moments. The tests based on  I  2 n ( h ) for di ff  erent values of   h  are compared with thecorresponding ones of the quadratic EDF tests of Anderson-Darling (  A 2 ) and Cram´er-von Mises ( W  2 ). The results we obtain suggest that a large bandwidth is adequate fordetection of location alternatives whereas a small bandwidth is adequate for detectionof alternatives for scale, skewness and kurtosis. A simulation study indicating that finitesample propertiesof tests  I  2 are in good accordance with the theoretical propertiesbasedon the Bahadur local slopes is also presented. Moreover, if one does not know muchabout the unknown density function it suggests that a test based on a combination of bandwidths,thatestablishacompromisebetweenthetwooppositee ff  ectsthatthechoiceof   h  has in the detection of location and nonlocation alternatives, is a good practicalrecommendation in alternative to traditional  A 2 or  W  2 tests.For convenience of presentation the proofs of some results in this article are givenin Section 5. We denote by  asn → + ∞ −→  the convergence with probability 1 and by  d n → + ∞ −→  theconvergence in distribution. 2 Asymptotic null distribution and consistency Consider the following assumptions on  K   which ensure that  d  (  f  , g )  =    { K  h    f  (  x  )  − K  h    g (  x  ) } 2 dx   1 / 2 , where    denotes the convolution product, is a distance on the set of integrable functions (see Anderson  et al.  (1994)). Assumptions on  K   (K) K   is a bounded and integrable function on R d  with Fourier transform  φ K   such that { t   ∈ R d  :  φ K  ( t  )  =  0 } has Lebesgue measure zero.In order to derive the asymptotic distribution of   I  2 n ( h ) under  H  0  for a fixed  h  >  0, wefirst note that  I  2 n ( h ) is a  V  -statistic, that is,  I  2 n ( h )  =  1 n n  i ,  j = 1 Q h (  X  i ,  X   j ) ,  (2)with kernel Q h ( u , v )  =     k  (  x  , u ; h ) k  (  x  , v ; h ) dx  , where k  (  x  , u ; h )  =  K  h (  x  − u ) − K  h    f  0 (  x  ) ,  (3)for  u , v ,  x   ∈  R d  . From the hypothesis on  K  , the kernel  Q h  is bounded. Therefore the  204  On the role played by the fixed bandwidth in the Bickel-Rosenblatt goodness-of-fit test  functions  u → Q h ( u , u ) and  Q h  are  P 0  and  P 0  ⊗  P 0  integrable, respectively, where  P 0  =  f  0 λ and λ istheLebesguemeasurein B ( R d  ).Moreover,  Q h  issymmetricanddegenerate,i.e.,    Q h ( · , v ) dP 0 ( v ) =  0 ,  a . e .  ( P 0 ) .  From Gregory (1977), we know that the asymptoticdistribution of   I  2 n ( h ) under  H  0  can be characterized in terms of the eigenvalues of thesymmetric Hilbert-Schmidt operator  A h  defined, for  q  ∈  L 2 ( R d  , B ( R d  ) , P 0 )  = :  L 2 ( P 0 ),by(  A h q )( u )  =     Q h ( u , v ) q ( v ) dP 0 ( v ) .  (4)In view of the degeneracy property of   Q h ,  q 0 , h  =  1 is an eigenfunction of   A h corresponding to the eigenvalue  λ 0 , h  =  0. Denoting by   1   the subspace generatedby  q 0 , h  and  H  ( P 0 )  =  g  ∈  L 2 ( P 0 ) :    gdP 0  =  0   the tangent space of   P 0 , we have  L 2 ( P 0 )  =   1 ⊕  H  ( P 0 ). The operator  A h  is positive definite on  H  ( P 0 ) as follows from theintegral form (3) of   Q h  and assumption (K). In fact, if    A h q , q   =  0, for some  q  ∈  H  ( P 0 ),where · , · denotes the usual inner product in  L 2 ( P 0 ), we have0  =     q ( u ) k  ( · , u ; h ) dP 0 ( u ) =  K     ( qf  0 )( · ) ,  a . e .  ( λ ) , yielding  φ K  ( t  ) φ qf  0 ( t  )  =  0, for all  t   ∈ R d  . From assumption (K) and the continuity of the Fourier transform, we deduce that φ qf  0 ( t  )  =  0,  t   ∈ R d  , i.e.,  q  =  0 ,  a . e .  ( P 0 ).Finally, using the the infinite-dimensionality of   H  ( P 0 ) and the positivity of   A h  on  H  ( P 0 ) we can conclude that  A h  has a countable infinite collection { λ k  , h , k   ∈ N } of strictlypositive eigenvalues (see Dunford and Schwartz (1963), Corollary X.4.5).The following result follows from the limit distribution of degenerate V-statistics(cf. Theorem 4.3.2 of Koroljuk and Borovskich (1989)). Remark that the asymptoticdistribution presented by Fan (1998) in Theorem 4.2, is not correct. In general the  P 0 -integrability of   u → Q h ( u , u ) is not a su ffi cient condition for   λ k  , h  <  ∞ . Theorem 1  If assumption  (K)  is fulfilled then, under H  0  we have I  2 n ( h )  d n → + ∞ −→  I  ∞ , with I  ∞  =     Q h ( u , u ) dP 0 ( u ) + ∞  k  = 1 λ k  , h (  Z  2 k   − 1) , where the sequence  ( λ k  , h )  , with  λ 1 , h  ≥  λ 2 , h  ≥  ...  and   λ k  , h  → 0 , k   →  + ∞  , is described above and   (  Z  k  )  are i.i.d. standard normal variables. Moreover, the test I  2 ( h )  =  (  I  2 n ( h ))  Carlos Tenreiro   205 defined by the critical regions  {  I  2 n ( h )  >  c α }  , where  P(  I  ∞  >  c α )  =  α  , is asymptotically of level  α  and consistent to test H  0  against H  a . Remark 1  If the density  f  0  has a compact support  S   and  Q h  is continuousin  S  × S  , fromthe Mercer’s expansion for  Q h  (see Dunford and Schwartz (1963), p. 1088) it followsthat    Q h ( u , u ) dP 0 ( u )  =  ∞ k  = 1 λ k  , h  and therefore  I  ∞  takes the form  I  ∞  =  ∞ k  = 1 λ k  , h  Z  2 k  . 3 Bahadur local efficiency In order to compare the test  I  2 ( h ) with other test procedures, or to compare  I  2 ( h ) testsobtained for di ff  erent values of   h , we derive in the following its Bahadur exact slopes C   I  2 ( h ) (  f  ), for  f   in a neighbourhood of   f  0 . They coincide with the Bahadur approximateslopes (and then with the Bahadur local approximate slopes) derived by Gregory (1980).For the description of Bahadur’s concept of e ffi ciency, see Bahadur (1967, 1971) orNikitin (1995).Throughout, ||·||  p  denotes the norm of the Lebesgue space  L  p ( R d  , B ( R d  ) ,λ )  = :  L  p ( λ ).The proof of the following result is given in Section 5. Theorem 2  We haveC   I  2 ( h ) (  f  )  = b  I  2 ( h ) (  f  ) λ 1 , h (1 + o (1)) ,  as ||  f   −  f  0 || 1 → 0 , whereb  I  2 ( h ) (  f  )  =     { K  h    f  (  x  ) − K  h    f  0 (  x  ) } 2 dx  , and   λ 1 , h  is the largest eigenvalue of the operator A h  defined by (4). If   f  0  belongs to a family of probability density functions of the form {  f  ( · ; θ  ) :  θ   ∈  Θ } ,where  Θ  is a nontrivial closed real interval and  f  0  =  f  ( · ; θ  0 ), for some  θ  0  ∈  Θ , it isnatural to compare a set of competitor tests through its Bahadur local exact slopes when θ  → θ  0 .Consider the following assumptions on the previous parametric family: Assumptions on {  f  ( · ; θ  ) :  θ   ∈  Θ }  (P)For all  x   ∈  R d  the function  θ   →  f  (  x  ; θ  ) is continuously di ff  erentiable on  Θ , andthere exists a neighbourhood  V   ⊂  Θ  of   θ  0  such that the function  x  → sup θ  ∈ V   ∂  f  ∂θ  (  x  ; θ  )   isintegrable on R d  .The following result comes easily from Theorem 2, assumption (P) and thedominated convergence theorem.
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