\section{Introduction}
Mixture models are a flexible tool for modeling heterogeneous data, e.g. from a population consisting of multiple hidden  homogeneous subpopulations. Finite mixture models are models containing distribution of the form
\begin{equation}
\label{eq:p_w_f}
P_{w,F} = \suml_{k=1}^K w_k F_k,
\end{equation}
where $K\geq 2$, each $F_k$ belongs to a specific class of probability distributions (e.g. normal distributions in the case of Gaussian mixture models) and $w$ belongs to the simplex $\mcal{W}_K = \left\{ w \in [0,1]^K; w_1 + \dots + w_k = 1 \right\}$. For a complete introduction to mixture models and an overview of the different applications we refer to the books of Mclachlan \& Peel \cite{maclachlan} and Frühwirth-Schnatter \cite{fruhwirth}.\par
Assume we dispose of a sample $\mbf{X} := (X_1,\dots,X_n)$ of i.i.d. data, each coordinate following the probability distribution $P^*$. The majority of the statistical methods based on finite mixture models aim to solve one of the following problems: density estimation (estimation of $P^*$), parameter estimation (estimation of $w^*$ and/or $F^*$ assuming $P^*=P_{w^*,F^*}$) and clustering. The monographs of Everitt \& Hand \cite{everitt} or Titterington \emph{et al} \cite{titterington1985statistical} provide a good overview of the different estimation methods that have been developed for mixture models such as maximum likelihood, minimum chi-square, moments method and Bayesian approaches. Although algorithms are numerous, theoretical guarantees are mostly asymptotic and restricted to very specific situations. To our knowledge, only a few non-asymptotic results have been established in the case of density estimation based on Gaussian Mixture Models (GMMs). The approximation and entropy properties of Gaussian mixture sieves have been investigated by Kruijer \emph{et al} \cite{kruijer}, Ghosal \& van der Vaart \cite{ghosal2001} and Genovese \& Wasserman \cite{genovese} where bounds on the convergence rate are given for the MLE and Bayesian estimators. Similarly, Maugis \& Michel \cite{maugis2011} use a penalized version of the MLE to build a Gaussian mixture estimator with non asymptotic adaptive properties proven in \cite{maugis2012}. However, those results rely on relatively strong assumptions and estimators are not proved to be robust to small departures from those assumptions.\\
This paper aims to provide non-asymptotic results in a very general setting. In our framework, the data are assumed to be independent but not necessarily i.i.d. Our mixture model consists of probabilities of the form (\ref{eq:p_w_f}) where the $F_k$ admit densities, called \emph{emission densities}, that belong to classes of function that are VC-subgraph. We investigate the performances of $\rho$-estimators, as defined by Baraud and Birgé \cite{baraudrevisited},  on finite mixture models. Our main result is an exponential deviation inequality  for the risk of the estimator $\hat{P}$, which is measured with an Hellinger-type loss. We get an upper bound on the risk that is the sum of two terms. The first one is an approximation term which provides a measure of the distance between the true distribution of the data and our mixture model. The second term is a complexity term that depends on the classes containing the emission densities and which is proportional to the sum of their VC-indices. We deduce from this deviation bound that the estimator is not only robust with respect to model misspecification but also to contamination and the presence of outliers among the data set. Dealing with models that may be approximate allows to build estimators that possess properties over wider classes of distribution. Ghosal \& Van der Vaart \cite{ghosal2001} used finite location-scale Gaussian mixtures to approximate continuous Gaussian mixtures with compactly supported mixing distribution. They consider mixtures with scale parameters lying between two constants that depend on the true distribution. By using a similar approximation, we show that our estimator achieves the same rate of convergence but without any restriction on the scale parameters so that the model we consider does not depend on the true mixing distribution. In particular, our result is insensitive to translation or rescaling. \par
Under suitable identifiability assumptions and when the distribution of the data belongs to our model, hence is of the form (\ref{eq:p_w_f}), we also analyze the performance of our estimators of the parameters $w_1,\dots,w_K$ and $F_1,\dots,F_K$. In order to establish convergence rates, we relate the Hellinger distance between the distribution of the data and its estimator to a suitable distance between the corresponding parameters. A general technique is using Fisher's information and results of Ibragimov \& Has’minski\u{\i} \cite{Ibragimov} for regular parametric models. We can also use other results specific to parameter estimation in mixture models such as what Gadat \emph{et al} \cite{gadat2018parameter} proved in the context of two component mixtures with one known component. In both situations, we obtain, up to a logarithmic parameter, the usual $1/\sqrt{n}$-rate of convergence for regular parametric models. We also provide the example of a parametric model for which our technics allow us to establish faster convergence rates while classical methods based on the likelihood or the least-squares fail to apply and hence give nothing.\par
In many applications, starting with a single mixture model may be restrictive and a more reasonable approach is to consider candidate ones for estimating the number of components of the mixture and proposing suitable models for the emission densities.  To tackle this problem, we design a model selection procedure from which we establish, under suitable assumptions, an oracle-type inequality. We consider several illustrations of this strategy. For example, we use a penalized estimator to select the number of components of a Gaussian mixture estimator and obtain similar adaptivity results as Maugis \& Michel \cite{maugis2012}. We also consider a model with a fixed number of components but each emission density can either belong to the Gaussian or to the Cauchy location-scale family. We prove that if we know the number of components, we can estimate consistently the proportions of Gaussian and Cauchy components as well as their location and scale parameters. To our knowledge, this result is the first of its kind.\par
The extension of the theory of $\rho$-estimation to mixture models is based on Proposition \ref{prop:rho_dim_vc} below. The proof of this result relies on an upper bound for the expectation of the supremum of an empirical process over a mixture of VC-subgraph classes. It generalizes the result that was previously established for a single VC-subgraph class. The key argument in the proof is the uniform entropy property of VC-subgraph classes that still holds for the overall density mixture model with lower bounded weights.\par
The paper is organized as follows. We describe our statistical framework in Section \ref{sec:framework}. In Section \ref{sec:main_result}, we present the construction of the estimator on a single mixture model. We state the general result for density estimation on a single model and illustrate the performance of the estimator on the specific example of GMMs. The problem of estimating the parameters of the mixture is addressed in the subsection \ref{sec:param_recover}. Finally, Section \ref{sec:model_selection} is devoted to model selection criterion and the properties of the estimator on the selected model. The appendix contains all the proofs that are gathered in the same sections when they are related. Those sections include the main results, density estimation, the parametric estimation in regular parametric models, the case of two-component mixtures with one known component and the lemmas.
\section{The statistical framework}
\label{sec:framework}
We observe $n$ independent random variables $X_1,X_2,\dots,X_n$ with  respective marginal distributions $P^*_1,P^*_2,\dots,P^*_n$ on the measurable space $(\mscr{X},\mcal{X})$. We model the joint distribution $\mbf{P^*} = P^*_1 \otimes P^*_2 \otimes \dots \otimes P ^*_n$ of $\mbf{X}=(X_1,X_2,\dots,X_n)$ by a probability of the form $\overline{P}^{\otimes n}$ doing as if the observations were i.i.d. with common distribution $\overline{P}$. We assume that $\overline{P}$ is a mixture of the form (\ref{eq:p_w_f}) where $K$ is a positive integer, the $w_k$ some positive weights that satisfy $\sum_{k=1}^K w_k=1$, and $F_k$ probability  distributions. In order to model each of these probabilities we introduce a collection $\left\{ \overline{\mscr{F}}_{k,\lambda}; k\geq 1,\lambda \in \Lambda_k \right\}$ of possible models and assume that for each $k\in\{1,\dots,K\}$, $F_k$ belongs to $\cup_{\lambda\in\Lambda_k} \overline{\mscr{F}}_{k,\lambda}$. We denote by $\mscr{Q}_K$ the family of distributions of the previous form. For each $k\geq 1$, we call $F_k$ an emission probability, $\overline{\mscr{F}}_{k,\lambda}$ an emission model, and $\mscr{E}_k=\left\{ \overline{\mscr{F}}_{k,\lambda}; \lambda\in\Lambda_k\right\}$ an emission family. Based on the observation of $\mbf{X}$, our aim is to design an estimator $\hat{P}$ of $\overline{P}$ of the form
\begin{equation}
\hat{P} = \suml_{k=1}^{\hat{K}} \hat{w}_k \hat{F}_k \in \bigcup_{K\geq 1} \mscr{Q}_K
\end{equation}
where $\hat{K}$, $(\hat{w}_k)_{1\leq k\leq \hat{K}}$ and $(\hat{F}_k)_k$ are estimators of $K$, $(w_k)_k$ and $(F_k)_k$ respectively. The classical situation that has been considered in the literature corresponds to the case where the collection $\left\{ \overline{\mscr{F}}_{k,\lambda}; k\geq 1, \lambda\in\Lambda_k \right\}$ reduces to a single emission model $\mscr{F}$, for example the family of Gaussian distributions, and the problem is to estimate $K$ and the emission probabilities $F_k$ under the assumption that they all belong to $\mscr{F}$. This assumption is quite restrictive and we rather consider a collection $\mscr{E}_k$  of candidate models for $F_k$ that may even depend on $k$. We say that $\mscr{E}_k$ is simple when it reduces to a single emission model $\overline{\mscr{F}}_k$ and composite otherwise.\par
In order to evaluate the performance of the estimator $\hat{P}$, we introduce on the set $\pmb{\mscr{P}}$ of all product probabilities on $(\mscr{X}^n,\mcal{X}^{\otimes n})$ the Hellinger-type distance $\mbf{h}$ defined by
\begin{equation}
\label{eq:h_dist}
\mbf{h}(\mbf{Q},\mbf{Q'}) = \sqrt{ \suml_{i=1}^n h^2(Q_i,Q'_i)}, \quad\text{for } \mbf{Q} = \bigotimes_{i=1}^n Q_i, \mbf{Q'}=\bigotimes_{i=1}^n Q'_i \in\pmb{\mscr{P}},
\end{equation}
where $h$ is the Hellinger distance on the set $\mscr{P}$ of probability distributions on $(\mscr{X},\mcal{X})$. We recall that for $Q$, $Q'$ in $\mscr{P}$
\begin{equation*}
h^2(Q,Q') = \frac{1}{2} \int \left(\sqrt{\frac{dQ}{d\mu}} -\sqrt{\frac{dQ'}{d\mu}}\right)^2 d\mu,
\end{equation*}
where $\mu$ is a measure that dominates both $Q$ and $Q'$, the result being independent of $\mu$. \par
\begin{hypothese}
\label{hyp:countable}
For all $k\geq 1$, the set $\Lambda_k$ is at most countable (which means finite or countable) and that for all $\lambda$ in $\Lambda_k$, $\overline{\mscr{F}}_{k,\lambda}$ contains an at most countable subset $\mscr{F}_{k,\lambda}$ which is dense in $\overline{\mscr{F}}_{k,\lambda}$ with respect to the Hellinger distance $h$.
\end{hypothese}
This condition implies that there exists a $\sigma$-finite measure $\mu$ that dominates all the $\overline{\mscr{F}}_{k,\lambda}$ for $k\geq 1$ and $\lambda\in\Lambda_k$. Throughout this paper, we fix such a measure $\mu$ and associate to each emission model $\overline{\mscr{F}}_{k,\lambda}$ a family of density distributions $\overline{\mcal{F}}_{k,\lambda}$ such that $\overline{\mscr{F}}_{k,\lambda} = \left\{ f \cdot \mu; f\in\overline{\mcal{F}}_{k,\lambda} \right\}$. In all the different examples considered $\mu$ is the Lebesgue measure. We assume the following.
\begin{hypothese}
\label{hyp:vc_density_model}
For all $k\geq 1$ and $\lambda\in\Lambda_k$, the family of density distributions $\overline{\mcal{F}}_{k,\lambda}$ is VC-subgraph with VC-index $V_{k,\lambda}\geq 1$.
\end{hypothese}
For more details on VC-subgraph classes we  refer the reader to Van der Vaart \& Wellner \cite{VanDerVaart} (Section 2.6.5) and Baraud \emph{et al} \cite{baraudinventiones} (Section 8). Throughout this paper we shall use the following notation. For $\mbf{P}=P_1\otimes\dots\otimes P_n \in\pmb{\mscr{P}}$ and $\mscr{A}\subset \mscr{P}$, we write
\begin{equation*}
\mbf{h}^2\left(\mbf{P},\mscr{A}\right) = \inf_{Q\in \mscr{A}} \mbf{h}^2\left(\mbf{P},Q^{\otimes n}\right) = \inf_{Q\in\mscr{A}} \suml_{i=1}^n h^2(P_i,Q).
\end{equation*}
For $x\in\mbb{R}$, $\lfloor x\rfloor$ is the only integer satisfying $\lfloor x\rfloor \leq x < \lfloor x\rfloor +1$ and similarly $\lceil x \rceil$ denotes the integer satisfying $\lceil x\rceil - 1 < x \leq \lceil x\rceil$. Moreover, if $x >0$ we write $\log_+(x)=\log(x)\vee 0$. If $A$ is a finite set, we denote its cardinal by $|A|$ and if $A$ is infinite, we write $|A|=\infty$. The notation $C(\theta)$ will mean that the constant $C=C(\theta)$ depends on the parameter or set of parameters $\theta$.
\section{Estimation on a mixture model based on simple emission families}
\label{sec:main_result}
In this section, we assume that the $\mscr{E}_k=\left\{ \overline{\mscr{F}}_k \right\}$ are simple for all $k\geq 1$ and that $\overline{P}$ belongs to $\mscr{Q}_K$ for some known value of $K\geq 1$. This means that we know that $\overline{P}$ is a mixture of at most $K$ emission probabilities $F_1,\dots,F_K$ and that $F_k$ belongs to $\overline{\mscr{F}}_k$ for all $k\in\{1,\dots,K\}$.  Under Assumption \ref{hyp:vc_density_model}, we denote by $V_k$ the VC-index of $\overline{\mscr{F}}_k$.
\subsection{Construction of the estimator on \texorpdfstring{$\mscr{Q}_K$}{QK}}
For $\delta$ in $(0,1/K]$, we define the subset $\mscr{Q}_{K,\delta}$ of $\mscr{Q}_K$ by
\begin{equation}
\label{eq:mscr_q_delta}
\mscr{Q}_{K,\delta} := \left\{ \suml_{k=1}^K w_k F_k \in\mscr{Q} ; w\in\mcal{W}_K, w_k\geq\delta, w_k\in\mbb{Q}, F_k\in\mscr{F}_k \right\}
\end{equation}
where the $\mscr{F}_k$ are the countable and dense subsets of $\overline{\mscr{F}}_k$ provided by Assumption \ref{hyp:countable}. We associate to $\mscr{Q}_{K,\delta}$ the family $\mcal{Q}_{K,\delta}$ of densities  with respect to $\mu$ and the $\rho$-estimator $\hat{P}_{\delta}$ of $\overline{P}$ based on the family $\mcal{Q}_{K,\delta}$. We recall that $\hat{P}_{\delta}$ is defined as follows. Given 
\begin{equation}
\label{eq:psi_rho}
\psi: \begin{array}{|lcl}
[0,+\infty] & \rightarrow & [-1,1]\\
x & \mapsto & \frac{x-1}{x+1}
\end{array},
\end{equation}
we set for $\mbf{x} = (x_1,...,x_n) \in \mscr{X}^n$ and $q,q'\in \mcal{Q}_{K,\delta}$
\begin{equation}
\label{eq:t}
\mbf{T}(\mbf{x},q,q') := \suml_{k=1}^{n}\psi\left(\sqrt{\frac{q'\left(x_i\right)}{q\left(x_i\right)}}\right),
\end{equation}
with the convention $0/0 = 1$ and $a/0 = +\infty$ for all $a >0$, and
\begin{equation}
\label{eq:upsilon_no_pen}
\mbf{\Upsilon}(\mbf{X},q) := \sup_{q'\in\mcal{Q}_{\delta}} \mbf{T}(\mbf{X},q,q').
\end{equation}
The $\rho$-estimator $\hat{P}_{\delta}$ is any measurable element of the closure (with respect to the Hellinger distance) of the set
\begin{equation}
\label{eq:rho_est}
\pmb{\mscr{E}}(\psi,\mbf{X}) := \left\{ Q = q\cdot\mu ; q\in\mcal{Q}_{\delta}, \mbf{\Upsilon}(\mbf{X},q) < \inf_{q'\in\mcal{Q}_{\delta}} \mbf{\Upsilon}(\mbf{X},q')  + 8.24 \right\}.
\end{equation}
This construction follows \cite{baraudinventiones} and the constant 8.24 is given by the choice of $\psi$.
\subsection{The performance of the estimator}
\label{sec:performance_estimator}
The following result holds.
\begin{theoreme} 
\label{th:single_model}
Let $\delta\in(0,1/K]$ and $\xi>0$. Assume that Assumptions \ref{hyp:countable} and \ref{hyp:vc_density_model} hold and set $\overline{V}=V_1+\dots+V_K$. Any $\rho$-estimator $\hat{P}_{\delta}$ on $\mscr{Q}_{K,\delta}$ satisfies with probability at least $1-e^{-\xi}$,
\begin{align}
\label{eq:th_single_model}
\mbf{h}^2 \left(\mbf{P^*}, \left(\hat{P}_{\delta}\right)^{\otimes n}\right) &\leq c_0 \left[ \mbf{h}^2\left(\mbf{P^*},\mscr{Q}_K\right) + n(K-1)\delta \right]\\
&+ c_1 \overline{V} \left[ 5.82 + \log\left( \frac{(K+1)^2}{\delta} \right) + \log_+ \left( \frac{n}{\overline{V}} \right) \right]\nonumber\\
&+ c_2 (1.49+\xi).\nonumber
\end{align}
where $c_0=300$, $c_1=8.8\times 10^5$ and $c_2=5014$. In particular, for the choice $\delta=1$ for $K=1$ and $\delta=\frac{\overline{V}}{n(K-1)}\bigwedge \frac{1}{K}$ otherwise, the resulting estimator $\hat{P}=\hat{P}_{\delta}$ satisfies 
\begin{equation}
\label{eq:th1_2}
C \mbf{h}^2 \left(\mbf{P^*},\hat{P}^{\otimes n}\right) \leq \mbf{h}^2\left(\mbf{P^*},\mscr{Q}_K\right) +  \overline{V}\left[ 1 + \log\left( \frac{K n}{\overline{V}\wedge n} \right) \right] + \xi,
\end{equation}
with probability at least $1-e^{-\xi}$, where $C$ is a universal constant.
\end{theoreme}
Inequality (\ref{eq:th_single_model}) shows the influence of the choice of the parameter $\delta$ on the performance of the estimator $\hat{P}_{\delta}$. Hereafter, we shall choose $\delta$ as in the second part of Theorem \ref{th:single_model} and therefore only comment on inequality (\ref{eq:th1_2}). Given $\overline{P}$ in $\mscr{Q}_K$, it follows from the triangle inequality and the fact that $(a+b)^2\leq 2a^2 +2b^2$ for all non-negative numbers $a$ and $b$, that
\begin{equation*}
n h^2\left( \overline{P}, \hat{P} \right) =  \mbf{h}^2\left( \overline{P}^{\otimes n}, \hat{P}^{\otimes n} \right) \leq 2 \mbf{h}^2\left( \mbf{P^*}, \hat{P}^{\otimes n}\right) + 2 \mbf{h}^2\left( \mbf{P^*}, \overline{P}^{\otimes n} \right).
\end{equation*}
We immediately derive from (\ref{eq:th1_2}) that on a set of probability at least $1-e^{-\xi}$ 
\begin{equation}
\label{eq:independent}
 C h^2 \left( \overline{P},\hat{P} \right) \leq \frac{1}{n}\suml_{i=1}^n  h^2(P^*_i,\overline{P}) + \frac{\overline{V}\log\left( \slfrac{Kn}{\overline{V}}\right)+\xi}{n} .
\end{equation}
In the ideal situation where the observations are i.i.d. with common distribution $\overline{P}\in\mscr{Q}_K$, we obtain that
\begin{equation*}
 C h^2 \left( \overline{P}, \hat{P} \right) \leq \frac{\overline{V}\log\left( \slfrac{Kn}{\overline{V}}\right)+\xi}{n}.
\end{equation*}
Integrating this result with respect to $\xi$ and the fact that $\overline{P}$ is arbitrary in $\mscr{Q}_K$ leads to the uniform risk bound 
\begin{equation}
\label{eq:simple_esperance}
\sup_{\overline{P}\in\mscr{Q}_K} \mbb{E} \left[ h^2 \left( \overline{P}, \hat{P} \right) \right] \leq C' \frac{\overline{V}\log\left( \slfrac{Kn}{\overline{V}}\right)}{n}.
\end{equation}
where $C'$ is a positive universal constant. This means that up to a logarithmic factor, the estimator $\hat{P}$ uniformly converges over $\mscr{Q}_K$ at the rate $1/\sqrt{n}$ with respect to the Hellinger distance.
Our assumption that the families of density functions $\overline{\mcal{F}}_k$ are VC-subgraph is actually weak since it includes situations where these models consist of unbounded densities or densities which are not in $L_2$ which to our knowledge have never been considered in the literature. A concrete example of such situations is the following one. Let $g$ be some non-increasing function on $(0,+\infty)$ which is unbounded and satisfies $\int_0^{+\infty} g(x) dx = \frac{1}{2}$ and $\overline{\mscr{F}}_k$ is the translation model associated to the family of densities $\left\{ x\mapsto g(|x-\theta|) \mathbbm{1}_{|x-\theta|>0} ; \theta\in\mbb{R} \right\}$ for all $k\in\{1,\dots,K\}$. It follows from Proposition 42-(vi) of Baraud \emph{et al} \cite{baraudinventiones} that the VC-index of $\overline{\mcal{F}}_k$ is not larger than $10$.\par
When the data are independent but not i.i.d., we derive from inequality (\ref{eq:independent}) that the estimator $\hat{P}$ performs almost as well as in the i.i.d. case as long as the marginals $P^*_1,\dots,P^*_n$ are close enough to $\overline{P}$. This means that the estimator is robust with respect to a possible misspecification of the model and the departure from the assumption that the data are i.i.d. In particular, this includes the situations where the dataset contains some outliers or has been contaminated. Consider Hüber’s contamination model where a proportion $\epsilon$ of the data is contaminated, i.e. we have $P^*=(1-\epsilon) \overline{P} + \epsilon Q$, where $\overline{P}$ is the probability distribution we want to estimate and $Q$ is the distribution of the contaminated data. In this situation, for any probability distribution $Q$, using (\ref{eq:independent}) we get
\begin{equation*}
 C h^2 \left( \overline{P},\hat{P} \right) \leq \epsilon + \frac{\overline{V}\log\left( n\right)+\xi}{n}.
\end{equation*}
We can see that there is no perturbation of the convergence rate as long as the contamination rate $\epsilon$ remains small as compared to $\overline{V}\log(n)/n$. Inequality (\ref{eq:hell_mix_parameter}), stated later, also allows to consider misspecification for the emission models for example.
\subsection{The case of totally bounded emission models}
\label{sec:models_with_entropy}
We might also consider emission models for which we do not have any bound on the VC-index. For a subset $\mscr{N}$ of $\mscr{P}$ and $\eta\in[0,1]$, the $\eta$-covering number $N(\eta,\mscr{N},h)$ of $\mscr{N}$, with respect to the Hellinger distance, is the minimum number of balls $\mcal{B}_h(P_i,\eta)$, $i=1,\dots,N$, necessary to cover $\mscr{N}$. In that case, the set $\mscr{N}[\eta]=\{P_i;i=1,\dots,N\}$ constitutes a finite approximation of $\mscr{N}$, i.e. for all $Q$ in $\mscr{N}$ there exists $i\in\{1,\dots,N\}$ such that  $h\left( Q,P_i \right) \leq \eta$. We say that $\mcal{N}$ is totally bounded (for the Hellinger distance) if its $\eta$-covering number is finite for all $\eta\in(0,1]$. A direct consequence of the definition of VC-subgraph classes is that any finite set $\mcal{F}$ of real-valued functions is VC-subgraph with VC-index at most $V(\mcal{F})\leq \log_2 \left( |\mcal{F}| \right)$. Consequently, we can still use $\rho$-estimation for models that are not proven to satisfy Assumption \ref{hyp:vc_density_model} but still are such that emission models are totally bounded.
\begin{theoreme}
\label{th:totally_bounded}
Let $\overline{\mscr{F}}_k$ be a totally bounded class of distributions for all $k\in\{1,\dots,K\}$ with $K\geq 2$. Let $\mscr{Q}$ be the mixture model defined by
\begin{equation*}
\mscr{Q}_K = \left\{ \suml_{k=1}^K w_k F_k ; w\in\mcal{W}_K, F_k \in \overline{\mscr{F}}_k, \forall k \in\{1,\dots,K\} \right\}.
\end{equation*}
Assume there are positive constants $A_k$ and $\alpha_k$ such that $\log_2 N(\eta,\mscr{F}_k,h) \leq \left( \frac{A_k}{\eta} \right)^{\alpha_k}$ for all $k$ in $\{1,\dots,K\}$ and for all $\eta\in(0,1)$. Let $\epsilon$ be in $(0,1)$. For $k$ in $\{1,\dots,K\}$, let $\mscr{F}_k[\epsilon]$ be a minimal $\epsilon$-net of $\overline{\mscr{F}}_k$ such that $|\mscr{F}_k[\epsilon]|=N(\epsilon,\mscr{F}_k,h)$. Let $\mscr{Q}_{K,\delta}[\epsilon]$ be the countable model defined by
\begin{equation*}
\mscr{Q}_{K,\delta}[\epsilon] = \left\{ P_{w,F} ; w\in\mcal{W}_K, w_k\geq \delta, w_k\in\mbb{Q}, F_k\in\mscr{F}_k[\epsilon], \forall k \in\{1,\dots,K\} \right\}.
\end{equation*}
Take $\epsilon = n^{-\frac{1}{\alpha_{\infty}+2}}$ and $\delta=\frac{\overline{V}}{n(K-1)}\wedge \frac{1}{K}$ with $\alpha_{\infty}=\max_{1\leq k\leq K} \alpha_k$. There exists a positive constant $C$ such that for any $\rho$-estimator $\hat{P}=\hat{P}_{\delta}$ on $\mscr{Q}_{K,\delta}[\epsilon]$, for all $\xi>0$, we have
\begin{equation*}
C h^2\left( P^*, \hat{P} \right) \leq h^2\left( P^*, \mscr{Q}_K \right) + n^{-\frac{2}{\alpha_{\infty}+2}} \left( 1 + \suml_{k=1}^K A_k^{\alpha_k} \right) \left[ 1 + \log\left( Kn \right) \right] + \xi,
\end{equation*}
with probability at least $1-e^{-\xi}$.
\end{theoreme}
We illustrate this lemma with the following example. Doss \& Wellner \cite{doss2016} provide a bound on the entropy for classes of $\log$-concave and $s$-concave densities. Let $\mcal{C}=\left\{ \varphi : \mbb{R} \rightarrow [-\infty,\infty); \varphi \text{  is a closed, proper concave function}\right\}$ where \emph{proper} and \emph{closed} are defined in \cite{Rockafellar2015} (Sections 4 and 7). For $0<M<\infty$ and $s>-1$, let $\mcal{P}_{M,s}$ be the class of densities defined by
\begin{equation*}
\mcal{P}_{M,s} = \left\{ p\in\mcal{P}_s ; \sup_{x\in\mbb{R}} p(x) \leq M, 1/M\leq p(x) \text{  for all  } |x|\leq 1 \right\},
\end{equation*} 
where $\mcal{P}_s=\left\{ p : \int pd\lambda =1 \right\} \bigcap h_s \circ\mcal{C}$, $\lambda$ is the Lebesgue measure on $\mbb{R}$ and $h_s:\mbb{R}\rightarrow\mbb{R}$ is defined by
\begin{equation*}
h_s(y) = \begin{cases} e^y, & s=0\\
(-y)_+^{1/s}, &s\in (-1,0),\\
y_+^{1/s}, &s>0.
\end{cases}
\end{equation*}
We fix such values of $M$ and $s$. Let $\mcal{Q}_K$ be the density model of mixtures of $s$-concave densities (or $\log$-concave for $s=0$) defined by
\begin{equation*}
\mcal{Q}_K = \left\{ \suml_{k=1}^K w_k f_k ; w\in\mcal{W}_K, f_k\in\mcal{P}_{M,s} \right\},
\end{equation*}
with $K\geq 2$. Let $\mscr{Q}_K$ be the class of distributions associated to $\mcal{Q}_K$. The class $\mcal{P}_{M,s}$ is not proven to be VC-subgraph but it is totally bounded. As a direct consequence of Theorem 3.1 of Doss \& Wellner \cite{doss2016}, there exists a positive constant $A$ such that for all $\epsilon$ in $(0,1]$, we have 
\begin{equation*}
\log_2 N(\epsilon,\mcal{P}_{M,s},h) \leq A \epsilon^{-1/2}.
\end{equation*}
In particular, it means there exists a $\epsilon$-net $\mcal{P}_{M,s}[\epsilon]$ such that $|\mcal{P}_{M,s}[\epsilon]|\leq 2^{C/\epsilon^{-1/2}}$.  Let $\mcal{Q}_{K,\delta}[\epsilon]$ be the countable density model given by  
\begin{equation*}
\mcal{Q}_{K,\delta}[\epsilon] = \left\{ \suml_{k=1}^K w_k f_k ; w\in\mcal{W}_K, w_k\geq \delta, w_k\in\mbb{Q}, f_k\in\mcal{P}_{M,s}[\epsilon] \right\}.
\end{equation*}
One can check that $\mscr{Q}_{K,\delta}[\epsilon]$ is also a $\epsilon$-net of $\mscr{Q}_{K,\delta}$ with respect to the Hellinger distance.
\begin{coro}
\label{coro:log_concave}
Assume there exists $P^*$ in $\mscr{P}$ such that $\mbf{P^*}=(P^*)^{\otimes n}$. Take $\epsilon= n^{-2/5}$ and $\delta=n^{-4/5}\wedge K^{-1}$. Let $\hat{P}=\hat{P}_{\delta}$ be a $\rho$-estimator on $\mscr{Q}_{K,\delta}[\epsilon]$. For all $\xi>0$, we have
\begin{align*}
C h^2 \left( P^*, \hat{P} \right) &\leq h^2\left( P^*,\mscr{Q}_K\right)+ \frac{K}{n^{4/5}} \left[ 1 + \log\left( K n \right) \right] + \frac{\xi}{n},
\end{align*}
with probability at least $1-e^{-\xi}$.
\end{coro}
This result provides a risk bound over the class of distributions associated to mixtures of $s$-concave densities. Up to a logarithmic factor, the estimator $\hat{P}$ uniformly converges over $\mscr{Q}_K$ at the rate $n^{-2/5}$ with respect to the Hellinger distance, which is the same rate given in Theorem 3.2 of Doss \& Wellner \cite{doss2016} for the MLE over $\mcal{P}_{M,s}$.
\subsection{Application to the estimation of a continuous Gaussian mixture}
\label{sec:gmm_vdv}
We denote by $\phi_{\sigma}$ the density function of the 	normal distribution (with respect to the Lebesgue measure on $\mbb{R}$) with mean $0$ and variance $\sigma^2 >0$, i.e.
\begin{equation}
\label{eq:phi_gauss}
\phi_{\sigma} : x \mapsto \frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{x^2}{2 \sigma^2}}.
\end{equation}
We assume $P^*$ is of the following form or is close enough to a distribution of the form
\begin{equation*}
p_H(x) = \int \phi_{\sigma}(x-z) dH(z,\sigma), \forall x\in\mbb{R}.
\end{equation*}
We say that $p_H$ is the Gaussian mixture density with  mixing distribution $H$. We want to approximate any distribution of this form with finite Gaussian mixtures, i.e. distribution with densities of the same form with mixing distribution supported on a finite set. For a mixing measure $H$ on $\mbb{R}\times\mbb{R}^{+*}$, we denote by $\text{supp}(H)$ its support. To obtain an approximation result, we need to consider mixing measures $H$ that are supported on a compact set, i.e. there exist $A\geq 0$ and $R\geq 1$ such that $\text{supp}(H)\subset [-A,A]\times[1,R]$. The Hellinger distance being invariant to translation and rescaling, we consider the following class of densities. For $A>0$ and $R\geq 1$ we define
\begin{equation*}
\mcal{C}(A,R) = \bigg\{ p_H ; \exists l\in\mbb{R},  \exists s >0,  \text{supp}(H) \subset \left[ l-sA, l+sA \right] \times [s,sR]  \bigg\}
\end{equation*}
and we denote by $\mscr{C}(A,R)$ the associated class of distributions. We denote by $\mcal{G}$ the location-scale Gaussian family of probability density functions, i.e.
\begin{equation}
\label{eq:gauss_location_scale}
\mcal{G} = \left\{ x \mapsto \phi_{\sigma}(x-\mu) ; \mu \in\mbb{R}, \sigma > 0 \right\}.
\end{equation}
We denote by $\mscr{G}_K$ the Gaussian mixture model with $K$ components associated to class of densities $\mcal{G}_K$ defined by
\begin{equation*}
\mcal{G}_K := \left\{\suml_{k=1}^K w_k \phi_{\sigma_k}(\cdot-z_k) ; w\in\mcal{W}_K, \sigma_k\in (0,+\infty), z_k\in\mbb{R}, \forall k\in\{1,\dots,K\} \right\}.
\end{equation*}
This situation corresponds to $\overline{\mcal{F}}_k=\mcal{G}$ for all $k\in \{1,\dots,K\}$. We can approximate the class $\mscr{C}(A,R)$ with the model $\mscr{G}_K$ as indicated by the following result.
\begin{prop}
\label{prop:ghosal_approximation}
For $K \geq (2/3)^3 A^4$, we have
\begin{equation*}
\sup_{p_H\in\mcal{C}(A,R)} h^2(P_H, \mscr{G}_K) \leq \frac{1}{2} \exp\left( - K^{1/2} \frac{3 \sqrt{3}}{\sqrt{2} R^2} \right) \left[ K^{1/4} \frac{3 \sqrt{2}}{ \sqrt{e\pi} 7^{1/4}} + R \right].
\end{equation*}
\end{prop}
We can deduce a deviation bound on the estimation over $\mscr{C}(A,R)$ from this last result and Theorem \ref{th:single_model}.
\begin{theoreme}
\label{th:gmm}
Assume  $n\geq \exp(2 (A/R)^2)$ and $\frac{n}{\log^2(n)} \geq 2 R^2 /27$. Let $\hat{P}$ be a $\rho$-estimator on $\mscr{G}_{K,\delta}$ with $\delta$ as in (\ref{eq:th1_2}) and $K= \lceil 2 R^4 \log^2(n) /27\rceil$. Assume the true distribution is i.i.d., i.e. $\mbf{P^*}=(P^*)^{\otimes n}$. There exists a numeric constant $C>0$ such that for all $\xi>0$, with probability at least $1-e^{-\xi}$, we have
\begin{equation}
\label{eq:th_continuous_gaussian_mixture}
C h^2 \left( P^*, \hat{P} \right) \leq h^2(P^*,\mscr{C}(A,R)) + \frac{R^4 \log^3(n)+\xi}{n}.
\end{equation}
\end{theoreme}
Therefore, for a fixed $R$, we obtain a rate of $\log^{3/2}(n)/\sqrt{n}$ over $\mscr{C}(A,R)$ with respect to the Hellinger distance. We can also consider larger classes of distributions, with $R$ increasing as $n$ increases but it would deteriorate this rate. Our result is still an improvement of Theorem 4.2 from \cite{ghosal2001} as it requires weaker assumptions. Their result is sensitive to translation or scaling and they have to specify bounds $0<\underline{\sigma}<\overline{\sigma}$ in the model such that $H^*$ is supported on a compact set $[-a,a]\times [\underline{\sigma},\overline{\sigma}]$. Moreover, our estimator is robust, to contamination for instance. Assume we have an $\epsilon$ contamination rate of our data, i.e. $P^*$ is of the form $P^*=(1-\epsilon)\overline{P} + \epsilon Q$ with $\epsilon\in(0,1)$, $\overline{P}\in\mscr{C}(A,R)$ and $Q$ is any probability distribution. Then, our estimator satisfies $Ch^2(P^*,\hat{P}) \leq \epsilon + \frac{R^4 \log^3(n)+\xi}{n}$ on an event of probability $1-e^{-\xi}$. As long as $\epsilon$ remains small as compared to $R^4\log^3(n)/n$, the rate is not deteriorated by the contamination.
\subsection{Parameter estimation}
\label{sec:param_recover}
We say that $\hat{w}$ and $\hat{F}$ are $\rho$-estimators if the resulting mixture distribution $\hat{P}$ given by
\begin{equation*}
\hat{P}=\suml_{k=1}^K \hat{w}_k \hat{F}_k
\end{equation*}
is a $\rho$-estimator. We have a general result for the performance of $\hat{P}$ but not for $\hat{w}$ and $\hat{F}$. In order to evaluate the performance of these estimators, we first need to ensure that the parameters $w=(w_1,\dots,w_K)$ and $F=(F_1,\dots,F_K)$ are identifiable.
\begin{exemple} Let $\overline{\mscr{F}}$ be the set of uniform distributions $\mcal{U}(a,b)$ the uniform distribution on the interval $(a,b)$ of positive lengths. Then the parameters $w$ and $F$ in the mixture model 
\begin{equation*}
\mscr{Q}_2 = \left\{ w_1 F_1 + (1-w_1) F_2 ; w_1\in(0,1), F_1,F_2\in\overline{F} \right\}
\end{equation*}
are not identifiable since 
\begin{equation*}
\frac{3}{4}\mcal{U}(0,1) + \frac{1}{4}\mcal{U}(1/3,2/3) = \frac{1}{2}\mcal{U}(0,2/3) + \frac{1}{2}\mcal{U}(1/3,1).
\end{equation*}
\end{exemple}
We shall say that $P=P_{w,F}$ is identifiable (with respect to the model) if for all $v$ in $\mcal{W}_K$ and all $G$ in $\mscr{F}_1\times\dots\times\mscr{F}_K$, we have 
\begin{equation*}
P_{w,F}=P_{v,G} \Rightarrow \exists \tau\in\mscr{S}_K, \forall k\in\{1,\dots,K\},w_k=v_{\tau(k)} \text{  and  } F_k=G_{\tau(k)},
\end{equation*}
where $\mscr{S}_K$ denotes the set of all permutations of $\{1,\dots,K\}$.
There is a wide literature about identifiability that includes the works of Teicher \cite{teicher1961}, Sapatinas \cite{sapatinas1995}  and Allman \emph{et al} \cite{allman2009} for example. Identifiability is a minimum requirement for the parameter estimators to be meaningful but we can hardly get more than consistency with it. As mentioned in the introduction, we are looking for a lower bound on the Hellinger distance between mixture distributions. Convexity properties ensure that we always have the upper bound
\begin{equation}
\label{eq:hell_mix_parameter}
h\left( P_{w,F}, P_{v,G} \right) \leq \inf\limits_{\tau\in\mscr{S}_K} \left\{ h( w, v \circ \tau) + \max_{k\in[K]} h\left( F_k, G_{\tau(k)} \right) \right\},
\end{equation}
for all mixing weights and emission distributions (see Lemma \ref{lem:upper_bound_parameters}). On the other hand, obtaining a lower bound on $h^2\left( P_{\pi,F}, P_{\nu,G} \right)$ is quite more complicated unfortunately. There are still some situations where we do have such a lower bound.
\subsubsection*{Regular parametric model}
Let $K$ be an integer larger than 1. We consider parametric emission models associated to density models $\overline{\mcal{F}}_k=\left\{ f_k(\cdot;\alpha), \alpha \in A_k \right\}$, where $A_k$ is a subset of $\mbb{R}^{d_k}$ for all $k\in\{1,\dots,K\}$. It is always possible to find a countable dense subset of $A_k$ with respect to the Euclidean distance on $\mbb{R}^{d_k}$. We assume there is a reasonably good connection between the Hellinger distance on the emission models and the Euclidean distances on the parameter spaces such that a dense subset of $A_k$ would translate into a dense subset of the emission model with respect to the Hellinger distance. This assumption is very weak and does not seem to be restrictive in any way. In the different examples we consider we can always consider $A_k\cap\mbb{Q}^{d_k}$ as a dense subset of $A_k$. Therefore Assumption \ref{hyp:countable} is satisfied with $\mcal{F}_k=\left\{ f_k(\cdot;\alpha), \alpha \in B_k \right\}$. We denote by $\mscr{Q}_K$ the distribution model associated to the mixture density model
\begin{equation*}
\mcal{Q}_K = \left\{ p(\cdot;\theta) = \suml_{k=1}^{K-1} w_k f_k(\cdot;z_k) + (1-w_1-\dots-w_{K-1}) f_K(\cdot;\alpha_K) ; \theta=(w,\alpha) \in \Theta \right\},
\end{equation*}
where $\Theta$ is an open convex subset of $\left\{ w \in (0,1)^{K-1} ; \suml_{k=1}^{K-1} w_k < 1 \right\} \times A_1 \times \dots \times A_K$. We make the following assumptions.
\begin{hypothese}
\label{hyp:regular}
\begin{itemize}
\item[a)] The function $z\mapsto f_k(x;z)$ is continuous on $A_k$ (with respect to the Euclidean distance) for $\mu$-almost all $x\in\mscr{X}$, for all $k\in\{1,\dots,K\}$.
\item[b)] For all $k\in\{1,\dots,K\}$, for $\mu$-almost all $x\in\mscr{X}$ the function $u\mapsto f_k(x;u)$ is differentiable at the point $u=\alpha$ and for all $j\in\{1,\dots,d_k\}$, we have
\begin{equation*}
\int_{\mscr{X}} \left| \frac{\partial f_k(x;\alpha)}{\partial \alpha_j} \right|^2 \frac{\mu(dx)}{f_k(x;\alpha)} <\infty.
\end{equation*}
\item[c)] The function $\theta\mapsto \psi(\cdot;\theta)=\frac{\partial}{\partial \theta} p^{1/2}(\cdot;\theta)$ is continuous in the space $L_2(\mu)$.
\item[d)] The class of densities $\overline{\mcal{F}}_k$ is VC-subgraph with VC-index not larger than $V_k$ for all $k\in\{1,\dots,K\}$. We write $\overline{V}=V_1=\dots+V_k$.
\end{itemize}
\end{hypothese}
We use the approach of Ibragimov and Has’minski\u{\i} \cite{Ibragimov} for regular parametric models to obtain a deviation inequality on the Euclidean distance between parameters using Fisher's information.
\begin{theoreme}{(Theorem 7.6 \cite{Ibragimov})}\\
\label{th:regular_parametric}
Let $\overline{\theta}$ be in $\Theta$. Assume the Fisher's information matrix
\begin{equation*}
I\left(\overline{\theta}\right) = \int_{\mscr{X}} \frac{\partial p\left(x;\overline{\theta}\right)}{\partial \theta} \left( \frac{\partial p\left(x;\overline{\theta}\right)}{\partial \theta} \right)^T \frac{\mu(dx)}{p\left(x;\overline{\theta}\right)}
\end{equation*}
is definite positive and $\inf_{ \substack{||\overline{\theta}-\theta||\geq a\\ \theta \in\Theta}} h^2\left(P_{\overline{\theta}},P_{\theta}\right) > 0$ for all $a>0$. Let $\hat{P}=P_{\hat{w},\hat{F}}$ be a $\rho$-estimator on $\mscr{Q}_{K,\delta}$, with $\delta$ as in (\ref{eq:th1_2}). There exists a positive constant $C\left(\overline{\theta}\right)$ such that for all $\xi>0$, with probability at least $1-e^{-\xi}$, we have
\begin{equation}
\label{eq:th_regular_parametric}
C\left(\overline{\theta}\right) \left( || \overline{w} - \hat{w}||^2 +  \suml_{k=1}^K 1\wedge ||\overline{\alpha}_k-\hat{\alpha}_k||^2 \right) \leq \frac{1}{n} \left[ \mbf{h}^2\left(\mbf{P^*},P_{\overline{\theta}}^{\otimes n}\right) + \overline{V}\log(n)+\xi \right].
\end{equation}
And assuming $P^*=P_{\overline{\theta}}$, we obtain the usual parametric convergence rate up to a logarithmic factor for the parameter estimators.
\end{theoreme}
Inequality (\ref{eq:th_regular_parametric}) proves that even if "true parameters" might not exist the parameter estimators can be meaningful as long as $\mbf{P^*}$ is relatively close to the model. The Gaussian mixture model is the most common mixture model and it is a regular parametric model. Let $K\geq 2$ and take $\mcal{F}_k=\mcal{G}$ for all $k\in\{1,\dots,K\}$. We define a binary relation on $\mbb{R}\times(0,\infty)$ by
\begin{equation}
\label{eq:order_parameter}
(z_1,\sigma_1) > (z_2,\sigma_2) \Leftrightarrow \begin{cases}
\sigma_1 >\sigma_2;\\
\text{or  }\sigma_1=\sigma_2 \text{ and  } z_1>z_2.
\end{cases}
\end{equation}
We consider the parameters $\theta=(w_1,\dots,w_{K-1},z_1,\sigma_1^2,\dots,z_K,\sigma_K^2)$ belonging to the set
\begin{equation*}
\Theta=\left\{ \theta \in(0,1)^{K-1}\times  \left( \mbb{R}\times\mbb{R^*} \right)^K; \suml_{k=1}^{K-1} w_k <1, \left(z_1,\sigma_1\right) > \dots > \left(z_K,\sigma_K\right)  \right\}.
\end{equation*}
\begin{theoreme}
\label{th:parameter_recovery_gmm}
Assume $P^*=P_{\overline{\theta}}=\suml_{k=1}^K \overline{w}_k \mcal{N}(\overline{z}_k,\overline{\sigma}^2_k)$ such that $(\overline{z}_1,\overline{\sigma}_1)>\dots>(z_K,\sigma_K)$ are all distinct and $\inf\limits_{1\leq k\leq K} \overline{w}_k >0$. Let $\hat{P}$ be a $\rho$-estimator on $\mscr{G}_{K,\delta}$, with $\delta$ as in (\ref{eq:th1_2}). There exists a positive constant $C\left(\overline{\theta}\right)$ such that, for all $\xi>0$, we have
\begin{equation}
\label{eq:th_gmm}
C\left(\overline{\theta}\right) \left( \suml_{k=1}^{K-1} ||\overline{w}_k-\hat{w}_k||^2 + \suml_{k=1}^K \left|\left|\left(\overline{z}_k,\overline{\sigma}_k^2\right) - \left(\hat{z}_k,\hat{\sigma}_k^2\right)\right|\right|^2 \wedge 1 \right) \leq \frac{5K\log(n)+\xi}{n},
\end{equation}
with probability at least $1-e^{-\xi}$.
\end{theoreme}
Our estimator reaches the optimal rate of convergence up to a logarithmic factor. One can notice that the assumption of ordered couples of parameters $(z_j,\sigma_j^2)$ can be replace by considering distinct couples only and taking the infimum over permutation of the hidden states in (\ref{eq:th_gmm}).
\subsubsection*{Connection with the $L_2$-distance}
We can use results from the literature that do not apply to the Hellinger distance but to other ones such as the $L_2$-distance between densities. There is a general inequality  between the $L_2$ and Hellinger distances when the density functions are bounded, i.e.
\begin{equation}
\label{eq:l2_hellinger}
||p-q||_2^2 \leq 4 \left(||p||_{\infty} + ||q||_{\infty}\right) h^2(P,Q).
\end{equation}
Assume one can prove an inequality of the following type. For any $w,v$ in $\mcal{W}_K$ and any $f_k,g_k$ in $\overline{\mcal{F}}_k$ for all $k\in\{1,\dots,K\}$ such that the resulting mixtures belong to our model, we have
\begin{equation}
\label{eq:L2_param}
\underline{c} \left( d_{\Pi}^2(w,v) + \max_{k\in[K]} d_F^2(f_k,g_k) \right) \leq \left|\left| \suml_{k=1}^K w_k f_k - \suml_{k=1}^K v_k g_k \right|\right|^2,
\end{equation}
where $d_{\Pi}$ is a distance on $\mcal{W}_K$ and $d_F$ is a distance on $\bigcup_{1\leq k\leq K} \overline{\mcal{F}}_k$. Moreover, assuming the density models $\overline{\mcal{F}}_k$ are uniformly bounded, i.e.
\begin{equation}
\label{eq:bounded_densities}
\sup_{k\in[K]} \sup_{f\in\overline{\mcal{F}}_k} ||f||_{\infty} =: U <\infty,
\end{equation}
we get
\begin{equation*}
d_{\Pi}^2(w,v) + \max_{k\in[K]} d_F^2(f_k,g_{\tau(k)}) \leq \frac{8U}{\underline{c}} h^2 \left( \suml_{k=1}^K w_k F_k, \suml_{k=1}^K v_k G_k \right). 
\end{equation*}
Here again, a density estimation result implies a result for the parameter estimation. We can apply this method to the special case of two-component mixture model with one known component. Let $\phi$ be a density function on $\mbb{R}^d$ with respect to the Lebesgue measure. We consider the 2-component mixture model $\mscr{Q}$ associated to the class of densities
\begin{equation}
\label{eq:contamination_model}
\mcal{Q} = \left\{ x \mapsto p_{\lambda,z}(x) = (1-\lambda) \phi(x) + \lambda \phi(x-z) ; \lambda \in[0,1], z\in\mbb{R}^d \right\},
\end{equation}
with $\overline{\mcal{F}}_1=\left\{ \phi \right\}$ and $\overline{\mcal{F}}_2=\left\{ x\mapsto \phi(x-z) ; z \in \mbb{R}^d \right\}$. We make the following assumptions on $\phi$.
\begin{hypothese}
\label{hyp:phi_contamination}
The function $\phi$ belongs to $\mcal{C}^3\left(\mbb{R}^d\right)\cap\mbb{L}^2\left(\mbb{R}^d\right)$. For any $M>0$, there exists a function $g$ in $\mbb{L}^2\left(\mbb{R}^d\right)$ such that
\begin{equation*}
\forall x\in\mbb{R}^d, \forall z\in[-M,M]^d, |\phi(x)-\phi(x-z)|\leq ||z|| g(x)
\end{equation*}
and
\begin{equation*}
\int g^2(x) \phi^{-1}(x) dx < + \infty.
\end{equation*}
\end{hypothese}
Gadat \emph{et al} proved an inequality such as (\ref{eq:L2_param}) in this situation.
\begin{prop}{(inequality (7.11) in \cite{gadat2018parameter})}\\
\label{prop:gadat}
Under Assumption \ref{hyp:phi_contamination}, for all $M > 0$, there exists a positive constant $c(\phi,M)$ such that for all $z_1,z_2 \in[-M,M]^d$ and $\lambda_1,\lambda_2 \in [0,1]$,
\begin{equation*}
c(\phi,M) ||z_1||^2 \left( ||z_2||^2 \left(\lambda_1-\lambda_2\right)^2 + \left(\lambda_1\right)^2  \left|\left|z_1-z_2\right|\right|^2 \right) \leq || p_{\lambda_1,z_1} - p_{\lambda_2,z_2}||^2.
\end{equation*}
\end{prop}
One can notice that Assumption \ref{hyp:phi_contamination} implies that $\phi$ is bounded (see Assumption ($\mbf{H}_{\mcal{S}}$) in \cite{gadat2018parameter}). Hence, we can deduce a deviation inequality for $\rho$-estimators of parameters.
\begin{theoreme}
\label{th:gadat}
We assume $\overline{\mcal{F}}_2$ has a finite VC-index $V$, $\lambda^*\in(0,1]$ and $z^*\neq 0$. For $\delta$ as in (\ref{eq:th1_2}), there exists a positive constant $C(\phi,\lambda^*,z^*)$ and an integer $n_0=n_0(\phi,\lambda^*,z^*)$ such that for any $\rho$-estimator $\hat{P}=P_{\hat{\lambda},\hat{z}}$ on $\mscr{Q}_{\delta}$, $n\geq n_0$ and for all $\xi\in (0,\xi_n)$, we have
\begin{equation*}
C(\phi,z^*,\lambda^*) \left( \left(\lambda^*-\hat{\lambda}\right)^2 + \left( \left|\left|z^*-\hat{z}\right|\right|^2 \wedge 1 \right) \right) \leq \frac{\xi+(V+1)\log(n)}{n},
\end{equation*}
with probability at least $1-e^{-\xi}$, where $\xi_n= (1+V)[1+\log(2n/(1+V))])$.
\end{theoreme}
This implies the consistency of $\hat{z}$ and consequently the consistency of $\hat{\lambda}$ if $z^*\neq 0$. We can deduce a bound on the convergence rate for $\hat{z}$ and also for $\hat{\lambda}$ but only for $n$ large enough. It is similar to Theorem 3.1 of Gadat \emph{et al.} \cite{gadat2018parameter} with a smaller power for the logarithmic term. This slight improvement is allowed by the VC assumption. Furthermore, we do not need to know a value of $M$ such that $z^*\in[-M,M]$ or to specify it in the model. The examples of translation families taken by Gadat \emph{et al} \cite{gadat2018parameter} (Section 6) all satisfy the VC assumption.
\begin{lemme}
\label{lem:vc}
\begin{itemize}
\item The Cauchy location-scale family $\mcal{C}$ of density functions defined by (\ref{eq:cauchy_location_scale}) is VC-subgraph with VC-index $V(\mcal{C})\leq 5$. 
\item The family of densities $\mcal{G}$ defined by (\ref{eq:gauss_location_scale}) is VC-subgraph with VC-index at most $5$. This bound extend to $3+\frac{d(d+3)}{2}$ for multivariate normal distributions in dimension $d$.
\item The Laplace location family $\mcal{L}$ of density functions defined by
\begin{equation*}
\mcal{L} = \left\{ x\mapsto \frac{1}{2} e^{-|x-z|} ; z\in\mbb{R} \right\}
\end{equation*}
is VC-subgraph with VC-index $V(\mcal{L})\leq $.
\item The location family of densities $\mcal{SG}_{\alpha}$ associated to the skew Gaussian density defined by
\begin{equation*}
\mcal{SG}_{\alpha} = \left\{ x\mapsto 2 \phi_1(x-z) \int_{-\infty}^{x-z} \phi_1(\alpha t) dt ; z\in\mbb{R}\right\}
\end{equation*}
is VC-subgraph with VC-index $V(\mcal{SG}_{\alpha})\leq 10$ for all $\alpha\in\mbb{R}$, where $\phi_1$ is given by (\ref{eq:phi_gauss}).
\end{itemize}
\end{lemme}
By inclusion, if the bound holds for the location-scale family it also holds for the location family with fixed scale parameter.
\subsubsection*{Proving a lower bound for a specific example}
\label{sec:faster_rate}
In some specific situations, it is relatively easy to prove a lower bound on the Hellinger distance. This is what we do in the following example and it allows us to obtain faster rates than the usual parametric one. Let $\alpha$ be in $(0,1)$. We denote by $s_{\alpha}$ the probability density function with respect to the Lebesgue measure on $\mbb{R}$ defined by
\begin{equation*}
s_{\alpha} : x\in\mbb{R} \mapsto \frac{1-\alpha}{2|x|^{\alpha}} \mathbbm{1}_{|x|\in(0,1]}.
\end{equation*}
We consider $\mcal{Q}$ as in (\ref{eq:contamination_model}) with $\phi=s_{\alpha}$ and for $\lambda\in[0,1]$ and $z\in\mbb{R}$, we write
\begin{equation*}
p_{\lambda,z} = (1-\lambda) s_{\alpha} + \lambda s_{\alpha}(\cdot-z).
\end{equation*}
We can prove that the Hellinger distance $h(P_{\lambda,z},P_{\lambda',z'})$ is lower bounded by some distance between the parameters which leads to the following theorem.
\begin{theoreme}
\label{th:faster_rate}
For $\lambda^*>0$ and $z^*\neq 0$, there is a positive constant $C(\alpha,z^*,\lambda^*)$ such that, for any $\rho$-estimator $\hat{P}=P_{\hat{\lambda},\hat{z}}$ on $\mscr{Q}_{\delta}$ with $\delta=10/n$ and $n\geq 20$, for all $\xi>0$, with probability at least $1-e^{-\xi}$ we have
\begin{equation*}
 C(\alpha,z^*,\lambda^*) \left[ 1 \wedge |\hat{z}-z^*|^{1-\alpha} + \left( \lambda^* - \hat{\lambda} \right)^2 \right] \leq \frac{\log(n) + \xi}{n}.
\end{equation*}
\end{theoreme}
We derive from this inequality that our estimators $\hat{\lambda}=\hat{\lambda}_n$ and $\hat{z}=\hat{z}n$ estimate $\lambda^*$ and $z^*$ at a rate which is at least $\sqrt{(\log n)/n}$ and $(n^{-1}\log n)^{1/(1-\alpha)}$ respectively. This latter rate is faster than the usual $1/\sqrt{n}$-rate for all $\alpha\in(0,1)$. Up to the logarithmic factors, these rates are optimal. Moreover, one can notice that both maximum likelihood and least squares approaches do not apply here since we consider density functions that are unbounded, and not even square integrable for $\alpha\in[1/2,1)$.
\section{Model selection}
\label{sec:model_selection}
In Section \ref{sec:main_result} we consider estimation on a model with a fixed order $K$ and simple emission families. We use model selection to overcome this restriction in this section and consider composite emission families and/or models with different orders.  
\subsection{Construction of the estimator}
Let $\Theta$ be a subset of
\begin{equation*}
\bigcup_{K\geq 1} \{K\} \times \prod_{k=1}^K \Lambda_k.
\end{equation*}
Let $\delta : \Theta \rightarrow (0,1]$ be such that for $\theta=(K,\lambda_1,\dots,\lambda_K)\in \Theta$, $\delta(\theta)\in(0,1/K]$. We write
\begin{equation*}
\mscr{Q}_\delta ( \theta ) = \left\{ \suml_{k=1}^K w_k F_k ; w\in\mcal{W}_K, w_k\geq \delta, w_k \in \mbb{Q}, F_k\in\mscr{F}_k,\forall k\in[K] \right\}.
\end{equation*}
We define $\mscr{Q}_{\delta}$ by
\begin{equation*}
\mscr{Q}_{\delta} = \bigcup_{\theta \in \Theta} \mscr{Q}_{\delta}(\theta).
\end{equation*}
We associate to $\mscr{Q}_{\delta}$ the family $\mcal{Q}_{\delta}$ of densities  with respect to $\mu$ and the $\rho$-estimator $\hat{P}_{\delta}$ of $\overline{P}$ based on the family $\mcal{Q}_{\delta}$.
Assuming we have a penalty function $\pen : \mcal{Q}_{\delta} \rightarrow \mbb{R}$, we set
\begin{equation}
\label{eq:upsilon_pen}
\mbf{\Upsilon}(\mbf{X},q) = \sup_{q'\in\mcal{Q}_{\delta}} \left[ \mbf{T}(\mbf{X},q,q') - \pen(q') \right] + \pen(q),
\end{equation}
for all $q\in\mcal{Q}_{\delta}$. The $\rho$-estimator $\hat{P}_{\delta}$ is any measurable element of the closure (with respect to the Hellinger distance) of the set $\pmb{\mscr{E}}(\psi,\mbf{X})$, as defined by (\ref{eq:rho_est}). One can notice that a constant penalty function does not have any impact on the definition of $\mbf{\Upsilon}$ and brings us back to the previous situation.
\subsection{Estimation on a mixture model based on composite emission families}
Let $K$ be larger than or equal to 2. Let $L$ be a subset of $\prod_{k=1}^K \Lambda_k$ and define $\Theta$  by $\Theta=\{K\}\times L$, i.e. $K$ is fixed. For $\lambda=(\lambda_1,\dots,\lambda_K)\in L$, the model $\mscr{Q}(\lambda)$ is a subset of
\begin{equation*}
\left\{ \suml_{k=1}^K w_k F_k ; w\in\mcal{W}_K, F_k\in\overline{\mscr{F}}_{\lambda_k},\forall k\in[K] \right\}
\end{equation*}
and we define its countable subset $\mscr{Q}_{\delta} ( \lambda )$ by
\begin{equation*}
\mscr{Q}_{\delta} = \left\{ \suml_{k=1}^K w_k F_k \in\mscr{Q}(\lambda) ; w\in\mcal{W}_K, w_k\geq \delta(\lambda), w_k \in \mbb{Q}, F_k\in\mscr{F}_{\lambda_k},\forall k\in[K] \right\},
\end{equation*}
where $\delta$ is any function $L \rightarrow (0,1/K]$, and $\mscr{Q}_{\delta}=\bigcup_{\lambda\in L} \mscr{Q}_{\delta}(\lambda)$. Under Assumption \ref{hyp:vc_density_model}, we write $\overline{V}(\lambda) = V(\lambda_1) + \dots + V(\lambda_K)$.
\begin{theoreme}
\label{th:selection_emission_models}
Let $\Delta$ be a mapping $L \rightarrow \mbb{R}^+$ such that $\suml_{\lambda\in  L} e^{-\Delta(\lambda)} \leq 1$. Let $\pen$ be the penalty function defined by
\begin{equation*}
\pen(q) = \kappa \inf\limits_{\lambda \in L|Q\in\mscr{Q}(\lambda)} \left[ 174.1 \overline{V}(\lambda) \left[ 5.82 + \log\left( \frac{(K+1)^2}{\delta(\lambda)} \right) + \log_+ \left( \frac{n}{\overline{V}(\lambda)} \right) \right] + \Delta(\lambda) \right],
\end{equation*}
where $\kappa$ is given by (19) in \cite{baraudinventiones}. Assume there is $P^*$ in $\mscr{P}$ such that $\mbf{P^*}=(P^*)^{\otimes n}$. For the choice $\delta(\lambda)=\frac{\overline{V}(\lambda)}{n(K-1)}\bigwedge \frac{1}{K}$, there is a positive constant $C$ such that the resulting estimator $\hat{P}=\hat{P}_{\delta}$ satisfies the following. For all $\xi>0$, with probability at least $1-e^{-\xi}$ we have
\begin{equation*}
C h^2(P^*,\hat{P}) \leq \inf_{\lambda\in L} \left\{ h^2( P^*, \mscr{Q}(\lambda) ) + \frac{1}{n}  \left( \overline{V}(\lambda) \left[ 1 + \log\left( \frac{ K n }{ \overline{V}(\lambda) \wedge n} \right) \right] + \Delta(\lambda) + \xi \right) \right\}.
\end{equation*}
\end{theoreme}
This is a general result for the situation where you know the number $K$ of subpopulations, or at least want to fix it for the estimation, but are hesitating on the models for the emission distributions. For instance, let us consider Gaussian and Cauchy location-scale families for the composite emission families. For all $k\in\{1,\dots,K\}$, we take $\Lambda_k=\{1,2\}$ with $\overline{\mscr{F}}_1=\mscr{G}$ and $\overline{\mscr{F}}_2=\mscr{C}$, where $\mscr{C}$ is the Cauchy location-scale family of distributions associated to the density class
\begin{equation}
\label{eq:cauchy_location_scale}
\mcal{C}=\left\{ x\mapsto \frac{1}{\pi\sigma} \frac{1}{1+ \left(\frac{x-z}{\sigma} \right)^2}; z\in\mbb{R}, \sigma>0  \right\}.
\end{equation}
We consider the model $\mscr{Q}=\cup_{0\leq j\leq K} \mscr{Q}_j$ with
\begin{equation*}
\mscr{Q}_j = \left\{ \suml_{k=1}^j w_k \mcal{N}(z_k,\sigma^2_k) + \suml_{k=j(\lambda)+1}^K w_k \cauchy(z_k,\sigma_k); \begin{array}{l}
(z_1,\sigma_1)>\dots>(z_j,\sigma_j),\\
(z_{j+1},\sigma_{j+1})>\dots>(z_K,\sigma_K)
\end{array} \right\},
\end{equation*}
where the order $>$ on the parameters $(z_k,\sigma_k)$ is defined by (\ref{eq:order_parameter}) and allows to have identifiability properties again here. We consider a null penalty function.
\begin{theoreme}
\label{th:gauss_cauchy}
Assume $P^*=\suml_{k=1}^{j^*} \overline{w}_k \mcal{N}(\overline{z}_k,\overline{\sigma}^2_k) + \suml_{k=j^*+1}^K \overline{w}_k \cauchy(\overline{z}_k,\overline{\sigma}_k)\in\mscr{Q}_{j^*}$ with $(\overline{z}_1,\overline{\sigma}_1)>\dots>(\overline{z}_{l^*},\overline{\sigma}_{l^*})$ and $(\overline{z}_{l^*+1},\overline{\sigma}_{l^*+1})>\dots>(\overline{z}_K,\overline{\sigma}_K)$. Let $\hat{P}$ be a $\rho$-estimator on $\mscr{Q}_{\delta}$ with $\delta=\frac{5}{n}\bigwedge \frac{1}{K}$ and a null penalty. There exists an integer $n_0(P^*)$ and a positive constant $C(P^*)$ such that for $n\geq n_0(P^*)$ there exists an event of probability $1-(n(K+1))^{-K}$ on which such that $\hat{P}\in\mscr{Q}_{j^*}$ and 
\begin{align*}
C(P^*) \left( ||\overline{w}-\hat{w}||^2 + \suml_{k=1}^{j^*} \left|\left|(\overline{z}_k,\overline{\sigma}^2_k)- (\hat{z}_k,\hat{\sigma}^2_k)\right|\right|^2\wedge 1\right. & + \left. \suml_{k=j^*+1}^{K} \left|\left|(\overline{z}_k,\overline{\sigma}_k)- (\hat{z}_k,\hat{\sigma}_k)\right|\right|^2\wedge 1 \right)\\
&\leq \frac{K\log(n(K+1))}{n}.
\end{align*}
\end{theoreme}
This result shows that it is possible to identify the true emission models for $n$ large enough and if this identification is established we can also estimate the different parameters. This seems to be somehow original as we did not find any result of this kind in the literature.
\subsection{Selection of the order \texorpdfstring{$K$}{K}}
We consider $\Theta$ of the form $\Theta = \bigcup\limits_{K \in \mscr{K}} \{K\}\times \{ \lambda \}^K$, where $\mscr{K}$ is a subset of $\{1,\dots,n\}$. For $K \in \mscr{K}$, we write $\overline{\mscr{F}}=\overline{\mscr{F}}_{\lambda}$ and $\mscr{F}=\mscr{F}_{\lambda}$ its countable and dense subset given by Assumption \ref{hyp:countable}. For $K\in\mscr{K}$, the model $\mscr{Q}(K)$ is a subset of
\begin{equation*}
\left\{ \suml_{k=1}^K w_k F_k ; w\in\mcal{W}_K, F_k\in\overline{\mscr{F}},\forall k\in[K] \right\}.
\end{equation*}
We define $\mscr{Q}_\delta (K) := \left\{ \suml_{k=1}^K w_k F_k \in \mscr{Q}(K) ; w\in\mcal{W}_K, w_k\geq \delta, w_k \in \mbb{Q}, F_k\in\mscr{F},\forall k\in[K] \right\}$ and $\mscr{Q}_{\delta}=\bigcup_{K\in \mscr{K}} \mscr{Q}_{\delta}(K)$, where $\delta : \mscr{K} \rightarrow (0,1]$ satisfies $\delta(K)\leq 1/K$. 
Under Assumption \ref{hyp:vc_density_model}, we denote by $V$ the VC-index of $\overline{\mcal{F}}$. 
\begin{theoreme}
\label{th:selection_K}
Let $\Delta$ be a function $\mscr{K} \rightarrow \mbb{R}^+$ satisfying $\suml_{K \in \mscr{K}} e^{-\Delta(K)} \leq 1$. We consider the penalty function defined by
\begin{equation}
\label{eq:pen_selection_K}
\pen(q) = \kappa \inf\limits_{K \in\mscr{K}|Q\in\mscr{Q}(K)} \left[ 174.1 K V \left[ 5.82 + \log\left( \frac{(K+1)^2}{\delta(K)} \right) + \log_+ \left( \frac{n}{ K V} \right) \right] + \Delta(K) \right].
\end{equation}
Assume there exists $P^*$ in $\mscr{P}$ such that $\mbf{P^*}=(P^*)^{\otimes n}$. For the choice $\delta(1)=1$ and $\delta(K)=\frac{V}{n}\bigwedge \frac{1}{K}$ for $K\geq 2$, there is a positive constant $C$ such that the resulting estimator $\hat{P}=\hat{P}_{\delta}$ sat  Any $\rho$-estimator $\hat{P}_{\delta}$ on $\mscr{Q}_{\delta}$ satisfies the following. For all $\xi>0$, with probability at least $1-e^{-\xi}$ we have
\begin{equation}
\label{eq:th_order_selection_delta}
C h^2(P^*,\hat{P}) \leq \inf_{K \in \mscr{K}} \left\{ h^2( P^*, \mscr{Q}(K) ) + \frac{K V \log(n) + \xi + \Delta(K)}{n} \right\}.
\end{equation}
\end{theoreme}
This result gives an oracle inequality and it provides a way to determine the number of clusters if one wants to use mixture models in order to do clustering. It is also interesting in the context of density estimation. Once again, we take advantage of the approximation properties of GMMs to use our estimator for density estimation on a wider class. We use the approximation result proven by Maugis \& Michel \cite{maugis2012}. Let $\beta> 0$, $r = \lfloor \beta \rfloor$ and $k \in \mbb{N}$ such that $\beta \in (2k, 2k + 2]$. Let also $\mcal{P}$ be the 8-tuple of parameters $(\gamma, l^+, L, \epsilon, C, \alpha, \xi, M)$ where $L$ is a polynomial function on $\mbb{R}$ and the other parameters are positive constants. We define the density class $\mcal{H}(\beta, \mcal{P})$ of all densities $p$ satisfying the following conditions.
\begin{itemize}
\item For all $x$ and $y$ such that $|y - x| \leq \gamma$,
\begin{equation*}
(\log p )^{(r)}(x) - (\log p )^{(r)}(y) \leq r! L(x)|y - x|^{\beta-r}.
\end{equation*}
Furthermore for all $j \in \{0,\dots,r\}$,
\begin{equation*}
|(\log p )^{(j)}(0)| \leq l^+.
\end{equation*}
\item We have
\begin{equation*}
\max\limits_{1\leq j \leq r} \int_{\mbb{R}}{ \left| (\log p)^{(j)}(x)\right|^{\frac{2\beta+\epsilon}{j}} p(x) dx} \vee \int_{\mbb{R}} |L(x)|^{2+\frac{\epsilon}{\beta}} p (x)dx \leq C.
\end{equation*}
\item For all $x \in \mbb{R}, p (x) \leq M \psi(x)$.
\item The function $f$ is strictly positive, non-decreasing on $(-\infty, -\alpha)$ and non-increasing on
$(\alpha,\infty)$. For all $x \in [-\alpha, \alpha]$ we have $p (x) \geq \xi$ .
\end{itemize}
This class of functions can be approximated by Gaussian mixture models, the quality of the approximation depending on the regularity parameter $\beta$.
\begin{lemme}{(Lemma 6.1 in \cite{maugis2012})}\\
For $0 < \underline{\beta} < \overline{\beta}$, there exists a set of parameters $\mcal{P}(\underline{\beta},\overline{\beta})$ and a positive constant $c_{\underline{\beta},\overline{\beta}}$ such that for all $\beta\in \left[ \underline{\beta}, \overline{\beta} \right]$ and for all $p \in \mcal{H}\left(\beta,\mcal{P}(\underline{\beta},\overline{\beta})\right)$, 
\begin{equation*}
h^2 \left( P, \mcal{G}_K \right) \leq c_{\underline{\beta},\overline{\beta}} \frac{ \left( \log K \right)^{3\beta} }{ K^{2\beta} }.
\end{equation*}
\end{lemme}
We consider $\mscr{K}=\{2,\dots,n\}$, $\Delta(K) = K$ and the penalty function $\pen$ as in (\ref{eq:pen_selection_K}).
\begin{theoreme}
\label{th:maugis}
Let $\hat{P}=\hat{P}_{\delta}$ be a $\rho$-estimator on $\mscr{Q}_{\delta}$ with $\delta$ as in (\ref{eq:th_order_selection_delta}). For $0<\underline{\beta}<\overline{\beta}$, there exist a  positive constant $C_{\underline{\beta},\overline{\beta}}$ such that for any $p$ in $\mcal{H}\left(\beta,\mcal{P}(\underline{\beta},\overline{\beta})\right)$ with $\beta\in\left[\underline{\beta},\overline{\beta}\right]$, for all $\xi>0$, we have
\begin{equation*}
h^2(P^*,\hat{P}) \leq C_{\underline{\beta},\overline{\beta}} \left( \frac{(\log n)^{\frac{5\beta}{2\beta+1}}}{n^{\frac{2\beta}{2\beta+1}}} + \frac{\xi}{n}\right),
\end{equation*}
with probability at least $1-e^{-\xi}$.
\end{theoreme}
This theorem provides an upper bound on the convergence rate of our estimator of order $(\log n)^{5\beta/(4\beta+2)} n^{-\beta/(2\beta+1)}$. It is the same rate obtained in Theorem 2.9 of Maugis \& Michel \cite{maugis2012}. Therefore our estimator is minimax adaptive to the regularity $\beta$, up to a power of $\log(n)$.
\section{Acknowledgment}
The author would like to thank Yannick Baraud for his guidance in the redaction of this article.
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