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Zhang, Jun [Zhang, Jun24 ] (PRC-SZU-CMS) ; Lin, Bingqing (PRC-SZU-CMS) ;Zhou, Yan [Zhou, Yan11 ] (PRC-SZU-CMS)
Kernel density estimation for partial linear multivariate responses models.
(English summary)J. Multivariate Anal. 185 (2021), Paper No. 104768, 17 pp.

This article proposes an estimation procedure for the q-dimensional partial linear
multivariate response model   >    
g1 (U )
X β0Y1
1..   ..   ..   .. (1)++ . .
. =.
.
Yq
qX> β 0gq (U )In this model, the p-dimensional covariate vector X and parameter vector of interest β0 in the parametric component are common across q equations, whereas unknown
functions g1 (U ), . . . , gq (U ) with a common scalar covariate U in the nonparametric component vary across q equations. The errors (1 , . . . , q )> are assumed to be independentof (X, U ).
The kth equation for k = 1, . . . , q in (1) implies that
Yk − E(Yk |U ) = {X − E(X|U )}> β0 + k .
The conditional expectations, namely, sYk (u) := E(Yk |U = ut) for k = 1, . . . , q andsXs (u) := E(Xs |U = u) for s = 1, . . . , p, can be estimated by a local linear (LL) regression smoother using n iid observations, a univariate nonnegative symmetric kernel K(·),
and a bandwidth h1 . Denote their local linear estimates by bsYk (·) and bsXs (·). Then, theparameter of interest β0 can be estimated by the method of maximum likelihood basedon a kernel density estimate asnn
oX>
b = arg maxln fbnh Ỹ1i − X̃>
β,.
.
.
,Ỹ
−X̃
β,β
qiii
βi=1

where Ỹki := Yki − bsYk (Ui ), X̃si := Xsi − bsXs (Ui ), X̃i = X̃1i , . . . , X̃pi )> , and!
qn Y>
X
1Ỹ
−X̃
β−tkikiK+ cnfbnh (t1 , . . . , tq ) :=nhq i=1hk=1

is the joint density estimator of f (1 , . . . , q ) using the product kernel based on K(·), abandwidth h 6= h1 and a positive sequence cn → 0 as n → ∞.
In practice, βb can be computed via an iterative algorithm. The details for implementing the estimation procedure are as follows:√(1) The Gaussian kernel K(t) = exp(−t2 /2)/ 2π.
(2) The bandwidth h1 = σbU n−1/3 for LL estimation, where σbU is the sample standard
deviation of U .
(3) The bandwidth!1/qqY−1/(q+6)
h = (log n/n)σbbkk=1

for the joint density estimation, where σbbk is the square root of 10% trimmed

variance of bk = Ỹki − X̃> βb∗ and βb∗ is the estimate obtained from the previousiteration.
(4) The sequence cn = n−1000 .
√It is demonstrated that βb is n-consistent and asymptotically normal. A testingprocedure for the linear hypothesis of β0 is also proposed. Simulation results indicatethat the estimator performs well when the joint density f (1 , . . . , q ) is multimodal,
Masayuki Hirukawaasymmetric, or heavy-tailed.
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