# Adaptive lasso and Dantzig selector for spatial point processes intensity
estimation

Achmad<EMAIL_ADDRESS>[ Jean-François
Coeurjollylabel=e2 [<EMAIL_ADDRESS>Frédérique<EMAIL_ADDRESS>[ Department
of Statistics, Institut Teknologi Sepuluh Nopember (ITS), Indonesia,
Laboratoire Jean Kuntzmann, Université Grenoble Alpes, France,

###### Abstract

Lasso and Dantzig selector are standard procedures able to perform variable
selection and estimation simultaneously. This paper is concerned with
extending these procedures to spatial point process intensity estimation. We
propose adaptive versions of these procedures, develop efficient computational
methodologies and derive asymptotic results for a large class of spatial point
processes under an original setting where the number of parameters, i.e. the
number of spatial covariates considered, increases with the expected number of
data points. Both procedures are compared theoretically, in a simulation
study, and in a real data example.

estimating equations,

high-dimensional statistics,

linear programming,

regularization methods,

spatial point pattern,

###### keywords:

, and

##  1 Introduction

Spatial point processes are stochastic processes which model random locations
of points in space, such as random locations of trees in a forest, locations
of disease cases, earthquake occurrences and crime events [e.g. 2, 8, 13, 23].
To understand the arrangement of points, the intensity function is the
standard summary function [2, 13]. When one seeks to describe the probability
of observing a point at location $u\in\mathbb{R}^{d}$ in terms of covariates,
the most popular model for the intensity function, $\rho$, is

$\displaystyle\rho(u;\bm{\beta})=\exp\\{\bm{\beta}^{\top}\mathbf{z}(u)\\},\quad
u\in D\subset\mathbb{R}^{d},$ (1)

where, for $p\geq 1$, $\mathbf{z}(u)=\\{z_{1}(u),\ldots,z_{p}(u)\\}^{\top}$,
represent spatial covariates measured at location $u$ and
$\bm{\beta}=\\{\beta_{1},\ldots,\beta_{p}\\}^{\top}$ is a real $p$-dimensional
parameter.

The score of the Poisson likelihood, i.e. the likelihood if the underlying
process is assumed to be the Poisson process, remains an unbiased estimating
equation to estimate $\bm{\beta}$ even if the point pattern does not arise
from a Poisson process [25]. Such a method is well-studied in the literature
and extended in several ways to gain efficiency [e.g. 18, 19] when the number
of covariates is moderate. Standard results cover the consistency and
asymptotic normality of the maximum Poisson likelihood under the increasing
domain asymptotic (see e.g. [18] and references therein).

When a large number of covariates is available, variable selection is
unavoidable. Performing estimation and selection for spatial point processes
intensity models has received a lot of attention. Recent developments consider
techniques centered around regularization methods [e.g. 10, 14, 24, 27] such
as lasso technique. In particular, [10] consider several composite likelihoods
penalized by a large class of convex and non-convex penalty functions and
obtain asymptotic results under the increasing domain asymptotic framework.

The Dantzig selector is an alternative procedure to regularization techniques.
It was initially proposed for linear models by [7] and subsequently extended
to more complex models [e.g. 1, 15, 21]. In particular, [15] generalizes this
approach to general estimating equations. One of the main advantages of the
Dantzig selector is its implementation which, for linear models, results in a
linear programming. Since then, the Dantzig selector and lasso procedures have
been compared in different contexts [e.g. 3, 22].

In this paper, we compare lasso and Dantzig selector when they are applied to
intensity estimation for spatial point processes. We compare these procedures
in the complex asymptotic framework where the number of informative
covariates, say $s_{n}$ and the number of non-informative covariates, say
$p_{n}-s_{n}$, may increase with the mean number of points. Our asymptotic
results are developed under a setting which embraces both increasing domain
asymptotic and infill asymptotic which are often considered in the literature
(see also Section 2). Such a setting is almost never considered in the spatial
point processes literature (see again e.g. [18]).

It is well-known that the Poisson likelihood can be approximated as a quasi-
Poisson regression model, see e.g. [2]. For the adaptive lasso procedure, our
theoretical contributions can also be seen as extension of work such as [16]
which provides asymptotic results for estimators from regularized generalized
linear models. However, in our spatial framework, the more standard sample
size must be substituted by, here, a mean number of points. Furthermore,
observations are no more independent, i.e. our results are valid for a large
class of spatial dependent point processes. Note also that [16] assumes
$s_{n}=s$.

The theoretical contributions of the present paper are two-fold. First, for
the adaptive lasso procedure, the question stems from extending the work by
[10] which considers only an increasing domain asymptotic, and assumes
$s_{n}=s$ and $p_{n}=p$. For the Dantzig selector, our contributions are to
extend the standard methodology to spatial point processes, propose an
adaptive version and derive theoretical results. This yields different
computational and theoretical issues. As revealed by our main result, Theorem
4.1, the adaptive lasso and Dantzig selector procedures share several
similarities but also some slight differences. We first prove that both
procedures satisfy an oracle property, i.e. methods can correctly select the
nonzero coefficients with probability converging to one, and second that the
estimators of the nonzero coefficients are asymptotically normal. However, the
conditions under which results are valid for the Dantzig selector are slightly
more restrictive.

Our conducted simulation study and application to environmental data also
demonstrate that both procedures behave similarly.

##  2 Background and framework

Let $\mathbf{X}$ be a spatial point process on $\mathbb{R}^{d}$, $d\geq 1$. We
view $\mathbf{X}$ as a locally finite random subset of $\mathbb{R}^{d}$. Let
$D\subset\mathbb{R}^{d}$ be a compact set of Lebesgue measure $|D|$ which will
play the role of the observation domain. A realization of $\mathbf{X}$ in $D$
is thus a set $\mathbf{x}=\\{x_{1},\ldots,x_{m}\\}$, where $x\in D$ and $m$ is
the observed number of points in $D$. Suppose $\mathbf{X}$ has intensity
function $\rho$ and second-order product density $\rho^{(2)}$. Campbell
theorem states that, for any function $k:\mathbb{R}^{d}\to[0,\infty)$ or
$k:\mathbb{R}^{d}\times\mathbb{R}^{d}\to[0,\infty)$

$\displaystyle\mathbb{E}\sum_{u\in\mathbf{X}}k(u)={\int_{\mathbb{R}^{d}}k(u)\rho(u)\mathrm{d}u},\quad\mathbb{E}\sum_{u,v\in\mathbf{X}}^{\neq}k(u,v)=\int_{\mathbb{R}^{d}\times\mathbb{R}^{d}}k(u,v)\rho^{(2)}(u,v)\mathrm{d}u\mathrm{d}v.$
(2)

Based on the first two intensity functions, the pair correlation function $g$
is defined by

$\displaystyle g(u,v)=\frac{\rho^{(2)}(u,v)}{\rho(u)\rho(v)},\quad u,v\in D,$

when both $\rho$ and $\rho^{(2)}$ exist with the convention $0/0=0$. The pair
correlation function is a measure of departure of the model from the Poisson
point process for which $g=1$. For further background materials on spatial
point processes, see for example [2, 23].

For our asymptotic considerations, we assume that a sequence
$(\mathbf{X})_{n\geq 1}$ is observed within a sequence of bounded domains
$(D_{n})_{n\geq 1}$. We denote by $\rho_{n}$ and $g_{n}$ the intensity and
pair correlation of $\mathbf{X}_{n}$. With an abuse of notation, we denote by
$\mathbb{E}$ and $\mathrm{Var}$, the expectation and variance under
$\mathbf{X}_{n}$. we assume that the intensity $\rho_{n}$ writes
$\rho_{n}(u)=\exp\\{\bm{\beta}_{0}^{\top}\mathbf{z}(u)\\}$, $u\in D_{n}$. We
thus let $\bm{\beta}_{0}$ denote the true parameter vector and assume it can
be decomposed as
$\bm{\beta}_{0}=\\{\beta_{01},\ldots,\beta_{0s_{n}},\beta_{0(s_{n}+1)},\ldots,\beta_{0p_{n}}\\}^{\top}=(\bm{\beta}^{\top}_{01},\bm{\beta}^{\top}_{02})^{\top}=(\bm{\beta}_{01}^{\top},\mathbf{0}^{\top})^{\top}$.
Therefore, $\bm{\beta}_{01}\in\mathbb{R}^{s_{n}}$,
$\bm{\beta}_{02}=\mathbf{0}\in\mathbb{R}^{p_{n}-s_{n}}$ and
$\bm{\beta}_{0}\in\mathbb{R}^{p_{n}}$, where $s_{n}$ is the number of non-zero
coefficients, $p_{n}-s_{n}$ the number of zero coefficients and $p_{n}$ the
total number of parameters. We underline that it is unknown to us which
coefficients are non-zero and which are zero. Thus, we consider a sparse
intensity model where in particular $s_{n}$ and $p_{n}$ may diverge to
infinity as $n$ grows.

For any $\bm{\beta}\in\mathbb{R}^{p_{n}}$ or for the spatial covariates
$\mathbf{z}(u)$, we use a similar notation, i.e.
$\bm{\beta}=(\bm{\beta}_{1}^{\top},\bm{\beta}_{2}^{\top})^{\top}$ and
$\mathbf{z}(u)=\\{\mathbf{z}_{1}(u)^{\top},\mathbf{z}_{2}(u)^{\top}\\}^{\top}$,
$u\in D_{n}$. We let $\mu_{n}=\mathbb{E}\\{N(D_{n})\\}$, that is the expected
number of points in $D_{n}$. By Campbell theorem, we have

$\mu_{n}=\int_{D_{n}}\rho_{n}(u,\bm{\beta}_{0})\mathrm{d}u=\int_{D_{n}}\exp\\{\bm{\beta}_{0}^{\top}\mathbf{z}(u)\\}\mathrm{d}u=\int_{D_{n}}\exp\\{\bm{\beta}_{01}^{\top}\mathbf{z}_{1}(u)\\}\mathrm{d}u$

Note that $\mu_{n}$ is a function of
$D_{n},\bm{\beta}_{01},\mathbf{z}_{1}(u),s_{n}$. In this paper, we assume that
$\mu_{n}\to\infty$ as $n\to\infty$. That kind of assumption is very general
and embraces the well-known frameworks called increasing domain asymptotics
and infill asymptotics. For the increasing domain context,
$D_{n}\to\mathbb{R}^{d}$ and usually $\bm{\beta}_{01}$ depends on $n$ only
through $s_{n}$. For the infill asymptotics, $D_{n}=D$ is assumed to be a
bounded domain of $\mathbb{R}^{d}$ and usually $z_{1}(u)=1$,
$\bm{\beta}_{01}=\theta_{n}\to\infty$ as $n\to\infty$. In some sense, the
parameter $\mu_{n}$ plays the role of the sample size in standard inference.

To reduce notation in the following, unless it is ambiguous, we do not index
$\mathbf{X}$, $\rho$, $g$, $\bm{\beta}_{0}$, $\bm{\beta}$, $\mathbf{z}(u)$
with $n$.

##  3 Methodologies

###  3.1 Standard methodology

If $\mathbf{X}$ is a Poisson point process, then, on $D_{n}$, $\mathbf{X}$
admits a density with respect to the unit rate Poisson point process [23].
This yields the log-likelihood function for $\bm{\beta}$, which, for the
intensity model (1), is proportional to

$\ell_{n}(\bm{\beta})=\sum_{u\in\mathbf{X}\cap
D_{n}}\bm{\beta}^{\top}\mathbf{z}(u)-\int_{D_{n}}\rho(u;\bm{\beta})\mathrm{d}u.$
(3)

The gradient of (3) writes

$\mathbf{U}_{n}(\bm{\beta})=\frac{\mathrm{d}}{\mathrm{d}\bm{\beta}}\ell_{n}(\bm{\beta})={\sum_{u\in\mathbf{X}\cap
D_{n}}\mathbf{z}(u)}-{\int_{D_{n}}\mathbf{z}(u)\rho(u;\bm{\beta})\mathrm{d}u}.$
(4)

If $\mathbf{X}$ is not a Poisson point process, Campbell Theorem shows that
(4) remains an unbiased estimating equation. Hence, the maximum of (3), still
makes sense for non-Poisson models. Such an estimator, which can be viewed as
composite likelihood has received a lot of attention in the literature and
asymptotic properties are well-established when $p_{n}=p$ and $p$ is moderate
[e.g 18, 19, 25].

We end this section with the definition of the two following $p_{n}\times
p_{n}$ matrices

$\displaystyle\mathbf{A}_{n}(\bm{\beta})$
$\displaystyle={\int_{D_{n}}\mathbf{z}(u)\mathbf{z}(u)^{\top}\rho(u;\bm{\beta})\mathrm{d}u}$
(5) $\displaystyle\mathbf{B}_{n}(\bm{\beta})$
$\displaystyle=\mathbf{A}_{n}(\bm{\beta})+{\int_{D_{n}}\int_{D_{n}}\mathbf{z}(u)\mathbf{z}(v)^{\top}\\{g(u,v)-1\\}\rho(u;\bm{\beta})\rho(v;\bm{\beta})\mathrm{d}u\mathrm{d}v}.$
(6)

The matrix $\mathbf{A}_{n}(\bm{\beta})$ corresponds to the sensitivity matrix
defined by
$\mathbf{A}_{n}(\bm{\beta})=-\mathbb{E}\\{\mathrm{d}\mathbf{U}_{n}(\bm{\beta})/\mathrm{d}\bm{\beta}^{\top}\\}$
while $\mathbf{B}_{n}(\bm{\beta})$ corresponds to the variance of the
estimating equation, i.e.
$\mathbf{B}_{n}(\bm{\beta})=\mathrm{Var}\\{\mathbf{U}_{n}(\bm{\beta})\\}$. By
passing, we point out that
$\mathbf{A}_{n}(\bm{\beta})=-\mathrm{d}\mathbf{U}_{n}(\bm{\beta})/\mathrm{d}\bm{\beta}^{\top}$.
Let $\mathbf{M}_{n}$ be some $p_{n}\times p_{n}$ matrix, e.g.
$\mathbf{A}_{n}(\bm{\beta})$ or $\mathbf{B}_{n}(\bm{\beta})$. Such a matrix is
decomposed as

$\displaystyle\mathbf{M}_{n}=\begin{bmatrix}\mathbf{M}_{n,1}\\\
\mathbf{M}_{n,2}\end{bmatrix}=\begin{bmatrix}\mathbf{M}_{n,11}&\mathbf{M}_{n,12}\\\
\mathbf{M}_{n,21}&\mathbf{M}_{n,22}\end{bmatrix},$ (7)

where $\mathbf{M}_{n,1}$ (resp. $\mathbf{M}_{n,2}$) is the first $s_{n}\times
p_{n}$ (resp. the following $(p_{n}-s_{n})\times p_{n}$) components of
$\mathbf{M}_{n}$ and $\mathbf{M}_{n,11}$ (resp. $\mathbf{M}_{n,12}$,
$\mathbf{M}_{n,21}$, and $\mathbf{M}_{n,22}$) is the $s_{n}\times s_{n}$ top-
left corner (resp. the $s_{n}\times(p_{n}-s_{n})$ top-right corner, the
$(p_{n}-s_{n})\times s_{n}$ bottom-left corner, and the
$(p_{n}-s_{n})\times(p_{n}-s_{n})$ bottom-right corner) of $\mathbf{M}_{n}$.
In what follows, for a squared symmetric matrix $\mathbf{M}_{n}$,
$\nu_{\min}(\mathbf{M}_{n})$ and $\nu_{\max}(\mathbf{M}_{n})$ denote
respectively the smallest and largest eigenvalue of $\mathbf{M}_{n}$. Finally,
$\|\mathbf{y}\|$ denotes the Euclidean norm of a vector $\mathbf{y}$, while
$\|\mathbf{M}_{n}\|=\sup_{\|\mathbf{y}\|\neq
0}\|\mathbf{M}_{n}\mathbf{y}\|/\|\mathbf{y}\|$ denotes the spectral norm. We
remind that the spectral norm is subordinate and that for a symmetric definite
positive matrix $\|\mathbf{M}_{n}\|=\nu_{\max}(\mathbf{M}_{n})$.

###  3.2 Adaptive lasso (AL)

When the number of parameters is large, regularization methods allow one to
perform both estimation and variable selection simultaneously. When $p_{n}=p$,
[10] consider several regularization procedures which consist in adding a
convex or non-convex penalty term to (3). The proposed methods are unchanged
even when the number of covariates diverges. In particular, the adaptive lasso
consists in maximizing

$\displaystyle
Q_{n}(\bm{\beta})=\frac{1}{\mu_{n}}\ell_{n}(\bm{\beta})-{\sum_{j=1}^{p_{n}}\lambda_{n,j}|\beta_{j}|},$
(8)

where the real numbers $\lambda_{n,j}$ are non-negative tuning parameters. We
therefore define the adaptive lasso estimator as

$\displaystyle\hat{\bm{\beta}}_{\mathrm{AL}}=\arg\max_{\bm{\beta}\in\mathbb{R}^{p_{n}}}Q_{n}(\bm{\beta}).$
(9)

When $\lambda_{n,j}=0$ for $j=1,\dots,p_{n}$, the method reduces to the
maximum composite likelihood estimator and when $\lambda_{n,j}=\lambda_{n}$ to
the standard lasso estimator. If $\beta_{1}$ acts as an intercept, meaning
that $z_{1}(u)=1,\forall u\in D_{n}$, it is often desired to let this
parameter free. This can be done by setting $\lambda_{n,1}=0$ in the second
term of (8). Finally, the choice of $\mu_{n}$ as a normalization factor in (8)
follows the implementation of the adaptive lasso procedure for generalized
linear models in the standard software (e.g. R package glmnet [17]).

###  3.3 Adaptive (linearized) Dantzig selector (ALDS)

When applied to a likelihood [7, 21], the Dantzig selector estimate is
obtained by minimizing $\|\bm{\beta}\|_{1}$ subject to the infinite norm of
the score function bounded by some threshold parameter $\lambda$. In the
spatial point process setting, we propose an adaptive version of the Dantzig
selector estimate as the solution of the problem

$\displaystyle\min\|{\bm{\Lambda}}_{n}\bm{\beta}\|_{1}\mbox{ subject to
}|(\mathbf{U}_{n}(\bm{\beta}))_{j}|\leq\lambda_{n,j}\quad\text{ for
}j=1,\dots,p_{n}$ (10)

where
${\bm{\Lambda}}_{n}=\mathrm{diag}(\lambda_{n,1},\cdots,\lambda_{n,p_{n}})$ and
where $\mathbf{U}_{n}(\bm{\beta})$ is the estimating equation given by (4). It
is worth pointing out $\lambda_{n,j}=0$ for $j=1,\dots,p_{n}$ reduces the
criterion (10) to $\mathbf{U}_{n}(\bm{\beta})=0$, which leads to the maximum
composite likelihood estimator. Similarly to the adaptive lasso procedure, the
intercept can be let free by setting $\lambda_{n,1}=0$. However, in the
following, we assume, for the ALDS procedure, that $\lambda_{n,j}>0$ for
convenience, in order to rewrite (10) in the following matrix form

$\displaystyle\min\|{\bm{\Lambda}}_{n}\bm{\beta}\|_{1}\mbox{ subject to
}(\mu_{n})^{-1}\Big{\|}{\bm{\Lambda}}_{n}^{-1}\mathbf{U}_{n}({\bm{\beta}})\Big{\|}_{\infty}\leq
1.$ (11)

We claim that the whole methodology and the proofs could be redone without
involving the notation ${\bm{\Lambda}}_{n}^{-1}$ and so that Theorem 4.1
remains valid if one does not regularize the intercept term for example.

Due to the nonlinearity of the constraint vector, standard linear programming
can no more be used to solve (11). This results in a non-convex optimization
problem. In particular the feasible set
$\\{\bm{\beta}:\|{\bm{\Lambda}}_{n}^{-1}\mathbf{U}_{n}(\bm{\beta})\|_{\infty}\leq
1\\}$ is non-convex, which makes the method difficult to implement and to
analyze from a theoretical point of view. In the context of generalized linear
models, [21] consider the iterative reweighted least squares method and define
an iterative procedure where at each step of the algorithm the constraint
vector corresponds to a linearization of the updated pseudo-score. Such a
procedure is not straightforward to extend from (11) and remains complex to
analyze from a theoretical point of view. As an alternative, we follow [15,
Chapter 3] and propose to linearize the constraint vector by expanding
$\mathbf{U}_{n}(\bm{\beta})$ around $\tilde{\bm{\beta}}$, an initial estimate
of $\bm{\beta}_{0}$, using a first order Taylor approximation; i.e. we
substitute $\mathbf{U}_{n}(\bm{\beta})$ by
$\mathbf{U}_{n}(\tilde{\bm{\beta}})+\mathbf{A}_{n}(\tilde{\bm{\beta}})(\tilde{\bm{\beta}}-\bm{\beta})$.
Such a linearization enables now the use of standard linear programming. We
term adaptive linearized Dantzig selector (ALDS) estimate and denote it by
$\hat{\bm{\beta}}_{\mathrm{ALDS}}$ the solution to the optimization problem

$\displaystyle\min\|{\bm{\Lambda}}_{n}\bm{\beta}\|_{1}\mbox{ subject to
}(\mu_{n})^{-1}\;\Big{\|}{\bm{\Lambda}}_{n}^{-1}\big{\\{}\mathbf{U}_{n}(\tilde{\bm{\beta}})+\mathbf{A}_{n}(\tilde{\bm{\beta}})(\tilde{\bm{\beta}}-{\bm{\beta}})\big{\\}}\Big{\|}_{\infty}\leq
1.$ (12)

Properties of $\hat{\bm{\beta}}_{\mathrm{ALDS}}$ depend on properties of
$\tilde{\bm{\beta}}$ which are made precise in the next section.

##  4 Asymptotic results

Our main result relies upon the following conditions:

  1. ($\mathcal{C}$.1)

The intensity function has the log-linear specification given by (1) where
$\bm{\beta}\in\mathbb{R}^{p_{n}}$.

  2. ($\mathcal{C}$.2)

$(\mu_{n})_{n\geq 1}$ is an increasing sequence of real numbers, such that
$\mu_{n}\to\infty$ as ${n}\to\infty$.

  3. ($\mathcal{C}$.3)

The covariates $\mathbf{z}$ satisfy

$\sup_{n\geq
1}\;\sup_{i=1,\dots,p_{n}}\;\sup_{u\in\mathbb{R}^{d}}|z_{i}(u)|<\infty\qquad\mbox{
and }\qquad\inf_{n\geq
1}\inf_{\bm{\phi}\in\mathbb{R}^{p_{n}},\|\bm{\phi}\|=1}\inf_{u\in
D_{n}}\\{\bm{\phi}^{\top}\mathbf{z}(u)\\}^{2}>0$

  4. ($\mathcal{C}$.4)

The intensity and pair correlation satisfy

$\int_{D_{n}}\int_{D_{n}}\rho(u;\bm{\beta}_{0})\rho(v;\bm{\beta}_{0})|g(u,v)-1|\mathrm{d}u\mathrm{d}v=O(\mu_{n}).$

  5. ($\mathcal{C}$.5)

The matrix $B_{n,11}(\bm{\beta}_{0})$ satisfies

$\liminf_{n}\inf_{\bm{\phi}\in\mathbb{R}^{s_{n}},\|\bm{\phi}\|=1}\bm{\phi}^{\top}\big{\\{}(\mu_{n})^{-1}\mathbf{B}_{n,11}(\bm{\beta}_{0})\big{\\}}\bm{\phi}>0.$

  6. ($\mathcal{C}$.6)

For any $\bm{\phi}\in\mathbb{R}^{s_{n}}\setminus\\{0\\}$, the following
convergence holds in distribution as $n\to\infty$:

$\sigma_{\bm{\phi}}^{-1}\bm{\phi}^{\top}\mathbf{U}_{n,1}(\bm{\beta}_{0})\stackrel{{\scriptstyle
d}}{{\to}}N(0,1)$

where
$\sigma^{2}_{\bm{\phi}}=\bm{\phi}^{\top}\mathbf{B}_{n,11}(\bm{\beta}_{0})\bm{\phi}$.

  7. ($\mathcal{C}$.7)

The initial estimate $\tilde{\bm{\beta}}$ satisfies
$\|\tilde{\bm{\beta}}-{\bm{\beta}_{0}}\|=O_{\mathrm{P}}(\sqrt{p_{n}/{\mu_{n}}})$
and is such that
$\|\mathbf{A}_{n,11}(\tilde{\bm{\beta}})^{-1}\|=O_{\mathrm{P}}(\mu_{n}^{-1})$.

  8. ($\mathcal{C}$.8)

As $n\to\infty$, we assume that $s_{n},p_{n}$ and $\mu_{n}$ are such that as
$n\to\infty$

$\left\\{\begin{array}[]{ll}\max\left(\frac{p_{n}^{4}}{\mu_{n}},\frac{s_{n}^{2}p_{n}^{3}}{\mu_{n}}\right)\to
0&\quad\text{ for the AL estimate}\\\ \frac{s_{n}^{3}p_{n}^{4}}{\mu_{n}}\to
0&\quad\text{ for the ALDS estimate.}\end{array}\right.$

  9. ($\mathcal{C}$.9)

Let $a_{n}=\max_{j=1,\ldots,{s_{n}}}\lambda_{n,j}$ and
$b_{n}=\min_{j={s_{n}}+1,\ldots,p_{n}}\lambda_{n,j}$. We assume that these
sequences are such that, as $n\to\infty$

$\left\\{\begin{array}[]{lll}a_{n}\sqrt{s_{n}\mu_{n}}\to
0,&b_{n}\sqrt{\frac{\mu_{n}}{p_{n}^{2}}}\to\infty&\quad\text{ for the AL
estimate}\\\ a_{n}\sqrt{s_{n}^{3}\mu_{n}}\to
0,&b_{n}\sqrt{\frac{\mu_{n}}{p_{n}^{3}}}\to\infty&\quad\text{ for the ALDS
estimate.}\end{array}\right.$

Condition ($\mathcal{C}$.1) specifies the form of intensity models considered
in this paper. In particular, note that we do not assume that $\bm{\beta}$ is
an element of a bounded domain of $\mathbb{R}^{p_{n}}$. Condition
($\mathcal{C}$.2) specifies our asymptotic framework where we assume to
observe in average more and more points in $D_{n}$. As already mentioned, this
may cover increasing domain type or infill type asymptotics. To our knowledge,
only [11] consider a similar asymptotic framework in order to construct
information criteria for spatial point process intensity estimation. The
context is however very different here as we consider a large number of
covariates and we study methodologies (adaptive lasso or Dantzig) which are
able to produce a sparse estimate. Condition ($\mathcal{C}$.3) is quite
standard and is not too restrictive. Note that conditions
($\mathcal{C}$.1)-($\mathcal{C}$.3) allow us in Lemma A.2 to prove that in a
‘neighbordhood’ of $\bm{\beta}_{0}$, $\int\rho(u;\bm{\beta})=O(\mu_{n})$, a
useful result widely used in our proofs. The last part of Condition
($\mathcal{C}$.3) asserts that at any location the covariates are linearly
independent. Condition ($\mathcal{C}$.3) also implies first that
$\liminf_{n}\inf_{\bm{\phi},\|\bm{\phi}\|=1}\bm{\phi}^{\top}\big{\\{}(\mu_{n})^{-1}\mathbf{A}_{n,11}(\bm{\beta}_{0})\big{\\}}\bm{\phi}>0$
and second that
$\liminf_{n}\inf_{\bm{\phi},\|\bm{\phi}\|=1}\bm{\phi}^{\top}\big{\\{}(\mu_{n})^{-1}\mathbf{A}_{n}(\bm{\beta}_{0})\big{\\}}\bm{\phi}>0$.
Condition ($\mathcal{C}$.5) is a similar assumption but for the submatrix
$\mathbf{B}_{n,11}(\bm{\beta}_{0})$ which corresponds to
$\mathrm{Var}\\{\mathbf{U}_{n,1}(\bm{\beta}_{0})\\}$. Condition
($\mathcal{C}$.4) is also natural. Combined with Condition ($\mathcal{C}$.3),
this implies that
$\mathrm{\mathbf{missing}}B_{n}(\bm{\beta}_{0})=O(\mu_{n}p_{n})$. When
$p_{n}=p$ (and therefore $s_{n}=s$) and in the increasing domain framework,
such an assumption can be satisfied by a large class of spatial point
processes such as determinantal point processes, log-Gaussian Cox processes
and Neyman-Scott point processes [see 10]. When $p_{n}=p$ and in the infill
asymptotic framework, these assumptions are also valid for many spatial point
processes, as discussed by [11]. Condition ($\mathcal{C}$.6) is required to
derive the asymptotic normality of $\hat{\bm{\beta}}_{01}$. Under a specific
framework, such a result has already been obtained for a large class of
spatial point processes: by [4, 26] under the increasing domain framework and
$p_{n}=p$ and [9] when $p_{n}\to\infty$; by [11] in the infill/increasing
domain asymptotics frameworks and $p_{n}=p$.

Condition ($\mathcal{C}$.7) is very specific to the ALDS estimate which
requires a preliminary estimate of $\bm{\beta}$. That condition is not
unrealistic as a simple choice for $\tilde{\bm{\beta}}$ could be the maximum
of the composite likelihood function (3), see the remark after Theorem 4.1. Of
course, we do not require that $\tilde{\bm{\beta}}$ produces a sparse
estimate.

Condition ($\mathcal{C}$.8) reflects the restriction on the number of
covariates that can be considered in this study. For the AL estimate, this
assumption is very similar to the one required by [16] when $\mu_{n}$ is
replaced by $n$ and where the number of non-zero coefficients $s_{n}$ is
constant.

Condition ($\mathcal{C}$.9) contains the main ingredients to derive sparsity
properties, consistency and asymptotic normality. We first note that if
$\lambda_{n,j}=\lambda_{n}$, then $a_{n}=b_{n}=\lambda_{n}$, whereby it is
easily deduced that the two conditions on $a_{n}$ and $b_{n}$ cannot be
satisfied simultaneously even if $p_{n}=p$. This justifies the introduction of
an adaptive version of the Dantzig selector and motivates the use of the
adaptive lasso. The condition $a_{n}\sqrt{s_{n}\mu_{n}}\to 0$ for the adaptive
lasso is similar to the one imposed by [16] when $\mu_{n}$ is replaced by $n$
and $s_{n}=s$ in their context. However, we require a slightly stronger
condition on $b_{n}$ than the one required by [16]. In our setting, their
assumption would be written as $b_{n}\sqrt{\mu_{n}/p_{n}}\to\infty$. However,
we would have to assume that
$\nu_{\max}\big{(}\mathbf{A}_{n}(\bm{\beta}_{0})\big{)}=O(\mu_{n})$. Such a
condition is not straightforwardly satisfied in our setting since, for
instance, the conditions ($\mathcal{C}$.2)-($\mathcal{C}$.4) only imply that
$\nu_{\max}\big{(}\mathbf{A}_{n}(\bm{\beta}_{0})\big{)}=O({p_{n}\mu_{n}})$.

As already mentioned, we do not assume that $\tilde{\bm{\beta}}$ satisfies any
sparsity property. We believe this is the main reason why conditions
($\mathcal{C}$.8))-($\mathcal{C}$.9) contain slightly stronger assumptions for
the ALDS estimate than for the AL estimate. We now present our main result,
whose proof is provided in Appendices B-C.

######  Theorem 4.1.

Let $\hat{\bm{\beta}}$ denote either $\hat{\bm{\beta}}_{\mathrm{AL}}$ or
$\hat{\bm{\beta}}_{\mathrm{ALDS}}$. Assume that the conditions
($\mathcal{C}$.1)-($\mathcal{C}$.9) hold, then the following properties hold.

  1. (i)

$\hat{\bm{\beta}}$ exists. Moreover $\hat{\bm{\beta}}_{\mathrm{AL}}$
satisfies,
${\displaystyle\|\hat{\bm{\beta}}_{\mathrm{AL}}-\bm{\beta}_{0}\|=O_{\mathrm{P}}\left(\sqrt{\frac{p_{n}}{\mu_{n}}}\right)}$.

  2. (ii)

Sparsity: $\mathrm{P}(\hat{\bm{\beta}}_{2}=0)\to 1$ as $n\to\infty$.

  3. (iii)

Asymptotic Normality: for any $\bm{\phi}\in\mathbb{R}^{s_{n}}\setminus\\{0\\}$
such that $\|\bm{\phi}\|<\infty$

$\sigma_{\bm{\phi}}^{-1}\,\bm{\phi}^{\top}\mathbf{A}_{n,11}(\bm{\beta}_{0})(\hat{\bm{\beta}}_{1}-\bm{\beta}_{01})\xrightarrow{d}\mathcal{N}(0,1)$

in distribution, where
$\sigma^{2}_{\bm{\phi}}=\bm{\phi}^{\top}\mathbf{B}_{n,11}(\bm{\beta}_{0})\bm{\phi}$.

To derive the consistency of $\hat{\bm{\beta}}_{\mathrm{AL}}$, a careful look
at the proof in Appendix B shows that the condition
$a_{n}\sqrt{{\mu_{n}s_{n}}/p_{n}}\to 0$ could be sufficient. The convergence
rate, i.e. $O_{\mathrm{P}}(\sqrt{p_{n}/\mu_{n}})$, is $\sqrt{p_{n}}$ times the
convergence rate of the estimator obtained when $p_{n}$ is constant [see 10,
Theorem 1]. It also corresponds to the rate of convergence obtained by [16]
for generalized linear models when $p_{n}\to\infty$ and when $\mu_{n}$
corresponds to the standard sample size. It is worth pointing out that a
possible diverging number of non-zero coefficients $s_{n}$ does not affect the
rate of convergence. It does however impose a more restrictive condition on
$a_{n}$. Still on Theorem 4.1 (i), its proof shows that this result remains
valid when $\lambda_{n,j}=0$ for $j=1,\dots,p_{n}$. In other words, the
maximum composite likelihood estimator is consistent with the same rate of
convergence. Hence a simple choice for the initial estimate
$\widetilde{\bm{\beta}}$ defining the ALDS estimate is the maximum of the
Poisson likelihood given by (3).

Theorem 4.1 (iii) would be the result one would obtain if $p_{n}-s_{n}=0$.
Therefore, the efficiency of $\hat{\bm{\beta}}_{\mathrm{AL},1}$ and
$\hat{\bm{\beta}}_{\mathrm{ALDS},1}$ is the same as the estimator of
$\bm{\beta}_{01}$ obtained by maximizing (3) based on the sub model knowing
that $\bm{\beta}_{02}=\mathbf{0}$. In other words, when $n$ is sufficiently
large, both estimators are as efficient as the oracle one.

We end this section with the following remark. Despite the asymptotic
properties for $\hat{\bm{\beta}}_{\mathrm{AL}}$ and
$\hat{\bm{\beta}}_{\mathrm{ALDS}}$ and the conditions under which they are
valid, are (almost) identical, the proofs are completely different and rely
upon different tools. For $\hat{\bm{\beta}}_{\mathrm{AL}}$, our contribution
is to extend the proof by [10] where only the increasing domain framework was
considered, i.e. $\mu_{n}=O(|D_{n}|)$ and $D_{n}\to\mathbb{R}^{d}$ as
$n\to\infty$ and where $s_{n}=s$ and $p_{n}=p$. The results for
$\hat{\bm{\beta}}_{\mathrm{ALDS}}$ are the first ones available for spatial
point processes. To handle this estimator, we first have to study existence
and optimal solutions for the primal and dual problems.

##  5 Computational considerations

To lessen notation, we remove the index $n$ in different quantities such as
$\mathbf{U}_{n},\lambda_{n,j},\ell_{n}$, and $\mu_{n}$. For the practical
point of view, we define $\mu=N(D)$ the number of data points in $D$.

###  5.1 Berman-Turner approach

Before discussing how AL and ALDS estimates are obtained we first remind the
Berman-Turner approximation [2] used to derive the Poisson likelihood estimate
(3). The so-called Berman-Turner approximation consists in discretizing the
integral term in (3) as

$\displaystyle{\int_{D}\rho(u;\bm{\beta})\mathrm{d}u}\approx{\sum_{i=1}^{M}w(u_{i})\rho(u_{i};\bm{\beta})},$

where $u_{i},i=1,\ldots,M$ are points in $D$ consisting of the $m$ data points
and $M-m$ dummy points and where the quadrature weights $w(u_{i})>0$ are non-
negative real numbers such that ${\sum_{i}w(u_{i})}=|D|$. Using this specific
integral discretization, (3) is then approximated by

$\displaystyle\ell(\bm{\beta})\approx\tilde{\ell}(\bm{\beta})={\sum_{i=1}^{M}w_{i}\\{y_{i}\log\rho_{i}(\bm{\beta})-\rho_{i}(\bm{\beta})\\}},$
(13)

where $w_{i}=w(u_{i}),y_{i}=w_{i}^{-1}\mathbf{1}(u_{i}\in\mathbf{X}\cap D)$
and $\rho_{i}(\bm{\beta})=\rho(u_{i};\bm{\beta})$. Equation (13) is formally
equivalent to the weighted likelihood function of independent Poisson
variables $y_{i}$ with weights $w_{i}$. The method to approximate (4) and (5)
follows along similar lines which respectively results in

$\displaystyle\mathbf{U}(\bm{\beta})\approx\tilde{\mathbf{U}}(\bm{\beta})={\sum_{i=1}^{M}w_{i}\mathbf{z}_{i}\\{y_{i}-\rho_{i}(\bm{\beta})\\}},\quad\mathbf{A}(\bm{\beta})\approx\tilde{\mathbf{A}}(\bm{\beta})={\sum_{i=1}^{M}w_{i}\mathbf{z}_{i}\mathbf{z}_{i}^{\top}\rho_{i}(\bm{\beta})},$
(14)

where $\mathbf{z}_{i}=\mathbf{z}(u_{i})$. Thus, standard statistical software
for generalized linear models can be used to obtain the estimates. This fact
is implemented in the $\mathtt{spatstat}$ R package by ppm function with
option method="mpl" [2].

###  5.2 Adaptive lasso (AL)

First, given a current estimate $\check{\bm{\beta}}$, (13) is approximated
using second order Taylor approximation to apply iteratively reweighted least
squares,

$\displaystyle\tilde{\ell}(\bm{\beta})\approx\ell_{Q}(\bm{\beta})=-\frac{1}{2}{\sum_{i=1}^{M}\psi_{i}(y_{i}^{*}-\bm{\beta}^{\top}\mathbf{z}_{i})^{2}+C(\check{\bm{\beta}})},$
(15)

where $C(\check{\bm{\beta}})$ is a constant, $y_{i}^{*}$ and $\psi_{i}$ are
the new working response values and weights,
$y_{i}^{*}=\mathbf{z}_{i}^{\top}\check{\bm{\beta}}+\\{y_{i}-\exp(\check{\bm{\beta}}^{\top}\mathbf{z}_{i})\\}/\\{\exp(\check{\bm{\beta}}^{\top}\mathbf{z}_{i})\\},\;\psi_{i}=w_{i}\exp(\check{\bm{\beta}}^{\top}\mathbf{z}_{i}).$
Second, a penalized weighted least squares problem is obtained by adding
penalty term. Therefore, we solve

$\displaystyle{\displaystyle\min_{\bm{\beta}\in\mathbb{R}^{p}}\Omega(\bm{\beta})}={\displaystyle\min_{\bm{\beta}\in\mathbb{R}^{p}}\left\\{-\frac{1}{N(D)}\ell_{Q}(\bm{\beta})+\sum_{j=1}^{p}\lambda_{j}|\beta_{j}|\right\\}}$
(16)

using the coordinate descent algorithm [17]. The method consists in partially
minimizing (16) with respect to $\beta_{j}$ given $\check{\beta}_{l}$ for
$l\neq j$, $l,j=1,\ldots,p$, that is

$\displaystyle{\displaystyle\min_{\beta_{j}}\Omega(\check{\beta}_{1},\ldots,\check{\beta}_{j-1},\beta_{j},\check{\beta}_{j+1},\ldots,\check{\beta}_{p})}.$

With a few modifications (16) is solved using the $\mathtt{glmnet}$ R package
[17]. More detail about this implementation can be found in [10, Appendix C].

###  5.3 Adaptive (linearized) Dantzig selector (ALDS)

Given $\tilde{\bm{\beta}}$, (12) is a linear problem, simple to implement. The
main task is to compute the vectors $\mathbf{U}(\tilde{\bm{\beta}})$ and
$\mathbf{A}(\tilde{\bm{\beta}})(\tilde{\bm{\beta}}-{\bm{\beta}})$. Typically,
$\tilde{\bm{\beta}}$ is chosen from the maximum composite likelihood estimate.
Then, $\mathbf{U}(\tilde{\bm{\beta}})$ and
$\mathbf{A}(\tilde{\bm{\beta}})(\tilde{\bm{\beta}}-{\bm{\beta}})$ are
approximated by (14). This results in solving

$\displaystyle\min{\sum_{j=1}^{p}\lambda_{j}|\beta_{j}|}\mbox{ subject to
}{N(D)}^{-1}\Big{|}\tilde{\mathbf{U}}_{j}(\tilde{\bm{\beta}})+\big{\\{}\tilde{\mathbf{A}}(\tilde{\bm{\beta}})(\tilde{\bm{\beta}}-{\bm{\beta}})\big{\\}}_{j}\Big{|}\leq\lambda_{j},\quad\text{
for }j=1,\dots,p_{n},$

where $\tilde{\mathbf{U}}_{j}(\tilde{\bm{\beta}})$ and
$\\{\tilde{\mathbf{A}}(\tilde{\bm{\beta}})(\tilde{\bm{\beta}}-{\bm{\beta}})\\}_{j}$
are the $j$-th components of vectors $\tilde{\mathbf{U}}(\tilde{\bm{\beta}})$
and $\tilde{\mathbf{A}}(\tilde{\bm{\beta}})(\tilde{\bm{\beta}}-{\bm{\beta}})$.

###  5.4 Tuning parameter selection

Both AL and ALDS rely on the proper regularization parameters $\lambda_{j}$ to
avoid from unnecessary bias due to too small $\lambda_{j}$ selection and from
large variance due to too large $\lambda_{j}$ choice, so the selection of
$\lambda_{j}$ becomes an important task. To tune the $\lambda_{j}$, we follow
[10, 28] and define $\lambda_{j}=\lambda|\tilde{\beta}_{j}|^{-\nu}$, where
$\lambda\geq 0,\nu>0$ and $\tilde{\bm{\beta}}$ is the maximum composite
likelihood estimate. The weights $|\tilde{\beta}_{j}|^{-\nu}$ serve as a prior
knowledge to identify the non-zero coefficients since, for a constant
$\lambda$, large (resp. small) $\tilde{\beta}_{j}$ will force $\lambda_{j}$
close to zero (resp. infinity) [28],

Our theoretical results are not in line with this stochastic way of setting
the regularization parameters but we believe this choice is pertinent in our
context. Here is the intuition: let
$\lambda_{nj}=\lambda_{n}/|\tilde{\beta}_{j}|$. Using the
$\sqrt{\mu_{n}/p_{n}}$-consistency of $\tilde{\beta}_{j}$,
$\lambda_{nj}=O_{P}(\lambda_{n})$ for non-zero coefficients, while, for zero
coefficients, we may conjecture that
$\lambda_{nj}=O_{P}(\lambda_{n}\sqrt{\mu_{n}/p_{n}})$. Forgetting the $O_{P}$,
we may conjecture that $a_{n}\asymp\lambda_{n}$ and
$b_{n}\asymp\lambda_{n}\sqrt{\mu_{n}/p_{n}}$. Hence, considering the AL
procedure for instance, that would mean that we require $\lambda_{n}$ to be
such that $\lambda_{n}\sqrt{s_{n}\mu_{n}}\to 0$ and
$\lambda_{n}\mu_{n}/p_{n}^{3/2}\to\infty$, which constitutes a non empty
condition. For instance, assuming ($\mathcal{C}$.8), the sequence
$\lambda_{n}=(s_{n}\mu_{n})^{-\eta}$ satisfies ($\mathcal{C}$.9) as soon as
$1/2<\eta<5/9$, since as $n\to\infty$

$\lambda_{n}\sqrt{s_{n}\mu_{n}}=(s_{n}\mu_{n})^{1/2-\eta}\to 0\qquad\text{ and
}\qquad\frac{\lambda_{n}\mu_{n}}{p_{n}^{3/2}}=\left(\frac{\mu_{n}}{s_{n}^{2}p_{n}^{3}}\right)^{\eta/2}\left(\frac{\mu_{n}}{p_{n}^{4}}\right)^{1-3\eta/2}p_{n}^{5/2-9\eta/2}\to\infty.$

Considering rigorously the stochastic choice $\lambda_{n}/|\tilde{\beta}_{j}|$
for the regularization parameters is left for future research.

The remaining task is to specify $\lambda$. We follow here the literature,
mainly [11], and propose to select $\lambda$ as the minimum of the Bayesian
information criterion, BIC($\lambda$), for spatial point processes

$\displaystyle\mathrm{BIC}(\lambda)=-2\ell\\{\hat{\bm{\beta}}(\lambda)\\}+p_{*}\log
N(D),$

where $\ell\\{\hat{\bm{\beta}}(\lambda)\\}$ is the maximized composite
likelihood function, $p_{*}$ is the number of non-zero elements in
$\hat{\bm{\beta}}(\lambda)$ and $N(D)$ is the number of observed data points.

##  6 Numerical results

In Sections 6.1-6.2, we compare the AL and ALDS for intensity modeling of a
simulated and real data. In particular, the real data example comes from an
environmental data where 1146 locations of Acalypha diversifolia trees given
by Figure 1 are surveyed in a 50-hectare region ($D=1000m\times 500m$) of the
tropical forest of Barro Colorado Island (BCI) in central Panama [e.g. 20]. A
main question is how this tree species profits from environmental habitats
[12, 26] that could be related to the 15 environmental covariates depicted in
Figure 2 and their 79 interactions. With a total of 94 covariates, we perform
variable selection using the AL and ALDS to determine which covariates should
be included in the model. We center and scale the 94 covariates to sort the
important covariates according to the magnitudes of $\hat{\bm{\beta}}$. The
subset of such covariates are also considered to construct a realistic setting
for the simulation studies.

Figure 1: Plot of 1146 Acalypha diversifolia tree locations observed in the tropical forest of Barro Colorado Island |  |
---|---|---
|  |
|  |
|  |
|  |
Figure 2: Maps of covariates used in simulation study and in application. From
left to right: Elevation, slope, Aluminium (row 1), Boron, Calcium, Copper
(row 2), Iron, Potassium, Magnesium (row 3), Manganese, Phosporus, Zinc (row
4), and Nitrogen, Nitrigen mineralisation, pH (row 5).

###  6.1 Simulation study

The simulated point patterns are generated from Poisson and Thomas cluster
processes with intensity (1). To generate point patterns from Thomas process
with intensity (1) [e.g. 10], we first generate a parent point pattern from a
stationary Poisson point process $\mathbf{C}$ with intensity $\kappa=4\times
10^{-4}$. Given $\mathbf{C}$, offspring point patterns are generated from
inhomogeneous Poisson point process $\mathbf{X}_{c},c\in\mathbf{C}$ with
intensity

$\displaystyle\rho_{child}(u)=\exp\\{\bm{\beta}^{\top}\mathbf{z}(u)\\}k(u-c;\gamma)/\kappa,$

where $k(u-c;\gamma)=(2\pi\gamma^{2})^{-1}\exp(-\|u-c\|^{2}/(2\gamma^{2}))$.
The point process $\mathbf{X}=\cup_{c\in\mathbf{C}}\mathbf{X}_{c}$ is indeed
an inhomogeneous Thomas point process with intensity (1). We set $\gamma=5$
and $15$. The smaller the $\gamma$, the more clustered the point patterns,
leading to moderate clustering for $\gamma=15$ and highly clustering for
$\gamma=5$.

The covariates $\mathbf{z}(u)$ used for the simulation experiment are from the
BCI data. In addition to the 15 environmental factors depicted by Figure 2, we
add interaction between two covariates until we obtain the desired number of
covariates $p=21,41$ or $81$. For each $p$, we consider three mean numbers of
points ($\mu$) which increase when the observation domain expands. More
precisely, we set that $\mu_{1}=150$ (resp. $\mu_{2}=600,\mu_{3}=2400$) points
is generated in $D_{1}=[0,250]\times[0,125]$ (resp.
$D_{2}=[0,500]\times[0,250]$, $D_{3}=[0,1000]\times[0,500]$). When $D_{1}$ or
$D_{2}$ is considered, we simply rescale the covariates to fit the desired
observation domain. We fix $\beta_{2}=1$ and $\beta_{3}=-1$ while the rest are
set to zero, so $s=2$. The parameter $\beta_{1}$ acts as intercept and is
tuned to control the average number of points.

Table 1: True positive rate (TPR), false positive rate (FPR) in percentage, RMSE and average time in seconds obtained for AL and ALDS estimates based on 500 simulations from inhomogeneous Poisson point processes observed on different observation domains. | TPR | FPR | RMSE | Time
---|---|---|---|---
| AL | ALDS | AL | ALDS | AL | ALDS | AL | ALDS
$D_{1}$ ($\mu_{1}=150$) |  |  |  |  |  |  |  |
$p=20$ | 57 | 57 | 23 | 23 | 2.4 | 2.4 | 0.3 | 0.3
$p=40$ | 7 | 7 | 15 | 15 | 2.9 | 2.9 | 2.0 | 2.0
$p=80$ | 0 | 0 | 8 | 8 | 2.8 | 2.8 | 4.0 | 4.0
$D_{2}$ ($\mu_{2}=600$) |  |  |  |  |  |  |  |
$p=20$ | 100 | 100 | 3 | 4 | 0.3 | 0.3 | 0.3 | 0.3
$p=40$ | 97 | 96 | 4 | 5 | 0.5 | 0.5 | 0.8 | 0.6
$p=80$ | 86 | 86 | 8 | 8 | 0.9 | 0.9 | 3.0 | 3.0
$D_{3}$ ($\mu_{3}=2400$) |  |  |  |  |  |  |  |
$p=20$ | 100 | 100 | 0 | 0 | 0.1 | 0.1 | 0.4 | 0.3
$p=40$ | 100 | 100 | 0 | 0 | 0.1 | 0.1 | 0.9 | 0.9
$p=80$ | 100 | 100 | 0 | 0 | 0.1 | 0.1 | 3.0 | 3.0
Table 2: True positive rate (TPR), false positive rate (FPR) in percentage, RMSE and average time in seconds obtained for AL and ALDS estimates based on 500 simulations from inhomogeneous Thomas point processes with $\kappa=4\times 10^{-4}$ and $\gamma=15$ (moderate clustering) observed on different observation domains. | TPR | FPR | RMSE | Time
---|---|---|---|---
| AL | ALDS | AL | ALDS | AL | ALDS | AL | ALDS
$D_{1}$ ($\mu_{1}=150$) |  |  |  |  |  |  |  |
$p=20$ | 59 | 59 | 44 | 44 | 8.2 | 8.2 | 0.3 | 0.3
$p=40$ | 20 | 20 | 30 | 30 | 11.0 | 11.0 | 2.0 | 2.0
$p=80$ | 0 | 0 | 15 | 15 | 8.3 | 8.3 | 4.0 | 4.0
$D_{2}$ ($\mu_{2}=600$) |  |  |  |  |  |  |  |
$p=20$ | 91 | 89 | 53 | 47 | 2.6 | 2.2 | 0.3 | 0.3
$p=40$ | 88 | 86 | 48 | 43 | 4.9 | 3.8 | 1.0 | 0.6
$p=80$ | 80 | 80 | 35 | 35 | 7.7 | 7.7 | 5.0 | 5.0
$D_{3}$ ($\mu_{3}=2400$) |  |  |  |  |  |  |  |
$p=20$ | 100 | 100 | 59 | 43 | 1.0 | 0.8 | 0.9 | 1.0
$p=40$ | 100 | 100 | 56 | 39 | 1.7 | 1.1 | 2.0 | 3.0
$p=80$ | 100 | 100 | 52 | 52 | 3.2 | 3.2 | 8.0 | 8.0
Table 3: True positive rate (TPR), false positive rate (FPR) in percentage, RMSE and average time in seconds obtained for AL and ALDS estimates based on 500 simulations from inhomogeneous Thomas point processes with $\kappa=4\times 10^{-4}$ and $\gamma=5$ (high clustering) observed on different observation domains. | TPR | FPR | RMSE | Time
---|---|---|---|---
| AL | ALDS | AL | ALDS | AL | ALDS | AL | ALDS
$D_{1}$ ($\mu_{1}=150$) |  |  |  |  |  |  |  |
$p=20$ | 64 | 64 | 72 | 72 | 39.0 | 39.0 | 0.5 | 0.5
$p=40$ | 29 | 29 | 72 | 72 | 130.0 | 130.0 | 6.0 | 6.0
$p=80$ | 0 | 0 | 36 | 36 | 69.0 | 69.0 | 6.0 | 6.0
$D_{2}$ ($\mu_{2}=600$) |  |  |  |  |  |  |  |
$p=20$ | 91 | 90 | 66 | 61 | 4.1 | 3.6 | 0.4 | 0.3
$p=40$ | 84 | 80 | 70 | 64 | 13.0 | 10.0 | 2.0 | 0.7
$p=80$ | 80 | 80 | 74 | 74 | 53.0 | 53.0 | 10.0 | 10.0
$D_{3}$ ($\mu_{3}=2400$) |  |  |  |  |  |  |  |
$p=20$ | 100 | 100 | 66 | 51 | 1.4 | 1.1 | 1.0 | 1.0
$p=40$ | 100 | 100 | 69 | 51 | 2.6 | 1.9 | 3.0 | 4.0
$p=80$ | 100 | 100 | 70 | 70 | 7.1 | 7.1 | 10.0 | 10.0

For each model and setting, we generate 500 independent point patterns and
estimate the parameters for each of these using the AL and ALDS procedures.
The performances of AL and ALDS estimates are compared in terms of the true
positive rate (TPR), false positive rate (FPR), and root-mean squared error
(RMSE). We also report the computing time. The TPR (resp. FPR) are the
expected fractions of informative (resp. non-informative) covariates included
in the selected model, so we would expect to obtain a high (resp. low) TPR
(resp. FPR). The RMSE is defined for an estimate $\hat{\bm{\beta}}$ by

$\displaystyle\mathrm{RMSE}=\left\\{\sum_{j=2}^{p}{\hat{\mathbb{E}}(\hat{\beta}_{j}-\beta_{j})^{2}}\right\\}^{\frac{1}{2}}$

where $\hat{\mathbb{E}}$ is the empirical mean.

Tables 1-3 report results respectively for the Poisson model and the Thomas
model with moderate and high clustering. In the situation where point patterns
come from Poisson processes, AL and ALDS perform very similarly. In
particular, both methods do not work well in a small spatial domain with large
$p$. The performances improve significantly as $D$ expands even for large $p$.
When the point patterns exhibit clustering (Tables 2-3), in general AL and
ALDS tend to overfit the intensity model by selecting too many covariates
(indicated by higher FPR) which yields higher RMSE. ALDS sometimes performs
slightly better in terms of RMSE. Results tend to deteriorate in the high
clustering situation but remain very satisfactory: for the three considered
models, the TPR (resp. FPR, RMSE) increases (decreases) when $|D|$ grows for
given $p$ while for given $D$ (especially when $D=D_{2},D_{3}$), the results
remain quite stable when $p$ increases. In terms of computing time, no major
difference can be observed.

###  6.2 Application to the forestry dataset

We model the intensity function for the point pattern of the Acalypha
diversifolia using (1) depending on 94 environmental described previously. The
overall $\bm{\beta}$ estimates from the AL and ALDS procedures are presented
in Table 5. We only report in Table 4 the top 12 important variables.

Among 94 environmental variables, the AL and ALDS respectively selects 32 and
33 important covariates (most of them are similar). We sort the magnitude of
$\hat{\bm{\beta}}$ to identify the 12 most informative covariates. It turns
out that these 12 covariates are similar for both procedures (see Table 4).
For the rest of selected covariates, the rankings are slightly different but
the magnitudes are very similar (see Table 5).

Table 4: Twelve most important covariates selected by AL and ALDS for modeling the intensity of Acalypha diversifolia point pattern Covariates | AL | ALDS
---|---|---
Ca:N.min | -0.89 | -0.89
K:N.min | 0.58 | 0.53
Al:Mg | -0.51 | -0.47
pH | 0.48 | 0.46
B | -0.46 | -0.43
Ca | 0.45 | 0.42
Al:Fe | 0.38 | 0.39
Fe:K | -0.31 | -0.33
B:P | -0.30 | -0.29
P:Nz | 0.27 | 0.26
Fe | 0.26 | 0.25
Mn | -0.24 | -0.24
Number of selected covariates | 32 | 33

##  7 Discussion

In this paper, we develop the adaptive lasso and Dantzig selector for spatial
point processes intensity estimation and provide asymptotic results under an
original setting where the number of non-zero and zero coefficients diverge
with the mean number of points. We demonstrate that both methods share
identical asymptotic properties and perform similarly on simulated and real
data. This study supplements previous ones [see e.g. 3] where similar
conclusions on linear models and generalized linear models were addressed.

[10] considered extensions of lasso type methods by involving general convex
and non-convex penalties. In particular, composite likelihoods penalized by
SCAD or MC+ penalty showed interesting properties. To integrate such an idea
for the Dantzig selector, we could consider the optimization problem

$\displaystyle\min\sum_{i=1}^{p}{p_{\lambda_{j}}(\beta_{j})}\mbox{ subject to
}{\mu}^{-1}\;\Big{|}\mathbf{U}_{j}(\tilde{\bm{\beta}})+[\mathbf{A}(\tilde{\bm{\beta}})(\tilde{\bm{\beta}}-{\bm{\beta}})]_{j}\Big{|}\leq
p^{\prime}_{\lambda_{j}}(\beta_{j}),$ $\displaystyle j=1,\ldots,p$

where $p^{\prime}_{\lambda}(\theta)$ is the derivative with respect to
$\theta$ of a general penalty function $p_{\lambda}$. However, such an
interesting extension would make linear programming unusable and theoretical
developments more complex to derive. We leave this direction for further
study.

Another direction for further study is to derive results for the selection of
regularization parameters. As mentioned earlier, a challenging and definitely
interesting perspective would be the validity of Theorem 4.1 when we define
the regularization parameters in a stochastic way such as
$\lambda_{n,j}=\lambda_{n}/|\tilde{\beta}_{j}|$.

On a similar topic, [11] studied information criteria such as AIC, BIC, and
the composite-versions under similar asymptotic framework for selecting
intensity model of spatial point process. These criteria could be extended for
tuning parameter selection in the context of regularization methods for
spatial point process.

##  Appendix A Additional notation and auxiliary Lemmas

Lemmas A.1-A.2 are used for the proof of Theorem 4.1 in both cases
$\hat{\bm{\beta}}=\hat{\bm{\beta}}_{\mathrm{AL}},\hat{\bm{\beta}}_{\mathrm{ALDS}}$.
Throughout the proofs, the notation $\mathbf{X}_{n}=O_{\mathrm{P}}(x_{n})$ or
$\mathbf{X}_{n}=o_{\mathrm{P}}(x_{n})$ for a random vector $\mathbf{X}_{n}$
and a sequence of real numbers $x_{n}$ means that
$\|\mathbf{X}_{n}\|=O_{\mathrm{P}}(x_{n})$ and
$\|\mathbf{X}_{n}\|=o_{\mathrm{P}}(x_{n})$. In the same way for a vector
$\mathbf{V}_{n}$ or a squared matrix $\mathbf{M}_{n}$, the notation
$\mathbf{V}_{n}=O(x_{n})$ and $\mathbf{M}_{n}=O(x_{n})$ mean that
$\|\mathbf{V}_{n}\|=O(x_{n})$ and $\|\mathbf{M}_{n}\|=O(x_{n})$.

######  Lemma A.1.

Under conditions ($\mathcal{C}$.2), ($\mathcal{C}$.3) and ($\mathcal{C}$.4),
the following result hold as $n\to\infty$

$\displaystyle\max\left\\{\|\mathbf{U}_{n}(\bm{\beta}_{0})\|,\|\mathbf{U}_{n,2}(\bm{\beta}_{0})\|\right\\}=O_{\mathrm{P}}\left(\sqrt{p_{n}\mu_{n}}\right)\quad\text{
and
}\quad\mathbf{U}_{n,1}(\bm{\beta}_{0})=O_{\mathrm{P}}\left(\sqrt{{s_{n}\mu_{n}}}\right).$
(17)

###### Proof.

Using Campbell Theorems (2), the score vector $\mathbf{U}_{n}(\bm{\beta}_{0})$
is unbiased and has variance
$\mathrm{Var}\mathbf{U}_{n}(\bm{\beta}_{0})=\mathbf{B}_{n}(\bm{\beta}_{0})$.
By Condition ($\mathcal{C}$.3) for any $u\in D_{n}$,
$\mathbf{z}(u)\mathbf{z}(u)^{\top}=O(p_{n})$. Hence,
$\mathbf{A}_{n}(\bm{\beta}_{0})=O(p_{n}\mu_{n})$. By definition of (6) and
conditions ($\mathcal{C}$.2) and ($\mathcal{C}$.4), we deduce that
$\mathbf{B}_{n}(\bm{\beta}_{0})=O(p_{n}\mu_{n})$. We deduce that
$\mathrm{Var}\\{\mathbf{U}_{n}(\bm{\beta}_{0})\\}=O(p_{n}\mu_{n})$. In the
same way,
$\mathrm{Var}\\{\mathbf{U}_{n,2}(\bm{\beta}_{0})\\}=O\\{(p_{n}-s_{n})\mu_{n}\\}=O(p_{n}\mu_{n})$
and $\mathrm{Var}\\{\mathbf{U}_{n,1}(\bm{\beta}_{0})\\}=O(s_{n}\mu_{n})$. The
result is proved since for any centered real-valued stochastic process
${Y_{n}}$ with finite variance $\mathrm{Var}({Y_{n}})$,
${Y_{n}}=O_{\mathrm{P}}{\big{\\{}\sqrt{\mathrm{Var}({Y_{n}})}\big{\\}}}$. ∎

The next lemma states that in the vicinity of $\bm{\beta}_{0}$,
$\rho(u;\bm{\beta})$ and $\rho(u;\bm{\beta}_{0})$ have the same behaviour.

######  Lemma A.2.

(i) Let $(\zeta_{n})_{n\geq 1}$ be any sequence such that
$\zeta_{n}=o(1/\sqrt{p_{n}})$ and let $\kappa$ be any non-negative real
number, then under the Conditions ($\mathcal{C}$.1)-($\mathcal{C}$.3), we have

$\sup_{\|\bm{\beta}-\bm{\beta}_{0}\|\leq\kappa\zeta_{n}}\int_{D_{n}}\rho(u;\bm{\beta})\mathrm{d}u=O(\mu_{n}).$

(ii) Similary for any random vector $\bm{\beta}$ such that
$\|\bm{\beta}-\bm{\beta}_{0}\|=o_{\mathrm{P}}(1/\sqrt{p_{n}})$, then

$\int_{D_{n}}\rho(u;\bm{\beta})\mathrm{d}u=O_{\mathrm{P}}(\mu_{n}).$

(iii) In addition, under condition ($\mathcal{C}$.8), (i)-(ii) are valid for
the sequence defined by $\zeta_{n}=\sqrt{p_{n}/\mu_{n}}$.

###### Proof.

(i)-(ii) We only focus on (i) as (ii) follows along similar lines. For any
$u\in D_{n}$, there exists, by Condition ($\mathcal{C}$.3), a constant
$\kappa<\infty$ (independent of $u,\bm{\beta}$ and $\bm{\beta}_{0}$) such that

$-\kappa\sqrt{p_{n}}\|\bm{\beta}-\bm{\beta}_{0}\|\leq(\bm{\beta}-\bm{\beta}_{0})^{\top}\mathbf{z}(u)\leq\kappa\sqrt{p_{n}}\|\bm{\beta}-\bm{\beta}_{0}\|.$

Since,
$\int\rho(u;\bm{\beta})\mathrm{d}u=\int\exp\\{(\bm{\beta}-\bm{\beta}_{0})^{\top}\mathbf{z}(u)\\}\rho(u;\bm{\beta}_{0})\mathrm{d}u$,
we deduce that

$\mu_{n}\,\exp(-\kappa\sqrt{p_{n}}\|\bm{\beta}-\bm{\beta}_{0}\|)\leq\int_{D_{n}}\rho(u;\bm{\beta})\mathrm{d}u\leq\mu_{n}\,\exp(\kappa\sqrt{p_{n}}\|\bm{\beta}-\bm{\beta}_{0}\|)$

which yields the result by definition of $\zeta_{n}$.
(iii) Condition ($\mathcal{C}$.8) implies in particular that
$\sqrt{p_{n}^{2}/\mu_{n}}\to 0$ as $n\to\infty$. ∎

##  Appendix B Proof of Theorem 4.1 when
$\hat{\bm{\beta}}=\hat{\bm{\beta}}_{\mathrm{AL}}$

###  B.1 Existence of a root-$(\mu_{n}/p_{n})$ consistent local maximizer

The first result presented hereafter shows that there exists a local maximizer
of $Q_{n}(\bm{\beta})$ which is a consistent estimator of $\bm{\beta}_{0}$.

######  Proposition B.1.

Assume that the conditions ($\mathcal{C}$.2)-($\mathcal{C}$.4) hold. If in
addition $p_{n}^{4}/\mu_{n}\to 0$ and $a_{n}\sqrt{s_{n}\mu_{n}/p_{n}}\to 0$ as
$n\to\infty$, then there exists a local maximizer
${\hat{\bm{\beta}}_{\mathrm{AL}}}$ of $Q_{n}(\bm{\beta})$ such that

$\displaystyle{\bf\|\hat{\bm{\beta}}_{\mathrm{AL}}-\bm{\beta}_{0}\|}=O_{\mathrm{P}}\big{\\{}\sqrt{p_{n}/\mu_{n}}\big{\\}}.$

Note that the conditions on $a_{n},s_{n},\mu_{n}$ and $p_{n}$ are actually
implied by conditions ($\mathcal{C}$.8) and ($\mathcal{C}$.9). In the proof of
this result and the following ones, the notation $\kappa$ stands for a generic
constant which may vary from line to line. In particular this constant is
independent of $n$, $\bm{\beta}_{0}$ and $\mathbf{k}$.

###### Proof.

Let $\mathbf{k}\in\mathbb{R}^{p_{n}}$. We remind the reader that the estimate
of $\bm{\beta}_{0}$ is defined as the maximum of the function $Q_{n}$, given
by (8), over $\mathbb{R}^{p_{n}}$. To prove Proposition B.1, we aim at proving
that for any given $\epsilon>0$, there exists sufficiently large $K>0$ such
that for $n$ sufficiently large

$\mathrm{P}\bigg{\\{}\sup_{\|\mathbf{k}\|=K}\Delta_{n}(\mathbf{k})>0\bigg{\\}}\leq\epsilon,\quad\mbox{
where
}\Delta_{n}(\mathbf{k})=Q_{n}(\bm{\beta}_{0}+\sqrt{p_{n}/\mu_{n}}\mathbf{k})-Q_{n}(\bm{\beta}_{0}).$
(18)

Equation (18) will imply that with probability at least $1-\epsilon$, there
exists a local maximum in the ball
$\\{\bm{\beta}_{0}+\sqrt{p_{n}/\mu_{n}}\mathbf{k}:\|\mathbf{k}\|\leq K\\}$,
and therefore a local maximizer $\bm{\hat{\beta}}$ such that
$\|{\bm{\hat{\beta}}-\bm{\beta}_{0}}\|=O_{\mathrm{P}}(\sqrt{p_{n}/\mu_{n}})$.
We decompose $\Delta_{n}(\mathbf{k})$ as $\Delta_{n}(\mathbf{k})=T_{1}+T_{2}$
where

$\displaystyle T_{1}$
$\displaystyle=\mu_{n}^{-1}\left\\{\ell_{n}(\bm{\beta}_{0}+\sqrt{p_{n}/\mu_{n}}\mathbf{k})-\ell_{n}(\bm{\beta}_{0})\right\\}$
$\displaystyle T_{2}$
$\displaystyle=\sum_{j=1}^{p_{n}}\lambda_{n,j}\left(|\beta_{0j}|-|\beta_{0j}+\sqrt{p_{n}/\mu_{n}}k_{j}|\right).$

Since $\rho(u;\cdot)$ is infinitely continuously differentiable and
$\ell_{n}^{(2)}(\bm{\beta})=-\mathbf{A}_{n}(\bm{\beta})$, then using a second-
order Taylor expansion there exists $t\in(0,1)$ such that

$\displaystyle\mu_{n}T_{1}=$
$\displaystyle\,\sqrt{p_{n}/\mu_{n}}\mathbf{k}^{\top}\ell_{n}^{(1)}(\bm{\beta}_{0})+T_{11}+T_{12}$

where

$\displaystyle T_{11}=$
$\displaystyle-\frac{1}{2}\frac{p_{n}}{\mu_{n}}\mathbf{k}^{\top}\mathbf{A}_{n}(\bm{\beta}_{0})\mathbf{k}$
$\displaystyle T_{12}=$
$\displaystyle+\frac{1}{2}\frac{p_{n}}{\mu_{n}}\mathbf{k}^{\top}\left\\{\mathbf{A}_{n}(\bm{\beta}_{0})-\mathbf{A}_{n}(\bm{\beta}_{0}+t\sqrt{p_{n}/\mu_{n}}\mathbf{k})\right\\}\mathbf{k}.$

By condition ($\mathcal{C}$.3)

$T_{11}=-\frac{1}{2}p_{n}\frac{\mathbf{k}^{\top}\\{\mu_{n}^{-1}\mathbf{A}_{n}(\bm{\beta}_{0})\\}\mathbf{k}}{\|\mathbf{k}\|^{2}}\,\|\mathbf{k}\|^{2}\leq-\frac{\alpha}{2}p_{n}\|\mathbf{k}\|^{2}$

where $\alpha=\liminf_{n\geq
1}\inf_{\mathbf{\bm{missing}}\phi,\|\mathbf{\bm{missing}}\phi\|=1}\bm{\phi}^{\top}\\{\mu_{n}^{-1}\mathbf{A}_{n}(\bm{\beta}_{0})\\}\bm{\phi}>0$.
Now, for some $\tilde{\bm{\beta}}$ on the line segment between
$\bm{\beta}_{0}$ and $\bm{\beta}_{0}+t{\sqrt{p_{n}/\mu_{n}}}\mathbf{k}$

$T_{12}=\frac{1}{2}\,\frac{p_{n}}{\mu_{n}}\mathbf{k}^{\top}\left\\{\int_{D_{n}}\mathbf{z}(u)\mathbf{z}(u)^{\top}t\sqrt{\frac{p_{n}}{\mu_{n}}}\mathbf{k}^{\top}\mathbf{z}(u)\rho(u;\tilde{\bm{\beta}})\mathrm{d}u\right\\}\mathbf{k}.$

By conditions ($\mathcal{C}$.2)-($\mathcal{C}$.3) and Lemma A.2

$T_{12}=O\left(\|\mathbf{k}\|^{3}\frac{p_{n}}{\mu_{n}}p_{n}\sqrt{\frac{p_{n}}{\mu_{n}}}\sqrt{p_{n}}\mu_{n}\right)=O\left(p_{n}\sqrt{\frac{p_{n}^{4}}{\mu_{n}}}\right)=o(p_{n}).$

Hence, for $n$ sufficiently large

$\mu_{n}T_{1}\leq\sqrt{\frac{p_{n}}{\mu_{n}}}\mathbf{k}^{\top}\ell_{n}^{(1)}(\bm{\beta}_{0})-\frac{\alpha}{4}p_{n}\|\mathbf{k}\|^{2}.$

Regarding the term $T_{2}$ we have,

$T_{2}\leq\sum_{j=1}^{s_{n}}\lambda_{n,j}\left\\{|\beta_{0j}|-\left|\beta_{0j}+\sqrt{\frac{p_{n}}{\mu_{n}}}k_{j}\right|\right\\}\leq
a_{n}\sqrt{\frac{p_{n}}{\mu_{n}}}\sum_{j=1}^{s_{n}}|k_{j}|\leq
a_{n}\sqrt{\frac{s_{n}p_{n}}{\mu_{n}}}\|\mathbf{k}\|.$

We deduce that for $n$ large enough, there exists $\kappa$ such that
$T_{2}\leq\kappa d_{n}^{2}\|k\|$ whereby we deduce that

$\Delta_{n}(\mathbf{k})\leq\frac{1}{\mu_{n}}\sqrt{\frac{p_{n}}{\mu_{n}}}\mathbf{k}^{\top}\ell_{n}^{(1)}(\bm{\beta}_{0})-\frac{\alpha}{4}\frac{p_{n}}{\mu_{n}}\|\mathbf{k}\|^{2}+a_{n}\sqrt{\frac{s_{n}p_{n}}{\mu_{n}}}\|\mathbf{k}\|.$

By the assumption of Proposition B.1,
$a_{n}\sqrt{s_{n}p_{n}/\mu_{n}}=a_{n}\sqrt{s_{n}\mu_{n}/p_{n}}p_{n}/\mu_{n}=o(p_{n}/\mu_{n})$,
whereby we deduce that for $n$ sufficiently large

$\Delta_{n}(\mathbf{k})\leq\frac{1}{\mu_{n}}\sqrt{\frac{p_{n}}{\mu_{n}}}\mathbf{k}^{\top}\ell_{n}^{(1)}(\bm{\beta}_{0})-\frac{\alpha}{8}\frac{p_{n}}{\mu_{n}}\|\mathbf{k}\|^{2}.$

Now for $n$ sufficiently large,

$\displaystyle\mathrm{P}\bigg{\\{}{\sup_{\|\mathbf{k}\|=K}\Delta_{n}(\mathbf{k})>0}\bigg{\\}}$
$\displaystyle\leq\mathrm{P}\bigg{\\{}\|\ell_{n}^{(1)}(\bm{\beta}_{0})\|\geq\frac{\alpha}{8}Kp_{n}\sqrt{\frac{\mu_{n}}{p_{n}}}\bigg{\\}}$
$\displaystyle=\mathrm{P}\bigg{\\{}\|\ell_{n}^{(1)}(\bm{\beta}_{0})\|\geq\frac{\alpha}{8}K\sqrt{p_{n}\mu_{n}}\bigg{\\}}<\varepsilon$

for any given $\varepsilon>0$ since
$\ell_{n}^{(1)}(\bm{\beta}_{0})=\mathbf{U}_{n}(\bm{\beta}_{0})=O_{\mathrm{P}}(\sqrt{p_{n}\mu_{n}})$
by Lemma A.1.

∎

###  B.2 Sparsity property for
$\hat{\bm{\beta}}=\hat{\bm{\beta}}_{\mathrm{AL}}$

The sparsity property for $\hat{\bm{\beta}}_{\mathrm{AL}}$ follows directly
from Proposition B.1 and the following Lemma B.2.

######  Lemma B.2.

Assume the conditions ($\mathcal{C}$.2)-($\mathcal{C}$.4) and
($\mathcal{C}$.8)-($\mathcal{C}$.9) hold, then with probability tending to
$1$, for any $\bm{\beta}_{1}\in\mathbb{R}^{s_{n}}$ satisfying
$\|{\bm{\beta}_{1}-\bm{\beta}_{01}}\|=O_{\mathrm{P}}(\sqrt{p_{n}/\mu_{n}})$,
and for any constant $K_{1}>0$,

$\displaystyle
Q_{n}\Big{\\{}({\bm{\beta}_{1}}^{\top},\mathbf{0}^{\top})^{\top}\Big{\\}}=\max_{\|\bm{\beta}_{2}\|\leq
K_{1}\sqrt{p_{n}/\mu_{n}}}Q_{n}\Big{\\{}({\bm{\beta}_{1}}^{\top},{\bm{\beta}_{2}}^{\top})^{\top}\Big{\\}}.$

###### Proof.

Let $\varepsilon_{n}=K_{1}\sqrt{p_{n}/\mu_{n}}$. It is sufficient to show that
with probability tending to $1$ as ${n\to\infty}$, for any ${\bm{\beta}_{1}}$
satisfying
$\|{\bm{\beta}_{1}-\bm{\beta}_{01}}\|=O_{\mathrm{P}}(\sqrt{p_{n}/\mu_{n}})$,
we have for any $j=s_{n}+1,\ldots,p_{n}$

$\frac{\partial Q_{n}(\bm{\beta})}{\partial\beta_{j}}<0\quad\mbox{ for
}0<\beta_{j}<\varepsilon_{n},\mbox{ and}$ (19)

$\frac{\partial Q_{n}(\bf\bm{\beta})}{\partial\beta_{j}}>0\quad\mbox{ for
}-\varepsilon_{n}<\beta_{j}<0.$ (20)

From (3),

$\displaystyle\frac{\partial\ell_{n}(\bm{\beta})}{\partial\beta_{j}}=\frac{\partial\ell_{n}{(\bm{\beta}_{0})}}{\partial\beta_{j}}+R_{n},$

where
$R_{n}={-}\int_{D_{n}}z_{j}(u)\big{\\{}\rho(u;\bm{\beta})-\rho(u;\bm{\beta}_{0})\big{\\}}\mathrm{d}u$.
Let $u\in\mathbb{R}^{d}$. By Taylor expansion, there exists $t\in(0,1),$ such
that

$\displaystyle\rho(u;\bm{\beta})=\rho(u;\bm{\beta}_{0})+(\bm{\beta}-\bm{\beta}_{0})^{\top}\mathbf{z}(u)\rho\\{u;\bm{\beta}_{0}+t(\bm{\beta}-\bm{\beta}_{0})\\}.$

By condition ($\mathcal{C}$.1)-($\mathcal{C}$.3) and Lemma A.2, we have for
$n$ sufficiently large

$|R_{n}|\leq\kappa\|\bm{\beta}-\bm{\beta}_{0}\|\sqrt{p_{n}}\int_{D_{n}}\rho(u;\bm{\beta}_{0})\mathrm{d}u=O_{\mathrm{P}}\left(\sqrt{\frac{p_{n}}{\mu_{n}}}\sqrt{p_{n}}\mu_{n}\right)=O_{\mathrm{P}}\left({p_{n}}\sqrt{\mu_{n}}\right).$

Following the proof of Lemma A.1, we can derive
$\mathrm{Var}({\partial\ell_{n}{(\bm{\beta}_{0})}}/{\partial\beta_{j}})=\mathrm{Var}[\\{\mathbf{U}_{n}(\bm{\beta}_{0})\\}_{j}]=O(\mu_{n})$
whereby we deduce that

$\frac{\partial\ell_{n}(\bm{\beta})}{\partial\beta_{j}}=O_{\mathrm{P}}(p_{n}\sqrt{\mu_{n}}).$
(21)

Now, we want to prove (19). Let $0<\beta_{j}<\varepsilon_{n}$ and remind that
the sequence $b_{n}$ is given by ($\mathcal{C}$.9). Then, for $n$ sufficiently
large,

$\displaystyle\mathrm{P}\left\\{\frac{\partial
Q_{n}(\bm{\beta})}{\partial\beta_{j}}<0\right\\}$
$\displaystyle=\mathrm{P}\left\\{\frac{\partial\ell_{n}(\bm{\beta})}{\partial\beta_{j}}-\mu_{n}{\lambda_{n,j}}\operatorname{sign}(\beta_{j})<0\right\\}$
$\displaystyle=\mathrm{P}\left\\{\frac{\partial\ell_{n}(\bm{\beta})}{\partial\beta_{j}}<\mu_{n}{\lambda_{n,j}}\right\\}$
$\displaystyle\geq\mathrm{P}\left\\{\frac{\partial\ell_{n}(\bm{\beta})}{\partial\beta_{j}}<\mu_{n}b_{n}\right\\}$
$\displaystyle=\mathrm{P}\left\\{\frac{\partial\ell_{n}(\bm{\beta})}{\partial\beta_{j}}<p_{n}\sqrt{\mu_{n}}\;\sqrt{\frac{\mu_{n}}{p_{n}^{2}}}b_{n}\right\\}.$

The assertion (19) is therefore deduced from (21) and from the assumption that
$b_{n}\sqrt{\mu_{n}/p_{n}^{2}}\to\infty$ as $n\to\infty$. We proceed similarly
to prove (20). ∎

###  B.3 Asymptotic normality for
$\hat{\bm{\beta}}=\hat{\bm{\beta}}_{\mathrm{AL}}$

###### Proof.

As shown in Proposition B.1, there is a root-$(\mu_{n}/p_{n})$ consistent
local maximizer $\hat{\bm{\beta}}_{\mathrm{AL}}$ of $Q_{n}(\bm{\beta})$, and
it can be shown that there exists an estimator
$\hat{\bm{\beta}}_{\mathrm{AL},1}$ in Proposition B.1 that is a
root-$(\mu_{n}/p_{n})$ consistent local maximizer of
$Q_{n}\\{({\bm{\beta}_{1}}^{\top},\mathbf{0}^{\top})^{\top}\Big{\\}}$, which
is regarded as a function of $\bm{\beta}_{1}$, and that satisfies

$\displaystyle\frac{\partial
Q_{n}(\hat{\bm{\beta}}_{\mathrm{AL}})}{\partial\beta_{j}}=0\quad\mbox{ for
}j=1,\ldots,s_{n}\mbox{ and
}\hat{\bm{\beta}}_{\mathrm{AL}}=(\hat{\bm{\beta}}_{\mathrm{AL},1}^{\top},\mathbf{0}^{\top})^{\top}.$

There exists $t\in(0,1)$ and
$\bm{\check{\beta}}=\hat{\bm{\beta}}_{\mathrm{AL}}+t(\bm{\beta}_{0}-\hat{\bm{\beta}}_{\mathrm{AL}})$
such that for $j=1,\cdots,s_{n}$

$\displaystyle 0=$
$\displaystyle\frac{\partial\ell_{n}{(\hat{\bm{\beta}}_{\mathrm{AL}})}}{\partial\beta_{j}}-\mu_{n}{\lambda_{n,j}}\operatorname{sign}({\hat{\beta}_{\mathrm{AL},j}})$
$\displaystyle=$
$\displaystyle\frac{\partial\ell_{n}{(\bm{\beta}_{0})}}{\partial\beta_{j}}+{\sum_{l=1}^{s_{n}}\frac{\partial^{2}\ell_{n}{(\bm{\check{\beta}})}}{\partial\beta_{j}\partial\beta_{l}}}({\hat{\beta}_{\mathrm{AL},l}-\beta_{0l}})-\mu_{n}{\lambda_{n,j}}\operatorname{sign}({\hat{\beta}_{\mathrm{AL},j}})$
$\displaystyle=$
$\displaystyle\frac{\partial\ell_{n}{(\bm{\beta}_{0})}}{\partial\beta_{j}}+{\sum_{l=1}^{s_{n}}\frac{\partial^{2}\ell_{n}{(\bm{\beta}_{0})}}{\partial\beta_{j}\partial\beta_{l}}}({\hat{\beta}_{\mathrm{AL},l}-\beta_{0l}})+{\sum_{l=1}^{s_{n}}\Psi_{n,jl}({\hat{\beta}_{\mathrm{AL},l}}-\beta_{0l})}$
$\displaystyle-\mu_{n}\lambda_{n,j}\operatorname{sign}({\hat{\beta}_{\mathrm{AL},j}})$
(22)

where

$\displaystyle\Psi_{n,jl}=\frac{\partial^{2}\ell_{n}{(\bm{\check{\beta}})}}{\partial\beta_{j}\partial\beta_{l}}-\frac{\partial^{2}\ell_{n}{(\bm{\beta}_{0})}}{\partial\beta_{j}\partial\beta_{l}}.$

Let $\mathbf{U}_{n,1}(\bm{\beta}_{0})$ (resp.
$\ell^{(2)}_{n,1}(\bm{\beta}_{0})$) be the first $s_{n}$ components (resp.
$s_{n}\times s_{n}$ top-left corner) of $\mathbf{U}_{n}(\bm{\beta}_{0})$
(resp. $\ell^{(2)}_{n}(\bm{\beta}_{0})$). Let also $\bm{\Psi}_{n}$ be the
$s_{n}\times s_{n}$ matrix containing $\Psi_{n,jl},j,l=1,\ldots,s_{n}$.
Finally, let the vector $\mathbf{p}^{\prime}_{n}$

$\displaystyle\mathbf{p}^{\prime}_{n}$
$\displaystyle={\\{\lambda_{n,1}\operatorname{sign}({\hat{\beta}_{\mathrm{AL},1}}),\ldots,\lambda_{n,s_{n}}\operatorname{sign}({\hat{\beta}_{\mathrm{AL},s_{n}}})\\}^{\top}}.$

These notation allow us to rewrite (22) as

$\mathbf{U}_{n,1}(\bm{\beta}_{0})-\mathbf{A}_{n,11}(\bm{\beta}_{0})(\hat{\bm{\beta}}_{\mathrm{AL},1}-\bm{\beta}_{01})+\bm{\Psi}_{n}(\hat{\bm{\beta}}_{\mathrm{AL},1}-\bm{\beta}_{01})-\mu_{n}\mathbf{p}^{\prime}_{n}=0.$
(23)

Let $\bm{\phi}\in\mathbb{R}^{s_{n}}\setminus\\{0\\}$ and
$\sigma^{2}_{\phi}=\bm{\phi}^{\top}\mathbf{B}_{n,11}(\bm{\beta}_{0})\bm{\phi}$,
then

$\sigma_{\bm{\phi}}^{-1}\bm{\phi}^{\top}\mathbf{U}_{n,1}(\bm{\beta}_{0})-\sigma_{\bm{\phi}}^{-1}\bm{\phi}^{\top}\mathbf{A}_{n,11}(\bm{\beta}_{0})(\hat{\bm{\beta}}_{\mathrm{AL},1}-\bm{\beta}_{01})+\sigma_{\bm{\phi}}^{-1}\bm{\phi}^{\top}\bm{\Psi}_{n}(\hat{\bm{\beta}}_{\mathrm{AL},1}-\bm{\beta}_{01})-\mu_{n}\sigma_{\bm{\phi}}^{-1}\bm{\phi}^{\top}\mathbf{p}^{\prime}_{n}=0.$

Now, by condition ($\mathcal{C}$.5),
$\sigma_{\bm{\phi}}^{-1}=O(\mu_{n}^{-1/2})$ and by the definition of $a_{n}$,
$\mathbf{p}^{\prime}_{n}=O(a_{n}\sqrt{s_{n}})$. By conditions
($\mathcal{C}$.1)-($\mathcal{C}$.3), there exists some $\tilde{\bm{\beta}}$ on
the line segment between $\bm{\beta}_{0}$ and $\check{\bm{\beta}}$ such that

$\bm{\Psi}_{n}=\int_{D_{n}}\mathbf{z}_{1}(u)\mathbf{z}_{1}(u)^{\top}(\check{\bm{\beta}}-\bm{\beta}_{0})^{\top}\mathbf{z}(u)\rho(u;\tilde{\bm{\beta}})\mathrm{d}u$

whereby we deduce from conditions ($\mathcal{C}$.1)-($\mathcal{C}$.3) and
Lemma A.2 that

$\|\bm{\Psi}_{n}\|=O_{\mathrm{P}}\left(s_{n}\sqrt{\frac{p_{n}}{\mu_{n}}}\sqrt{p_{n}}\mu_{n}\right)=O_{\mathrm{P}}(s_{n}p_{n}\sqrt{\mu_{n}}).$

The last two results and conditions ($\mathcal{C}$.8)-($\mathcal{C}$.9) yield
that

$\displaystyle\sigma_{\bm{\phi}}^{-1}\bm{\phi}^{\top}\bm{\Psi}_{n}(\hat{\bm{\beta}}_{\mathrm{AL},1}-\bm{\beta}_{01})$
$\displaystyle=O_{\mathrm{P}}\left(\frac{1}{\sqrt{\mu_{n}}}s_{n}p_{n}\sqrt{\mu_{n}}\sqrt{\frac{p_{n}}{\mu_{n}}}\right)=O_{\mathrm{P}}\left(\sqrt{\frac{s_{n}^{2}p_{n}^{3}}{\mu_{n}}}\right)=o_{\mathrm{P}}(1)$
$\displaystyle\mu_{n}\sigma_{\bm{\phi}}^{-1}\bm{\phi}^{\top}\mathbf{p}^{\prime}_{n}$
$\displaystyle=O\left(\mu_{n}\frac{1}{\sqrt{\mu_{n}}}a_{n}\sqrt{s_{n}}\right)=O(a_{n}\sqrt{s_{n}\mu_{n}})=o(1).$

These results finally lead to

$\sigma_{\bm{\phi}}^{-1}\bm{\phi}^{\top}\mathbf{A}_{n,11}(\bm{\beta}_{0})(\hat{\bm{\beta}}_{\mathrm{AL},1}-\bm{\beta}_{01})=\sigma_{\bm{\phi}}^{-1}\bm{\phi}^{\top}\mathbf{U}_{n,1}(\bm{\beta}_{0})+o_{\mathrm{P}}(1)$

and finally to the proof of the result using Slutsky’s lemma and condition
($\mathcal{C}$.6).

∎

##  Appendix C Proof of Theorem 4.1 when
$\hat{\bm{\beta}}=\hat{\bm{\beta}}_{\mathrm{ALDS}}$

###  C.1 Existence and optimal solutions for the primal and dual problems

For $\bm{\beta}\in\mathbb{R}^{p_{n}}$, we let
$\bm{\Delta}_{n}(\bm{\beta})=\mathbf{U}_{n}(\tilde{\bm{\beta}})+\mathbf{A}_{n}(\tilde{\bm{\beta}})(\tilde{\bm{\beta}}-\bm{\beta})$.

######  Lemma C.1.

There exists a solution to the problem (12).

###### Proof.

Following [6], we state that (12) is equivalent to

$\displaystyle\min_{\bm{\beta},{\bm{u}}}\sum_{j}u_{j}\mbox{ subject to
}\begin{cases}{\bm{\Lambda}}_{n}\bm{\beta}\leq{\bm{u}}\\\
-{\bm{\Lambda}}_{n}\bm{\beta}\leq{\bm{u}}\\\
\mu_{n}^{-1}{\bm{\Lambda}}_{n}^{-1}\bm{\Delta}_{n}(\bm{\beta})-\bm{1}_{p_{n}}\leq{\mathbf{0}}\\\
-\mu_{n}^{-1}{\bm{\Lambda}}_{n}^{-1}\bm{\Delta}_{n}(\bm{\beta})-\bm{1}_{p_{n}}\leq{\mathbf{0}}\end{cases}$
(24)

where $\bm{u}\in\mathbb{R}^{p_{n}}$ is an additional parameter vector to be
optimized and $\tilde{\bm{\beta}}$ is the initial estimator. Note that (24) is
a linear problem with $4p_{n}$ linear inequality constraints. To prove the
existence of ALDS estimates, we need to derive dual problem of (24) and prove
that strong duality holds. To derive the dual problem, we first construct the
Lagrangian form associated with the problem (24) considering the main
arguments by [5, section 5.2]

$\displaystyle L(\bm{\beta};\bm{u};\bm{\alpha})$
$\displaystyle=\sum_{j}u_{j}+\bm{\alpha}_{1}^{\top}({\bm{\Lambda}}_{n}\bm{\beta}-\mathbf{u})+\bm{\alpha}_{2}^{\top}(-{\bm{\Lambda}}_{n}\bm{\beta}-\mathbf{u})$
$\displaystyle\quad+\bm{\alpha}_{3}^{\top}\Big{[}\mu_{n}^{-1}{\bm{\Lambda}}_{n}^{-1}\bm{\Delta}_{n}(\bm{\beta})-\bm{1}_{p_{n}}\Big{]}+\bm{\alpha}_{4}^{\top}\Big{[}-\mu_{n}^{-1}{\bm{\Lambda}}_{n}^{-1}\bm{\Delta}_{n}(\bm{\beta})-\bm{1}_{p_{n}}\Big{]}$
$\displaystyle=(\bm{1}_{p_{n}}-\bm{\alpha}_{1}-\bm{\alpha}_{2})^{\top}\bm{u}+\Big{\\{}(\bm{\alpha}_{1}-\bm{\alpha}_{2})^{\top}{\bm{\Lambda}}_{n}-\mu_{n}^{-1}(\bm{\alpha}_{3}-\bm{\alpha}_{4})^{\top}{\bm{\Lambda}}_{n}^{-1}\mathbf{A}_{n}(\tilde{\bm{\beta}})\Big{\\}}\bm{\beta}$
$\displaystyle\quad+(\bm{\alpha}_{3}-\bm{\alpha}_{4})^{\top}\Big{\\{}\mu_{n}^{-1}{\bm{\Lambda}}_{n}^{-1}\Big{(}\mathbf{U}_{n}(\tilde{\bm{\beta}})+\mathbf{A}_{n}(\tilde{\bm{\beta}})\tilde{\bm{\beta}}\Big{)}\Big{\\}}-(\bm{\alpha}_{3}+\bm{\alpha}_{4})^{\top}\bm{1}_{p_{n}},$

where
$\bm{\alpha}=(\bm{\alpha}_{1}^{\top},\bm{\alpha}_{2}^{\top},\bm{\alpha}_{3}^{\top},\bm{\alpha}_{4}^{\top})^{\top}\in\mathbb{R}^{4p_{n}}$
is the dual vector (which can be viewed as a Lagrange multiplier).

The dual function $h$ is defined by

$\displaystyle h(\bm{\alpha})$
$\displaystyle=\inf_{\bm{\beta},\bm{u}}L(\bm{\beta};\bm{u};\bm{\alpha})$ (25)
$\displaystyle=\begin{cases}(\bm{\alpha}_{3}-\bm{\alpha}_{4})^{\top}\Big{\\{}\mu_{n}^{-1}{\bm{\Lambda}}_{n}^{-1}\Big{(}\mathbf{U}_{n}(\tilde{\bm{\beta}})+\mathbf{A}_{n}(\tilde{\bm{\beta}})\tilde{\bm{\beta}}\Big{)}\Big{\\}}-(\bm{\alpha}_{3}+\bm{\alpha}_{4})^{\top}\bm{1}_{p_{n}},\text{
if}\\\
\quad\begin{cases}\bm{1}_{p_{n}}-\bm{\alpha}_{1}-\bm{\alpha}_{2}={\mathbf{0}}\\\
(\bm{\alpha}_{1}-\bm{\alpha}_{2})^{\top}{\bm{\Lambda}}_{n}-\mu_{n}^{-1}(\bm{\alpha}_{3}-\bm{\alpha}_{4})^{\top}{\bm{\Lambda}}_{n}^{-1}\mathbf{A}_{n}(\tilde{\bm{\beta}})={\mathbf{0}}\end{cases}\\\
-\infty\text{ otherwise}.\end{cases}$

For any
$\bm{\alpha}=(\bm{\alpha}_{1}^{\top},\bm{\alpha}_{2}^{\top},\bm{\alpha}_{3}^{\top},\bm{\alpha}_{4}^{\top})^{\top}\in\mathbb{R^{+}}^{4p_{n}}$,
$h(\bm{\alpha})$ is a lower bound to the optimality problem (24) (see [5,
p.216]). To find the best lower bound comes to solve the dual problem:
$\max_{\bm{\alpha}\geq{\mathbf{0}}}h(\bm{\alpha}).$

Recall that problem (24) is a linear program with linear inequality
constraints, so that strong duality holds if the dual problem is feasible [see
5, p.227], that is to say if there exists some
$\bm{\alpha}=(\bm{\alpha}_{1}^{\top},\bm{\alpha}_{2}^{\top},\bm{\alpha}_{3}^{\top},\bm{\alpha}_{4}^{\top})^{\top}\in\mathbb{R^{+}}^{4p_{n}}$
such that

$\displaystyle\bm{1}_{p_{n}}-\bm{\alpha}_{1}-\bm{\alpha}_{2}$ $\displaystyle=$
$\displaystyle{\mathbf{0}}$
$\displaystyle(\bm{\alpha}_{1}-\bm{\alpha}_{2})^{\top}{\bm{\Lambda}}_{n}-\mu_{n}^{-1}(\bm{\alpha}_{3}-\bm{\alpha}_{4})^{\top}{\bm{\Lambda}}_{n}^{-1}\mathbf{A}_{n}(\tilde{\bm{\beta}})$
$\displaystyle=$ $\displaystyle{\mathbf{0}}.$

Moreover, we remark that

$\displaystyle\bm{\alpha}_{1}\geq{\mathbf{0}},\;\bm{\alpha}_{2}\geq{\mathbf{0}},\;\bm{\alpha}_{3}\geq{\mathbf{0}},\;\bm{\alpha}_{4}$
$\displaystyle\geq$ $\displaystyle{\mathbf{0}}$
$\displaystyle\bm{1}_{p_{n}}-\bm{\alpha}_{1}-\bm{\alpha}_{2}$ $\displaystyle=$
$\displaystyle{\mathbf{0}}$
$\displaystyle(\bm{\alpha}_{1}-\bm{\alpha}_{2})^{\top}{\bm{\Lambda}}_{n}-\mu_{n}^{-1}(\bm{\alpha}_{3}-\bm{\alpha}_{4})^{\top}{\bm{\Lambda}}_{n}^{-1}\mathbf{A}_{n}(\tilde{\bm{\beta}})$
$\displaystyle=$ $\displaystyle{\mathbf{0}}$

is equivalent to

$\displaystyle\bm{\alpha}_{1}\geq{\mathbf{0}},\;\bm{\alpha}_{2}=\bm{1}_{n}-\bm{\alpha}_{1}\geq{\mathbf{0}},\;\bm{\alpha}_{3}\geq{\mathbf{0}},\;\bm{\alpha}_{4}$
$\displaystyle\geq$ $\displaystyle{\mathbf{0}}$
$\displaystyle(2\bm{\alpha}_{1}-\bm{1}_{p_{n}})^{\top}{\bm{\Lambda}}_{n}-\mu_{n}^{-1}(\bm{\alpha}_{3}-\bm{\alpha}_{4})^{\top}{\bm{\Lambda}}_{n}^{-1}\mathbf{A}_{n}(\tilde{\bm{\beta}})$
$\displaystyle=$ $\displaystyle{\mathbf{0}}$

which is also equivalent to

$\displaystyle\bm{\alpha}_{1}=\frac{1}{2}\Big{\\{}\bm{1}_{p_{n}}+\mu_{n}^{-1}{\bm{\Lambda}}_{n}^{-1}\mathbf{A}_{n}(\tilde{\bm{\beta}}){\bm{\Lambda}}_{n}^{-1}(\bm{\alpha}_{3}-\bm{\alpha}_{4})\Big{\\}}$
$\displaystyle\geq$ $\displaystyle{\mathbf{0}}$
$\displaystyle\bm{\alpha}_{2}=\bm{1}_{p_{n}}-\bm{\alpha}_{1}=\frac{1}{2}\Big{\\{}\bm{1}_{p_{n}}-\mu_{n}^{-1}{\bm{\Lambda}}_{n}^{-1}\mathbf{A}_{n}(\tilde{\bm{\beta}}){\bm{\Lambda}}_{n}^{-1}(\bm{\alpha}_{3}-\bm{\alpha}_{4})\Big{\\}}$
$\displaystyle\geq$ $\displaystyle{\mathbf{0}}$
$\displaystyle\bm{\alpha}_{3}\geq{\mathbf{0}},\;\bm{\alpha}_{4}$
$\displaystyle\geq$ $\displaystyle{\mathbf{0}}.$

This comes to the condition that there exists
$(\bm{\alpha}_{3}^{\top},\bm{\alpha}_{4}^{\top})^{\top}\in\mathbb{R^{+}}^{2p_{n}}$
such that

$\mu_{n}^{-1}\|{\bm{\Lambda}}_{n}^{-1}\mathbf{A}_{n}(\tilde{\bm{\beta}}){\bm{\Lambda}}_{n}^{-1}(\bm{\alpha}_{3}-\bm{\alpha}_{4})\|_{\infty}\leq
1.$ (26)

Therefore, the dual problem associated with (24) is

$\displaystyle\max_{\bm{\alpha}_{3},\bm{\alpha}_{4}\geq
0}(\bm{\alpha}_{3}-\bm{\alpha}_{4})^{\top}\Big{[}\mu_{n}^{-1}{\bm{\Lambda}}_{n}^{-1}\Big{\\{}\mathbf{U}_{n}(\tilde{\bm{\beta}})+\mathbf{A}_{n}(\tilde{\bm{\beta}})\tilde{\bm{\beta}}\Big{\\}}\Big{]}-(\bm{\alpha}_{3}+\bm{\alpha}_{4})^{\top}\bm{1}_{p_{n}}$
$\displaystyle\text{subject to
}\mu_{n}^{-1}\|{\bm{\Lambda}}_{n}^{-1}\mathbf{A}_{n}(\tilde{\bm{\beta}}){\bm{\Lambda}}_{n}^{-1}(\bm{\alpha}_{3}-\bm{\alpha}_{4})\|_{\infty}\leq
1.$ (27)

The condition (26) is always true as far as the matrix
${\bm{\Lambda}}_{n}^{-1}\mathbf{A}_{n}(\tilde{\bm{\beta}}){\bm{\Lambda}}_{n}^{-1}$
is non zero. Indeed, let $\mathbf{y}\in\mathbb{R}^{p_{n}}$ such that
$\mathbf{\\{}{\bm{\Lambda}}_{n}^{-1}\mathbf{A}_{n}(\tilde{\bm{\beta}}){\bm{\Lambda}}_{n}^{-1}y\\}\neq
0$. Now define

$\displaystyle\alpha_{3j}$ $\displaystyle=$
$\displaystyle\frac{y_{j}}{\mu_{n}^{-1}\|{\bm{\Lambda}}_{n}^{-1}\mathbf{A}_{n}(\tilde{\bm{\beta}}){\bm{\Lambda}}_{n}^{-1}y\|_{\infty}}\mathbf{1}(y_{j}>0),$
$\displaystyle\alpha_{4j}$ $\displaystyle=$
$\displaystyle\frac{-y_{j}}{\mu_{n}^{-1}\|{\bm{\Lambda}}_{n}^{-1}\mathbf{A}_{n}(\tilde{\bm{\beta}}){\bm{\Lambda}}_{n}^{-1}y\|_{\infty}}\mathbf{1}(y_{j}<0).$

Clearly,
$(\bm{\alpha}_{3}^{\top},\bm{\alpha}_{4}^{\top})^{\top}\in\mathbb{R^{+}}^{2p_{n}}$,
(26) is always verified. This ends the proof. ∎

Note that the dual problem (27) can be unequivocally reparameterized in terms
of $\bm{\gamma}=\bm{\alpha}_{3}-\bm{\alpha}_{4}$ as follows

$\displaystyle\max_{\bm{\gamma}\in\mathbb{R}^{p_{n}}}\bm{\gamma}^{\top}\Big{[}\mu_{n}^{-1}{\bm{\Lambda}}_{n}^{-1}\Big{\\{}\mathbf{U}_{n}(\tilde{\bm{\beta}})+\mathbf{A}_{n}(\tilde{\bm{\beta}})\tilde{\bm{\beta}}\Big{\\}}\Big{]}-\|\bm{\gamma}\|_{1}$
$\displaystyle\text{subject to
}\mu_{n}^{-1}\|\bm{\gamma}^{\top}{\bm{\Lambda}}_{n}^{-1}\mathbf{A}_{n}(\tilde{\bm{\beta}}){\bm{\Lambda}}_{n}^{-1}\|_{\infty}\leq
1$ (28)

due to complementary slackness conditions. Now, we derive the following
optimality conditions and obtain optimal primal and dual solutions.

We derive the following optimality conditions ensuring the Karush-Kuhn-Tucker
(KKT) conditions and thus obtain optimal primal and dual solutions.

######  Lemma C.2.

Consider the primal and dual problems defined by (12) and (28). Suppose that
the matrix
${\bm{\Lambda}}_{n}^{-1}\mathbf{A}_{n}(\tilde{\bm{\beta}}){\bm{\Lambda}}_{n}^{-1}$
is non zero and that ${\hat{\bm{\beta}}}$ and $\hat{\bm{\gamma}}$ verify

$\displaystyle\mu_{n}^{-1}\Big{\|}{\bm{\Lambda}}_{n}^{-1}\bm{\Delta}_{n}({\hat{\bm{\beta}}})\Big{\|}_{\infty}$
$\displaystyle\leq 1$ (29)
$\displaystyle\mu_{n}^{-1}\Big{\|}\hat{\bm{\gamma}}^{\top}{\bm{\Lambda}}_{n}^{-1}\mathbf{A}_{n}(\tilde{\bm{\beta}}){\bm{\Lambda}}_{n}^{-1}\Big{\|}_{\infty}$
$\displaystyle\leq 1$ (30)
$\displaystyle\mu_{n}^{-1}\hat{\bm{\gamma}}^{\top}{\bm{\Lambda}}_{n}^{-1}\mathbf{A}_{n}(\tilde{\bm{\beta}})\hat{\bm{\beta}}$
$\displaystyle=\|{\bm{\Lambda}}_{n}\hat{\bm{\beta}}\|_{1}$ (31)
$\displaystyle\mu_{n}^{-1}\hat{\bm{\gamma}}^{\top}{\bm{\Lambda}}_{n}^{-1}\bm{\Delta}_{n}({\hat{\bm{\beta}}})$
$\displaystyle=\|\hat{\bm{\gamma}}\|_{1}.$ (32)

Then the Karush-Kuhn-Tucker (KKT) conditions for (12) are fulfilled and
${\hat{\bm{\beta}}}$ and $\hat{\bm{\gamma}}$ are the optimal primal and dual
solutions.

###### Proof.

We start by writing the Karush-Kuhn-Tucker (KKT) conditions for the problem
(12):

$\displaystyle{\bm{\Lambda}}_{n}\bm{\beta}$ $\displaystyle\leq{\bm{u}}$ (33)
$\displaystyle-{\bm{\Lambda}}_{n}\bm{\beta}$ $\displaystyle\leq{\bm{u}}$ (34)
$\displaystyle\mu_{n}^{-1}{\bm{\Lambda}}_{n}^{-1}{\bm{\Delta}_{n}(\bm{\beta})}-\bm{1}_{p_{n}}$
$\displaystyle\leq{\mathbf{0}}$ (35)
$\displaystyle-\mu_{n}^{-1}{\bm{\Lambda}}_{n}^{-1}{\bm{\Delta}_{n}(\bm{\beta})}-\bm{1}_{p_{n}}$
$\displaystyle\leq{\mathbf{0}}$ (36) $\displaystyle\bm{\alpha}_{1}\geq
0,\bm{\alpha}_{2}$ $\displaystyle\geq 0,$ (37)
$\displaystyle\bm{\alpha}_{3}\geq 0,\bm{\alpha}_{4}$ $\displaystyle\geq 0,$
(38) $\displaystyle\forall
i\;\alpha_{1i}\\{({\bm{\Lambda}}_{n}\bm{\beta})_{i}-u_{i}\\}$
$\displaystyle=0$ (39) $\displaystyle\forall
i\;\alpha_{2i}\\{-({\bm{\Lambda}}_{n}\bm{\beta})_{i}-u_{i}\\}$
$\displaystyle=0$ (40) $\displaystyle\forall
i\;\alpha_{3i}[\mu_{n}^{-1}\\{{\bm{\Lambda}}_{n}^{-1}{\bm{\Delta}_{n}(\bm{\beta})}\\}_{i}-1]$
$\displaystyle=0$ (41) $\displaystyle\forall
i\;\alpha_{4i}[-\mu_{n}^{-1}\\{{(}{\bm{\Lambda}}_{n}^{-1}{\bm{\Delta}_{n}(\bm{\beta})}\\}_{i}-1]$
$\displaystyle=0$ (42) $\displaystyle 1-\bm{\alpha}_{1}-\bm{\alpha}_{2}$
$\displaystyle=0$ (43)
$\displaystyle(\bm{\alpha}_{1}-\bm{\alpha}_{2})^{T}{\bm{\Lambda}}_{n}-\mu_{n}^{-1}(\bm{\alpha}_{3}-\bm{\alpha}_{4})^{T}{\bm{\Lambda}}_{n}^{-1}\mathbf{A}_{n}(\tilde{\bm{\beta}})$
$\displaystyle=0$ (44)

Let ${\hat{\bm{\beta}}}$ and $\hat{\bm{\gamma}}$ satisfy (29)-(32). (35) and
(36) are obviously satisfied under (29). If one defines
$\hat{\bm{\alpha}}_{3}$ and $\hat{\bm{\alpha}}_{4}$ such that
$\hat{\alpha}_{3i}=(0,\hat{\gamma}_{i})_{+}$ and
$\hat{\alpha}_{4i}=(0,-\hat{\gamma}_{i})_{+}$, then
$\hat{\bm{\gamma}}=\hat{\bm{\alpha}}_{3}-\hat{\bm{\alpha}}_{4}$ and (38) is
satisfied.

Now, we define

$\displaystyle\hat{\bm{\alpha}}_{1}$
$\displaystyle=\frac{1}{2}\\{\mathbf{1}_{p_{n}}+\mu_{n}^{-1}{\bm{\Lambda}}_{n}^{-1}\mathbf{A}_{n}(\tilde{\bm{\beta}}){\bm{\Lambda}}_{n}^{-1}\\}(\hat{\bm{\alpha}}_{3}-\hat{\bm{\alpha}}_{4})$
$\displaystyle\hat{\bm{\alpha}}_{2}$
$\displaystyle=\frac{1}{2}\\{\mathbf{1}_{p_{n}}-\mu_{n}^{-1}{\bm{\Lambda}}_{n}^{-1}\mathbf{A}_{n}(\tilde{\bm{\beta}}){\bm{\Lambda}}_{n}^{-1}\\}(\hat{\bm{\alpha}}_{3}-\hat{\bm{\alpha}}_{4})$

which ensures (43) and (44). In addition, under (30) implies that (37) is also
true.

From the definition of $\bm{\alpha}_{3}$ and $\bm{\alpha}_{4}$, we rewrite
(32) as

$\sum_{i}\left(\hat{\alpha}_{3i}[\mu_{n}^{-1}\\{{\bm{\Lambda}}_{n}^{-1}{\bm{\Delta}_{n}(\hat{\bm{\beta}})}\\}_{i}-1]+\hat{\alpha}_{4i}[-\mu_{n}^{-1}\\{{\bm{\Lambda}}_{n}^{-1}{\bm{\Delta}_{n}(\hat{\bm{\beta}})}\\}_{i}-1]\right)=0.$

From (36)-(37) each term in the above sum is the sum of two negative terms,
whereby we dedude that (41)-(42) necessarily hold. With a similar argument, by
using (31), we also deduce that (39)-(40) are also true by setting in
particular $u_{i}=|({\bm{\Lambda}}_{n}\hat{\bm{\beta}})_{i}|$. And that latter
choice implies that (33)-(34) are also satisfied. ∎

###  C.2 A few auxiliary statements

Before tackling more specifically the proof of Theorem 4.1 for the ALDS
estimator, we present a few auxiliary results that will be used.

######  Lemma C.3.

Assume conditions ($\mathcal{C}$.1)-($\mathcal{C}$.7) hold.
(i)

$\|{\bm{\Lambda}}_{n,11}\|=a_{n},\qquad\|{\bm{\Lambda}}_{n,22}^{-1}\|=\frac{1}{b_{n}}.$
(45)

(ii) For any $t\in[0,1]$ and
$\check{\bm{\beta}}=\bm{\beta}_{0}+t(\tilde{\bm{\beta}}-\bm{\beta}_{0})$, we
have

$\displaystyle\mathbf{A}_{n}({\check{\bm{\beta}}})$
$\displaystyle=O_{\mathrm{P}}\left(p_{n}\mu_{n}\right)$
$\displaystyle\mathbf{A}_{n,1}(\check{\bm{\beta}})$
$\displaystyle=O_{\mathrm{P}}\left(\sqrt{p_{n}s_{n}}\mu_{n}\right)$
$\displaystyle\mathbf{A}_{n,2}(\check{\bm{\beta}})$
$\displaystyle=O_{\mathrm{P}}\left(p_{n}\mu_{n}\right)$
$\displaystyle\mathbf{A}_{n,11}(\check{\bm{\beta}})$
$\displaystyle=O_{\mathrm{P}}\left(s_{n}\mu_{n}\right)$
$\displaystyle\mathbf{A}_{n,21}(\check{\bm{\beta}})$
$\displaystyle=O_{\mathrm{P}}\left(\sqrt{p_{n}s_{n}}\mu_{n}\right)$
$\displaystyle\mathbf{A}_{n}(\check{\bm{\beta}})-\mathbf{A}_{n}(\tilde{\bm{\beta}})$
$\displaystyle=O_{\mathrm{P}}\left(p_{n}^{2}\sqrt{\mu_{n}}\right)$
$\displaystyle\mathbf{A}_{n,1}(\check{\bm{\beta}})-\mathbf{A}_{n,1}(\tilde{\bm{\beta}})$
$\displaystyle=O_{\mathrm{P}}\left(\sqrt{s_{n}p_{n}^{3}\mu_{n}}\right)$
$\displaystyle\mathbf{A}_{n,11}(\check{\bm{\beta}})-\mathbf{A}_{n,11}(\tilde{\bm{\beta}})$
$\displaystyle=O_{\mathrm{P}}\left({\sqrt{s_{n}^{2}p_{n}^{2}\mu_{n}}}\right).$

(iii)

$\max\left\\{\|\mathbf{U}_{n}(\tilde{\bm{\beta}})\|,\|\mathbf{U}_{n,2}(\tilde{\bm{\beta}})\|\right\\}=O_{\mathrm{P}}(\sqrt{p_{n}^{3}\mu_{n}}).$
(46)

###### Proof.

(i) follows from conditions on $a_{n}$ and $b_{n}$.
(ii) follows from conditions ($\mathcal{C}$.1)-($\mathcal{C}$.3),
($\mathcal{C}$.7) and Lemma A.2. We only prove the assertions for the matrices
$\mathbf{A}_{n}(\tilde{\bm{\beta}})$ and
$\mathbf{A}_{n}(\check{\bm{\beta}})-\mathbf{A}_{n}(\tilde{\bm{\beta}})$. The
other cases follow along similar lines. First,

$\displaystyle\|\mathbf{A}_{n}(\tilde{\bm{\beta}})\|\leq\int_{D_{n}}\|\mathbf{z}(u)\|^{2}\rho(u;\tilde{\bm{\beta}})\mathrm{d}u=O_{\mathrm{P}}(p_{n}\mu_{n}).$

Second, using Taylor expansion, there exists $\bm{\beta}^{\prime}$ on the
segment between $\tilde{\bm{\beta}}$ and $\check{\bm{\beta}}$ such that
$\rho(u;\check{\bm{\beta}})-\rho(u;\tilde{\bm{\beta}})=(\check{\bm{\beta}}-\tilde{\bm{\beta}})^{\top}\mathbf{z}(u)\rho(u;\bm{\beta}^{\prime})$
whereby we deduce that

$\displaystyle\|\mathbf{A}_{n}(\check{\bm{\beta}})-\mathbf{A}_{n}(\tilde{\bm{\beta}})\|$
$\displaystyle\leq\int_{D_{n}}\|\mathbf{z}(u)\|^{3}\|\check{\bm{\beta}}-\tilde{\bm{\beta}}\|\rho(u;\bm{\beta}^{\prime})\mathrm{d}u$
$\displaystyle=O_{\mathrm{P}}\left(\mu_{n}p_{n}^{3/2}\|\tilde{\bm{\beta}}-\bm{\beta}_{0}\|\right)=O_{\mathrm{P}}\left(p_{n}^{2}\sqrt{\mu_{n}}\right).$

(iii) We only have to prove it for $\|\mathbf{U}_{n}(\tilde{\bm{\beta}})\|$.
Using Taylor expansion, there exists $\check{\bm{\beta}}$ such that
$\mathbf{U}_{n}(\tilde{\bm{\beta}})=\mathbf{U}_{n}(\bm{\beta}_{0})-\mathbf{A}_{n}(\check{\bm{\beta}})(\tilde{\bm{\beta}}-\bm{\beta}_{0})$.
Using Lemma A.1, (ii) and condition ($\mathcal{C}$.7), we obtain

$\displaystyle\|\mathbf{U}_{n}(\tilde{\bm{\beta}})\|=O_{\mathrm{P}}\left(\sqrt{p_{n}\mu_{n}}+p_{n}\mu_{n}\sqrt{p_{n}/\mu_{n}}\right)=O_{\mathrm{P}}\left(\sqrt{p_{n}^{3}\mu_{n}}\right).$

∎

###  C.3 Sparsity property for
$\hat{\bm{\beta}}=\hat{\bm{\beta}}_{\mathrm{ALDS}}$

The sparsity property of $\hat{\bm{\beta}}_{\mathrm{ALDS}}$ follows from the
following Lemma.

######  Lemma C.4.

Let $\hat{\bm{\beta}}_{\mathrm{ALDS}}$ and $\hat{\bm{\gamma}}$ satisfy the
following conditions

$\displaystyle\hat{\bm{\beta}}_{\mathrm{ALDS,1}}$
$\displaystyle=\mathbf{A}_{n,11}(\tilde{\bm{\beta}})^{-1}\Big{\\{}\mathbf{U}_{n,1}(\tilde{\bm{\beta}})+\mathbf{A}_{n,1}(\tilde{\bm{\beta}})\tilde{\bm{\beta}}-\mu_{n}{\bm{\Lambda}}_{n,11}\operatorname{sign}(\hat{\bm{\gamma}}_{1})\Big{\\}}$
(47) $\displaystyle\hat{\bm{\beta}}_{\mathrm{ALDS,2}}$
$\displaystyle=\mathbf{0}$ (48) $\displaystyle\hat{\bm{\gamma}}_{1}$
$\displaystyle=\mu_{n}{\bm{\Lambda}}_{n,11}\mathbf{A}_{n,11}(\tilde{\bm{\beta}})^{-1}{\bm{\Lambda}}_{n,11}\operatorname{sign}(\hat{\bm{\beta}}_{\mathrm{ALDS,1}}).$
(49) $\displaystyle\hat{\bm{\gamma}}_{2}$ $\displaystyle=\mathbf{0}.$ (50)

Then, under the conditions ($\mathcal{C}$.1)-($\mathcal{C}$.9), the following
two statements hold.
(i)

$\displaystyle\hat{\bm{\beta}}_{\mathrm{ALDS,1}}-\bm{\beta}_{01}$
$\displaystyle=\mathbf{A}_{n,11}(\bm{\beta}_{0})^{-1}\mathbf{U}_{n,1}(\bm{\beta}_{0})+o_{\mathrm{P}}\left(\frac{1}{s_{n}\sqrt{\mu_{n}}}\right)=O_{\mathrm{P}}\left(\sqrt{\frac{s_{n}}{\mu_{n}}}\right).$

(ii) With probability tending to 1, $\hat{\bm{\beta}}_{\mathrm{ALDS}}$ and
$\hat{\bm{\gamma}}$ given by (47)-(50) satisfy conditions (29)-(32) and are
thus the primal and dual optimal solutions (whence the notation
$\hat{\bm{\beta}}_{\mathrm{ALDS}}$).

It is worth mentioning that the rate $o_{\mathrm{P}}(1/s_{n}\sqrt{\mu_{n}})$
in Lemma C.4 (i) is required to derive the central limit theorem proved in
Appendix C.4. That required rate of convergence imposes some stronger
restriction on the sequence $a_{n}$.

###### Proof.

(i) Using Taylor expansion, there exists $\check{\bm{\beta}}$ on the line
segment between $\tilde{\bm{\beta}}$ and $\bm{\beta}_{0}$ such that
$\mathbf{U}_{n,1}(\tilde{\bm{\beta}})=\mathbf{U}_{n,1}(\bm{\beta}_{0})-\mathbf{A}_{n,1}(\check{\bm{\beta}})(\tilde{\bm{\beta}}-\bm{\beta}_{0})$
which leads, by noticing that $\bm{\beta}_{02}=0$, to

$\displaystyle\hat{\bm{\beta}}_{\mathrm{ALDS,1}}-\bm{\beta}_{01}=$
$\displaystyle\mathbf{A}_{n,11}(\tilde{\bm{\beta}})^{-1}\bigg{[}\left\\{\mathbf{A}_{n,1}(\tilde{\bm{\beta}})-\mathbf{A}_{n,1}(\check{\bm{\beta}})\right\\}\left\\{\tilde{\bm{\beta}}-\bm{\beta}_{0}\right\\}$
$\displaystyle+\mathbf{U}_{n,1}(\bm{\beta}_{0})-\mu_{n}{\bm{\Lambda}}_{n,11}\operatorname{sign}(\hat{\bm{\gamma}}_{1})\bigg{]}.$

Condition ($\mathcal{C}$.3) ensures that
$\|\mathbf{A}_{n,11}(\tilde{\bm{\beta}})^{-1}\|=O_{\mathrm{P}}(\mu_{n}^{-1})$.
Let
$\hat{\bm{\beta}}_{\mathrm{ALDS,1}}-\bm{\beta}_{01}=\mathbf{A}_{n,11}(\bm{\beta}_{0})^{-1}\mathbf{U}_{n,1}(\bm{\beta}_{0})+T_{1}+T_{2}+T_{3}$
where

$\displaystyle T_{1}$
$\displaystyle=\left\\{\mathbf{A}_{n,11}(\tilde{\bm{\beta}})^{-1}-\mathbf{A}_{n,11}(\bm{\beta}_{0})^{-1}\right\\}\mathbf{U}_{n,1}(\bm{\beta}_{0})$
$\displaystyle T_{2}$
$\displaystyle=\mathbf{A}_{n,11}(\tilde{\bm{\beta}})^{-1}\left\\{\mathbf{A}_{n,1}(\tilde{\bm{\beta}})-\mathbf{A}_{n,1}(\check{\bm{\beta}})\right\\}\left\\{\tilde{\bm{\beta}}-\bm{\beta}_{0}\right\\}$
$\displaystyle T_{3}$
$\displaystyle=\mu_{n}\mathbf{A}_{n,11}(\tilde{\bm{\beta}})^{-1}{\bm{\Lambda}}_{n,11}\operatorname{sign}(\hat{\bm{\gamma}}_{1}).$

Regarding the term $T_{1}$ we have

$T_{1}=\mathbf{A}_{n,11}(\tilde{\bm{\beta}})^{-1}\left\\{\mathbf{A}_{n,11}(\bm{\beta}_{0})-\mathbf{A}_{n,11}(\tilde{\bm{\beta}})\right\\}\mathbf{A}_{n,11}(\bm{\beta}_{0})^{-1}\mathbf{U}_{n,1}(\bm{\beta}_{0}).$

Condition ($\mathcal{C}$.3) ensures that
$\max(\|\mathbf{A}_{n,11}(\tilde{\bm{\beta}})^{-1}\|,\|\mathbf{A}_{n,11}(\bm{\beta}_{0})^{-1}\|)=O_{\mathrm{P}}(\mu_{n}^{-1})$.
Using this, Lemma C.3 and Lemma A.1 we obtain

$T_{1}=O_{\mathrm{P}}\left(\frac{1}{\mu_{n}}\sqrt{s_{n}^{2}p_{n}^{2}\mu_{n}}\frac{1}{\mu_{n}}\sqrt{s_{n}\mu_{n}}\right)=O_{\mathrm{P}}\left(\frac{\sqrt{s_{n}^{3}p_{n}^{2}}}{\mu_{n}}\right).$

With similar arguments, we have

$T_{2}=O_{\mathrm{P}}\left(\frac{1}{\mu_{n}}\sqrt{s_{n}p_{n}^{3}\mu_{n}}\sqrt{p_{n}/\mu_{n}}\right)=O_{\mathrm{P}}\left(\frac{\sqrt{s_{n}p_{n}^{4}}}{\mu_{n}}\right).$

Condition ($\mathcal{C}$.8) ensures that

$T_{1}+T_{2}=O_{\mathrm{P}}\left(\frac{\sqrt{s_{n}p_{n}^{4}}}{\mu_{n}}\right)=o_{\mathrm{P}}\left(\frac{1}{s_{n}\sqrt{\mu_{n}}}\right).$

Now, regarding the last term

$T_{3}=O_{\mathrm{P}}\left(\mu_{n}\frac{1}{\mu_{n}}a_{n}\sqrt{s_{n}}\right)=O_{\mathrm{P}}(a_{n}\sqrt{s_{n}}).$

And we observe that condition ($\mathcal{C}$.9) is sufficient to establish
that $T_{3}=o_{\mathrm{P}}(1/s_{n}\sqrt{\mu_{n}})$ which proves (i) using
again condition ($\mathcal{C}$.3) and Lemma A.1.

(ii) We have to show that with probability tending to 1,
$\hat{\bm{\beta}}_{\mathrm{ALDS}}$ and $\hat{\bm{\gamma}}$ given by (47)-(50)
satisfy conditions (29)-(32). By (47)-(50),

$\displaystyle\mu_{n}^{-1}\hat{\bm{\gamma}}^{\top}{\bm{\Lambda}}_{n}^{-1}\mathbf{A}_{n}(\tilde{\bm{\beta}})\hat{\bm{\beta}}_{\mathrm{ALDS}}$
$\displaystyle=\mu_{n}^{-1}\hat{\bm{\gamma}}_{1}^{\top}{\bm{\Lambda}}_{n,11}^{-1}\mathbf{A}_{n,11}(\tilde{\bm{\beta}})\hat{\bm{\beta}}_{\mathrm{ALDS},1}$
$\displaystyle=\operatorname{sign}(\hat{\bm{\beta}}_{\mathrm{ALDS,1}})^{\top}{\bm{\Lambda}}_{n,11}\big{\\{}\mathbf{A}_{n,11}(\tilde{\bm{\beta}})\big{\\}}^{-1}{\bm{\Lambda}}_{n,11}{\bm{\Lambda}}_{n,11}^{-1}\mathbf{A}_{n,11}(\tilde{\bm{\beta}})\hat{\bm{\beta}}_{\mathrm{ALDS,1}}$
$\displaystyle=\operatorname{sign}(\hat{\bm{\beta}}_{\mathrm{ALDS,1}})^{\top}{\bm{\Lambda}}_{n,11}\hat{\bm{\beta}}_{\mathrm{ALDS,1}}$
$\displaystyle=\
\|{\bm{\Lambda}}_{n,11}\hat{\bm{\beta}}_{\mathrm{ALDS,1}}\|_{1}=\|{\bm{\Lambda}}_{n}\hat{\bm{\beta}}_{\mathrm{ALDS}}\|_{1},$

so, (31) is satisfied. Now, we want to show that (32) holds. We have

$\displaystyle\mu_{n}^{-1}\hat{\bm{\gamma}}^{\top}{\bm{\Lambda}}_{n}^{-1}\big{\\{}\mathbf{U}_{n}(\tilde{\bm{\beta}})+\mathbf{A}_{n}(\tilde{\bm{\beta}})(\tilde{\bm{\beta}}-\hat{\bm{\beta}}_{\mathrm{ALDS}})\big{\\}}=\mathbf{I}+\mathbf{II},$

where

$\displaystyle\mathbf{I}=\;$
$\displaystyle\mu_{n}^{-1}\hat{\bm{\gamma}}^{\top}{\bm{\Lambda}}_{n}^{-1}\mathbf{U}_{n}(\tilde{\bm{\beta}})=\mu_{n}^{-1}\hat{\bm{\gamma}}_{1}^{\top}{\bm{\Lambda}}_{n,11}^{-1}\mathbf{U}_{n,1}(\tilde{\bm{\beta}}),$
$\displaystyle\mathbf{II}=\;$
$\displaystyle\mu_{n}^{-1}\hat{\bm{\gamma}}_{1}^{\top}{\bm{\Lambda}}_{n,11}^{-1}\mathbf{A}_{n,1}(\tilde{\bm{\beta}})\tilde{\bm{\beta}}-\mu_{n}^{-1}\hat{\bm{\gamma}}_{1}^{\top}{\bm{\Lambda}}_{n,11}^{-1}\mathbf{A}_{n,11}(\tilde{\bm{\beta}})\hat{\bm{\beta}}_{\mathrm{ALDS,1}}$
$\displaystyle=\;$
$\displaystyle\mu_{n}^{-1}\hat{\bm{\gamma}}_{1}^{\top}{\bm{\Lambda}}_{n,11}^{-1}\mathbf{A}_{n,1}(\tilde{\bm{\beta}})\tilde{\bm{\beta}}$
$\displaystyle-\mu_{n}^{-1}\hat{\bm{\gamma}}_{1}^{\top}{\bm{\Lambda}}_{n,11}^{-1}\\{\mathbf{U}_{n,1}(\tilde{\bm{\beta}})+\mathbf{A}_{n,1}(\tilde{\bm{\beta}})\tilde{\bm{\beta}}-\mu_{n}{\bm{\Lambda}}_{n,11}\operatorname{sign}(\hat{\bm{\gamma}}_{1})\\}$
$\displaystyle=\;$
$\displaystyle\hat{\bm{\gamma}}_{1}^{\top}\operatorname{sign}(\hat{\bm{\gamma}}_{1})-\mu_{n}^{-1}\hat{\bm{\gamma}}_{1}^{\top}{\bm{\Lambda}}_{n,11}^{-1}\mathbf{U}_{n,1}(\tilde{\bm{\beta}}),$

from (47)-(50). By summing $\mathbf{I}$ and $\mathbf{II}$, we deduce that (32)
holds.

To prove (30) holds, we use (50) and decompose the vector
$\mu_{n}^{-1}{\bm{\Lambda}}_{n}^{-1}\mathbf{A}_{n}(\tilde{\bm{\beta}}){\bm{\Lambda}}_{n}^{-1}\hat{\bm{\gamma}}$
as

$\displaystyle\mu_{n}^{-1}{\bm{\Lambda}}_{n}^{-1}\mathbf{A}_{n}(\tilde{\bm{\beta}}){\bm{\Lambda}}_{n}^{-1}\hat{\bm{\gamma}}=\mu_{n}^{-1}\begin{bmatrix}\mathbf{I}^{\prime}\\\
\mathbf{II}^{\prime}\end{bmatrix}=\mu_{n}^{-1}\begin{bmatrix}{\bm{\Lambda}}_{n,11}^{-1}\mathbf{A}_{n,11}(\tilde{\bm{\beta}}){\bm{\Lambda}}_{n,11}^{-1}\hat{\bm{\gamma}_{1}}\\\
{\bm{\Lambda}}_{n,22}^{-1}\mathbf{A}_{n,21}(\tilde{\bm{\beta}}){\bm{\Lambda}}_{n,11}^{-1}\hat{\bm{\gamma}_{1}}\end{bmatrix}.$

By (49)

$\displaystyle\mu_{n}^{-1}\|\mathbf{I}^{\prime}\|_{\infty}$
$\displaystyle=\mu_{n}^{-1}\|{\bm{\Lambda}}_{n,11}^{-1}\mathbf{A}_{n,11}(\tilde{\bm{\beta}}){{\bm{\Lambda}}_{n,11}^{-1}}\hat{\bm{\gamma}}_{1}\|_{\infty}$
$\displaystyle=\|\operatorname{sign}(\hat{\bm{\beta}}_{\mathrm{ALDS,1}})\|_{\infty}=1.$

Regarding $\mathbf{II}^{\prime}$, by (49), conditions on $a_{n}$ and $b_{n}$,
conditions ($\mathcal{C}$.1)-($\mathcal{C}$.3), ($\mathcal{C}$.7) and Lemma
C.3(i)-(ii), we have

$\displaystyle\mu_{n}^{-1}\mathbf{II}^{\prime}$
$\displaystyle=\mu_{n}^{-1}{\bm{\Lambda}}_{n,22}^{-1}\mathbf{A}_{n,21}(\tilde{\bm{\beta}}){\bm{\Lambda}}_{n,11}^{-1}\hat{\bm{\gamma}}_{1}$
$\displaystyle={\bm{\Lambda}}_{n,22}^{-1}\mathbf{A}_{n,21}(\tilde{\bm{\beta}}){{\bm{\Lambda}}_{n,11}^{-1}}{\bm{\Lambda}}_{n,11}\mathbf{A}_{n,11}(\tilde{\bm{\beta}})^{-1}{\bm{\Lambda}}_{n,11}\operatorname{sign}(\hat{\bm{\beta}}_{\mathrm{ALDS,1}})$
$\displaystyle={\bm{\Lambda}}_{n,22}^{-1}\mathbf{A}_{n,21}(\tilde{\bm{\beta}})\mathbf{A}_{n,11}(\tilde{\bm{\beta}})^{-1}{\bm{\Lambda}}_{n,11}\operatorname{sign}(\hat{\bm{\beta}}_{\mathrm{ALDS,1}})$
$\displaystyle=O_{\mathrm{P}}\left(\frac{1}{b_{n}}\sqrt{p_{n}s_{n}}\mu_{n}\frac{1}{\mu_{n}}a_{n}\sqrt{s_{n}}\right)=O_{\mathrm{P}}\left(\frac{a_{n}\sqrt{s_{n}^{2}p_{n}}}{b_{n}}\right)$
$\displaystyle=O_{\mathrm{P}}\left(a_{n}\sqrt{s_{n}^{3}\mu_{n}}\frac{1}{b_{n}}\sqrt{\frac{p_{n}^{3}}{\mu_{n}}}\,\frac{1}{s_{n}p_{n}}\right).$

Hence, $\mu_{n}^{-1}\|\mathbf{II}^{\prime}\|_{\infty}=o_{\mathrm{P}}(1)$ by
condition ($\mathcal{C}$.8) and (30) is satisfied with probability tending to
1. We finally focus on (29). Note that

$\displaystyle\mu_{n}^{-1}{\bm{\Lambda}}_{n}^{-1}\\{\mathbf{U}_{n}(\tilde{\bm{\beta}})$
$\displaystyle+\mathbf{A}_{n}(\tilde{\bm{\beta}})(\tilde{\bm{\beta}}-\hat{\bm{\beta}}_{\mathrm{ALDS}})\\}=\mu_{n}^{-1}\begin{bmatrix}\tilde{\mathbf{I}}\\\
\tilde{\mathbf{II}}\end{bmatrix}$
$\displaystyle=\mu_{n}^{-1}\begin{bmatrix}{\bm{\Lambda}}_{n,11}^{-1}\\{\mathbf{U}_{n,1}(\tilde{\bm{\beta}})+\mathbf{A}_{n,1}(\tilde{\bm{\beta}})(\tilde{\bm{\beta}}-\hat{\bm{\beta}}_{\mathrm{ALDS}})\\}\\\
{\bm{\Lambda}}_{n,22}^{-1}\\{\mathbf{U}_{n,2}(\tilde{\bm{\beta}})+\mathbf{A}_{n,2}(\tilde{\bm{\beta}})(\tilde{\bm{\beta}}-\hat{\bm{\beta}}_{\mathrm{ALDS}})\\}\end{bmatrix}.$

Regarding $\tilde{\mathbf{I}}$, from (47)-(48),

$\displaystyle\mu_{n}^{-1}\|\tilde{\mathbf{I}}\|_{\infty}=$
$\displaystyle\mu_{n}^{-1}\|{\bm{\Lambda}}_{n,11}^{-1}\mathbf{U}_{n,1}(\tilde{\bm{\beta}})+{\bm{\Lambda}}_{n,11}^{-1}\mathbf{A}_{n,1}(\tilde{\bm{\beta}})\tilde{\bm{\beta}}-{\bm{\Lambda}}_{n,11}^{-1}\mathbf{A}_{n,11}(\tilde{\bm{\beta}})\hat{\bm{\beta}}_{\mathrm{ALDS,1}}\|_{\infty}$
$\displaystyle=$
$\displaystyle\mu_{n}^{-1}\|{\bm{\Lambda}}_{n,11}^{-1}\mathbf{U}_{n,1}(\tilde{\bm{\beta}})+{\bm{\Lambda}}_{n,11}^{-1}\mathbf{A}_{n,1}(\tilde{\bm{\beta}})\tilde{\bm{\beta}}$
$\displaystyle-{\bm{\Lambda}}_{n,11}^{-1}\\{\mathbf{U}_{n,1}(\tilde{\bm{\beta}})+\mathbf{A}_{n,1}(\tilde{\bm{\beta}})\tilde{\bm{\beta}}-\mu_{n}{\bm{\Lambda}}_{n,11}\operatorname{sign}(\hat{\bm{\gamma}}_{1})\\}\|_{\infty}$
$\displaystyle=$
$\displaystyle\|\operatorname{sign}(\hat{\bm{\gamma}}_{1})\|_{\infty}=1.$

Now, consider $\tilde{\mathbf{II}}$. By the sparsity of
$\hat{\bm{\beta}}_{\mathrm{ALDS}}$ and $\bm{\beta}_{0}$ we can write

$\mu_{n}^{-1}\tilde{\mathbf{II}}=\mu_{n}^{-1}\bm{\Lambda}_{n,22}^{-1}\left\\{\mathbf{U}_{n,2}(\tilde{\bm{\beta}})+\mathbf{A}_{n,2}(\tilde{\bm{\beta}})(\tilde{\bm{\beta}}-\bm{\beta}_{0})+\mathbf{A}_{n,21}(\tilde{\bm{\beta}})(\bm{\beta}_{01}-\hat{\bm{\beta}}_{\mathrm{ALDS,1}})\right\\}.$

We combine Lemma C.3 (i)-(iii) and Lemma B.2 to derive

$\displaystyle\mu_{n}^{-1}\tilde{\mathbf{II}}$
$\displaystyle=O_{\mathrm{P}}\left\\{\frac{1}{\mu_{n}}\frac{1}{b_{n}}\left(\sqrt{p_{n}\mu_{n}}+p_{n}\mu_{n}\sqrt{\frac{p_{n}}{\mu_{n}}}+\sqrt{p_{n}s_{n}}\mu_{n}\sqrt{\frac{s_{n}}{\mu_{n}}}\right)\right\\}$
$\displaystyle=O_{\mathrm{P}}\left(\frac{1}{b_{n}}\sqrt{\frac{p_{n}^{3}}{\mu_{n}}}\right)=o_{\mathrm{P}}(1)$

by condition ($\mathcal{C}$.9). Hence,
$\mu_{n}^{-1}\|\tilde{\mathbf{II}}\|_{\infty}=o_{\mathrm{P}}(1)$ and (29) is
satisfied with probability tending to 1. ∎

###  C.4 Asymptotic normality for
$\hat{\bm{\beta}}=\hat{\bm{\beta}}_{\mathrm{ALDS}}$

###### Proof.

By Lemma C.3,
$\mathbf{A}_{n,11}(\bm{\beta}_{0})=O_{\mathrm{P}}(s_{n}\mu_{n})$. This and
Lemma C.4 show that

$\mathbf{A}_{n,11}(\bm{\beta}_{0})\left(\hat{\bm{\beta}}_{\mathrm{ALDS,1}}-\bm{\beta}_{01}\right)=\mathbf{U}_{n,1}(\bm{\beta}_{0})+o_{\mathrm{P}}(\sqrt{\mu_{n}}).$

Let $\bm{\phi}\in\mathbb{R}^{s_{n}}\setminus\\{0\\}$ with
$\|\bm{\phi}\|<\infty$ and let
$\sigma^{2}_{\bm{\phi}}=\bm{\phi}^{\top}\mathbf{B}_{n,11}(\bm{\beta}_{0})\bm{\phi}$.
Now

$\sigma_{\bm{\phi}}^{-1}\bm{\phi}^{\top}\mathbf{A}_{n,11}(\bm{\beta}_{0})\left(\hat{\bm{\beta}}_{\mathrm{ALDS,1}}-\bm{\beta}_{01}\right)=\sigma_{\bm{\phi}}^{-1}\bm{\phi}^{\top}\mathbf{U}_{n,1}(\bm{\beta}_{0})+\sigma_{\bm{\phi}}^{-1}\bm{\phi}^{\top}o_{\mathrm{P}}(\sqrt{\mu_{n}}).$

By condition ($\mathcal{C}$.5), $\sigma_{\bm{\phi}}^{-1}=O(1/\sqrt{\mu_{n}})$.
The result is therefore deduced from condition ($\mathcal{C}$.6) and Slutsky’s
theorem. ∎

##  Appendix D Resulting $\bm{\beta}$ estimates for the BCI dataset

Table 5: Values of $\hat{\bm{\beta}}_{\mathrm{AL}}$ and $\hat{\bm{\beta}}_{\mathrm{ALDS}}$ for the real data example | Int | elev | grad | Al | B | Ca | Cu | Fe | K | Mg | Mn | P | Zn | N
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
AL | -6.252 | 0 | 0 | 0 | -0.461 | 0.448 | 0 | 0.260 | 0 | 0 | -0.244 | 0 | 0 | 0
ALDS | -6.245 | 0 | 0 | 0 | -0.429 | 0.419 | 0 | 0.245 | 0 | 0 | -0.235 | 0 | 0 | 0
| N.min | pH | AlB | AlCa | AlCu | AlFe | AlK | AlMg | AlMn | AlP | AlZn | AlN | AlN.min | AlpH
AL | 0.076 | 0.477 | -0.162 | 0 | 0 | 0.379 | 0 | -0.514 | 0 | -0.022 | 0 | 0.103 | 0 | -0.033
ALDS | 0.077 | 0.464 | -0.221 | 0 | 0 | 0.394 | 0 | -0.471 | 0 | -0.008 | 0 | 0.103 | 0 | 0
| BCa | BCu | BFe | BK | BMg | BMn | BP | BZn | BN | BN.min | BpH | CaCu | CaFe | CaK
AL | 0 | 0 | 0.152 | 0 | 0 | 0 | -0.299 | 0 | -0.071 | 0 | 0 | 0 | 0.144 | 0
ALDS | 0 | -0.093 | 0.183 | 0 | 0 | 0 | -0.286 | -0.080 | -0.027 | 0.053 | 0 | 0 | 0.142 | 0
| CaMg | CaMn | CaP | CaZn | CaN | CaN.min | CapH | CuFe | CuK | CuMg | CuMn | CuP | CuZn | CuN
AL | -0.155 | 0 | 0.104 | 0 | 0 | -0.888 | 0 | 0 | -0.091 | 0 | 0.134 | 0.148 | 0 | 0
ALDS | -0.125 | 0 | 0.095 | 0 | 0 | -0.890 | 0.042 | 0 | -0.013 | 0 | 0.130 | 0.148 | 0 | 0
| CuN.min | CupH | FeK | FeMg | FeMn | FeP | FeZn | FeN | FeN.min | FepH | KMg | KMn | KP | KZn
AL | 0 | 0 | -0.311 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -0.051
ALDS | 0 | 0 | -0.331 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
| KN | KN.min | KpH | MgMn | MgP | MgZn | MgN | MgN.min | MgpH | MnP | MnZn | MnN | MnN.min | MnpH
AL | 0.198 | 0.580 | 0.023 | 0 | 0 | -0.011 | 0 | 0 | 0 | 0 | 0 | -0.050 | 0.107 | 0
ALDS | 0.161 | 0.530 | 0 | -0.003 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -0.047 | 0.100 | 0
| PZn | PN | PN.min | PpH | ZnN | ZnN.min | ZnpH | NN.min | NpH | N.minpH | elevgrad |  |  |
AL | 0 | 0.269 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.054 | 0 |  |  |
ALDS | 0 | 0.258 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.054 | 0 |  |  |

## Acknowledgements

We thank the editor, associate editor, and two reviewers for the constructive
comments. The research of J.-F. Coeurjolly is supported by the Natural
Sciences and Engineering Research Council of Canada. J.-F. Coeurjolly would
like to thank Université du Québec à Montréal for the excellent research
conditions he received these last years. The research of A. Choiruddin is
supported by the Direktorat Riset, Teknologi, dan Pengabdian Kepada
Masyarakat, Direktorat Jenderal Pendidikan Tinggi, Riset, dan Teknologi,
Kementerian Pendidikan, Kebudayaan, Riset, dan Teknologi Republik Indonesia.
The BCI soils data sets were collected and analyzed by J. Dalling, R. John, K.
Harms, R. Stallard and J. Yavitt with support from NSF DEB021104,021115,
0212284,0212818 and OISE 0314581, and STRI Soils Initiative and CTFS and
assistance from P. Segre and J. Trani.

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