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\title{Convergence rate and consistency at zero for decreasing densities on $[0,1]$ }

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\begin{document}

\title{Convergence rate and consistency at zero for decreasing densities on $[0,1]$}
\author{J-B. Salomond}
\institution{CREST, Université Paris Dauphine}

\maketitle
\begin{abstract}
{In this paper, we consider the well known problem of estimating
a density function defined on a compact set under qualitative assumption.
More precisely, we study monotone non increasing densities in a Bayesian
setting, and derive convergence rate for the posterior distribution
for two types of priors, namely, a Dirichlet process and finite mixture
prior. We prove that the posterior distribution based on both prior
concentrates at the rate $(n/\log(n))^{-1/3}$, which is the minimax
rate of estimation up to a $\log(n)$ factor. We also prove that the
posterior probability of the continuous limit of the density at $0$
is consistent, which is of particular interest as the classical non
parametric maximum likelihood estimator is known to be inconsistent
at this point.}
\end{abstract}


\section{Introduction}

The non parametric problem of estimating monotone non increasing curve,
and monotone densities in particular, has been well studied in the
literature, both form a theoretical and applied perspective. For instance,
\citet{MR822052} and more recently, \citet{balabdaoui:wellner} studied
very precisely the asymptotic properties of the non parametric maximum
likelihood estimator (NPMLE) , namely, the Grenander estimator, which
is proved to be consistent and to converge at the minimax rate $n^{-1/3}$
when the support of the distribution is compact. Monotone non increasing
densities arise naturally when considering survival analysis, where it is 
natural to assume that the uncensored survival time have a monotone non increasing density.

In this paper, we study the asymptotic properties of a Bayesian estimator
of a monotone non increasing density function with a compact support.
Following \citet{KHAZAEI:2010:INRIA-00494692:1} we study two families
of non parametric priors on $\F$, the class of monotone non increasing
densities with support on $[0,1]$. We obtained the two families of prior
a concentration rate of order $(n/\log(n))^{-1/3}$. Interestingly, the NPMLE
 is not consistent at $0$, (see \citet{MR1392133} and \citet{balabdaoui:wellner}
 for instance). However, we prove that the posterior distribution of $f(0)$
 is still concentrate around the true value a the rate $(n/\log(n))^{-1/3}$.
The non parametric prior are constructed from the mixture representation of
 monotone non increasing density. It is known since \citet{will:56} that any
 monotone non increasing density on $\R^+$ has a mixture representation
% 
\begin{equation}\label{eq:Will}
f(x) = \int_{0}^\infty {\I_{[0,\theta]}(x) \over \theta } dP(\theta)
\end{equation}
%
where $P$ is a probability distribution on $\R^+$, namely a mixing distribution.
In order to indicate the dependence on $P$, we shall note $f_P$ the density
function admiting representation \eqref{eq:Will}. Let $\F$ be the set
$$
\F = \left \{f\text{ s.t. } f>0,\; f\searrow \, \;  \int_0^L f = 1\right\}
$$
%
Mixtures models have been well studied in a Bayesian framework and various types
of prior have been considered for the mixing distribution. The most popular non
parametric mixture prior model is the Dirichlet Process prior (DP) introduced by
\citet{MR736538} and in their article, \citet{WG08} studied properties of
general mixtures models. Using the mixture representation od monotone non
increasing densities \eqref{eq:Will} we construct non parametric priors on the
set $\F$ by considering a prior on the mixing distribution $P$. We thus fall in
the set up of non parametric mixture prior models. We consider two types of
prior 
%
\begin{description}
\item[Type 1 : Dirichlet Process prior] $P \sim DP(A,\alpha)$ where $A$ is a
positive constant and $\alpha$ a probability distribution on $[0,L]$ \\
\item[Type 2 : Finite mixture] $P = \sum_{j=1}^K p_j \delta_{x_j}$ with $K$ a
non zero integer and $\delta_x$ the dirac distribution on $x$. We define a
prior on $P$ by  a distribution $Q$ on $K$ and given $K$, distributions
$\pi_{x,K}$ on $(x_1, \dots, x_K)$ and $\pi_{p,K}$ on $(p_1, \dots, p_K)$ 
\end{description}
%
Each of these are prior on the set of probability distributions. For $\Xn =
(X_1, \dots, X_n)$, a sample of $n$ independent and identically distributes
random variables with a common probability distribution function $f_0$ in $\F$
with respect to the Lebesgue measure, we denote $\Pi(\cdot|\Xn)$ the posterior
probability measure associated with the prior $\Pi$. Most of the results on the
convergence rates of posterior distribution requires that the prior puts enough
mass on Kullback Leiber neighborhood of the true density 
%
\begin{equation}\label{eq:KL1}
 \Pi(\{P,KL(f_P,f_0)\leq \epsilon \}) > 0
\end{equation}
%
where $KL(f_1,f2)$ is the Kullback Leiber divergence between $f_1$ and $f_2$
defined by $KL(f_1,f_2) = \int f_1\log(f_1/f_2)$. In particular, \citet{WG08}
proved that this condition is satisfied by the Type 1 prior in the case of
monotone non increasing densities, under mild conditions on $f_0$. The study of
the Type 2 prior is more delicate as it does not satisfy \eqref{eq:KL1} in
general. Moreover, it has been proved (see \cite{KHAZAEI:2010:INRIA-00494692:1}
) that for this choice of prior, $\Pi(\{P,KL(f_P,f_0)= \infty\}) = 1$. However,
we achieve to prove that the posterior based on a Type 2 prior concentrate at
the same rate without the Kullback Leiber property.

The paper is organised as follow. the main results are given in section
\ref{sec:main}, where conditions on the priors are discussed. The proof are
given afterward. 

\section{Main results} \label{sec:main}

Convergence rate of the posterior distribution have been well studied in the
literature and some general results links the rate to the prior (see
\cite{ggv00}). We deduce from these results the optimality of the Bayesian
estimator (up to a $\log(n)$ factor).The following theorem gives general
conditions on the prior to achieve a posterior convergence rate of
$(n/\log(n))^{-1/3}$. 

\begin{theo}\label{th1}
\begin{subequations}\label{eq:cond_th1}
Let $\Xn = (X_1,\dots,X_n)$ be a iid sample with a common probability
distribution function $f_0 \in \F$; and let $\alpha$ be a positive probability
density on $[0,1]$ with respect to the Lebesgue measure that satisfy for
$\theta$ close to $0$, and $t > 1$
\begin{equation}\label{eq:cond_alpha}
 \alpha(\theta) \lesssim \theta^t
\end{equation}
Define also $Q$ a probability distribution on $\N$ and $\pi_{p,K}$ a probability
distribution on the simplex of $\R^K$ satisfying for some positive constants
$C_1, C_2, a_1, \dots, a_K, c$ 
\begin{eqnarray}
 e^{-C_1 K \log(K)} \geq Q(K) \geq e^{-C_2 K \log(K)} \label{eq:cond_Q}
 \\
\pi_{p,k}(p_1,\dots, p_K) \geq K^{-K} c^K p_1^{a_1}\dots p_K^{a_K}
\label{eq:cond_p}
\end{eqnarray}
\end{subequations}
and finally, let $(x_i)_i$ be the order statistics of $K$ iid random variables
from $\alpha$. If $d$ is either the $L^1$ or Hellinger distance, then for $\Pi$
a Type 1 or Type 2 prior, there exists a positive constant $C$ such that 
\begin{equation}
 \Pi\left(f,d(f,f_0)\geq C \left({n \over \log(n)} \right)^{-1/3}|\Xn\right)
\to 0, \qquad P_0 \, a.s. 
\end{equation}
when $n$ goes to infinity. 
\end{theo}
The proof of this theorem is given in section \ref{sec:proof}. Condition
\eqref{eq:cond_th1} are roughly the same than in \citet{KHAZAEI:2010:INRIA-00494692:1}. This Theorem is thus an extension of their result when the density's support is compact. Given that $f_0$ is in $\F$, we can uniquely define $f_0(0)$ by considering the left limit of $f_0(x)$ when $x \searrow 0$. Let $f_0(0) = \lim_{x\searrow 0} f_0(x)$, under some mild conditions on the first derivative of $f$, we get the consistency and the convergence rate of the posterior distribution and of the Bayesian estimator associated with the absolute loss, namely, the posterior median. 

\begin{theo}
Let $f_0 \in \F$ such that, $f'(0^+) < 0$. Let $X_i \overset{iid}{\sim} f_0$ for $i= 1 \dots n$, and let $\Pi$ be a prior satisfying \eqref{eq:cond_th1}, then, if $\epsilon_n = C (n/\log(n))^{-1/3}$ 
\begin{equation}
P^{\pi}\big(|f_P(0) - f_0(0)|> \epsilon_n | \Xn \big) \to 0  
\end{equation}
Consider the posterior median $\fha_n(0) = \inf \{ x, P^\pi\big[f_p(0) \leq x \big] > 1/2 \}$ thus 
\begin{equation}
P_0\big(|\fha_n(0) - f_0(0)| > \epsilon_n | \Xn \big) \to 0  
\end{equation} 
\end{theo}
%
Thus the Bayesian approach is yield a consistent estimator of $f_0(0)$, which is not the case of the maximum likelihood approach. It is known that integrating the parameter as done in Bayesian approaches induces a penalization. This is particularly useful in testing or model choice problems but can also be effective in estimation problems, see for instance \citet{RSSB:RSSB781}. The problem of estimating $f_0(0)$ under monotonicity constraints is another example of the effectiveness of penalization induced by integration on the parameters. However, contrariwise to \citet{RSSB:RSSB781}, we have not clearly identified how the penalisation act, and only observe that it leads to a consistent posterior distribution and a consistent estimator. Furthermore, we only studied here the consistency in probability and it is not clear whether or not stronger consistency results, such as almost sure consistency, should be expected. 
\section{Proofs} \label{sec:proof}
We present here the proof of the two main theorem. The proof of Theorem \ref{th1} adapted from \citet{ggv00}, and is based on a piecewise constant approximation of $f_0$ in the sense of Kullback Leiber divergence. We therefore detail the construction of this approximation in the following section. Note that this is not the construction proposed in \citet{KHAZAEI:2010:INRIA-00494692:1} which was adapted to the $L^1$ distance but not to the Kullback Leiber divergence. This construction is adapted from the one proposed by \citet{WnVwcep} in the proof of theorem 2.7.5. 

\subsection{Proof of Theorem \ref{th1}}

The piecewise constant approximation of $f_0$ is base on a sequential subdivision of the interval $[0,L]$ with more refined subdivision where $f_0$ is less regular. We then identify the piecewise constant density by a mixture of uniformes. The following Lemma gives the form of the probability distribution $P$ such that $f_P$ is in the Kullback Leiber neighbourhood of $f_0$. 
%
\begin{lemm} \label{lem:KL}
Let $f \in \F$ be such that $f(0) \leq M < + \infty$. For all $0<\epsilon<1$ there exists $m \lesssim M^{2/3} \epsilon^{-1}$, $p = (p_1, \dots, p_m)$ and $x = (x_1,\dots, x_m)$ such that $P = \sum_{i=1}^m \delta_{x_i} p_i$ satisfy 
\begin{equation}
KL(f,f_P) \lesssim \epsilon^2, \; \int f \log\left({f\over f_P} \right)^2 \lesssim \epsilon^2
\end{equation}
where $f_P$ is defined as in \eqref{eq:Will} and $KL$ denote the Kullback Leiber divergence. 
\end{lemm}
%
\begin{proof}
For a fixed $\epsilon$, let $f$ be in $\F$. Consider $\P_0$ the coarsest partition : 
$$
0=x_0^0 < x_1^0 = L
$$
at the $i^{th}$ step, let $\P_i$ be the partition 
$$ 
0=x_0^i<x_1^i<\cdots<x_{n_i}^i = L
$$ 
%
and define 
$$
\eps_i = \max_j \left\{ (f(x_{j-1}^i) - f(x_j^i))(x_j^i - x_{j-1}^i)^{1/2}  \right\}
$$
%
For each $j\leq 1$, if $
(f(x_{j-1}^i) - f(x_j^i))(x_j^i - x_{j-1}^i)^{1/2} \geq {\eps_i \over \sqrt{2}} $ we split the interval $[x_{j-1}, x_j]$ into two subsets of equal length. We then get a new partition $\P_{i+1}$. We continue the partitioning until the first $k$ such that $\eps_k^2 \leq \epsilon^3$.
%
At each step $i$, let $n_i$ be the number of intervals in $\P_i$, $s_i$ the number of interval in $\P_i$ that have been divided to obtain $\P_{i+1}$, and $c = 1/\sqrt{2}$. Thus, it is clear that $\eps_{i+1} \leq 2^{-1/2} \eps_i = c\eps_i$ 

\begin{eqnarray*}
 s_i(c\eps_i)^{2/3} & \leq & \sum_j (f(x_{j-1}^i) - f(x_{j}^i) )^{2/3}(x_{j}^i-x_{j-1}^i) ^ {1/3} \\
                    & \leq & \left( \sum_j f(x_{j-1}^i) - f(x_{j}^i) \right)^{2/3}  \left( \sum_j x_{j}^i-x_{j-1}^i\right)^{1/3} 
                     \leq  M^{2/3}L^{1/3}
\end{eqnarray*}

using H\"older inequality. We then deduce that 

\begin{align*}
\sum_{j=1}^k n_j  =  k + \sum_{j = 1}^k j  s_{k-j}   \leq  2 \sum_{j = 1}^k j  s_{k-j} 
                & \leq 2 \sum_{j = 1}^k j  M^{2/3} L^{1/3} (c\eps_{k-j})^{- 2/3} \\
                & \leq  2  M^{2/3} L^{1/3} \eps_k^{-2/3}  2^{1/3} \sum_{j=1}^k j 2^{-j/3} \\
                & \leq   K_0  M^{2/3} L^{1/3} \eps_k^{-2/3}
\end{align*}
where $K_0$ is a constant. Thus 
\begin{equation} \label{eq:nk}
n_k \leq K_0  M^{2/3} L^{1/3} \epsilon^{-1}
\end{equation}

Now, let $f$ be in $\F$, we prove that there exists a stepwise density with less than $C \frac{1}{\epsilon}$ pieces, where C is a constant depending on $f$, such that 
\begin{equation} \label{eq:KL_fP}
KL(f,h) \leq \O(\epsilon^2) \text{  and  } \int f \log({f_0 \over f_P})^2(x) dx \leq \O(\epsilon^2)
\end{equation}

In order to lighten notations, we define 

$$
x_i = x_i^k, \quad
l_i = x_i - x_{i-1}, \quad  
g_i = f(x_{i-1})^{1/2} \quad 
$$

Thus consider the relation defined above with $f^{1/2}$ which is also monotone nonincreasing based on $g = \sum \I_{[x_{i-
1},x_i]} g_i $  

\begin{equation*}
\begin{split}
||f^{1/2}-g ||_2^2 = \int (f^{1/2}-g)^2(x) dx &= \sum_{i=1}^{n_k} \int_{I_i} (f^{1/2}-g)^2(x) dx \\
&\leq  \sum_{i=1}^{n_k} \int_{I_i} (f^{1/2}(x_{i-1}^k) - f^{1/2}(x_{i}^k))^2 dx \\
&\leq  \sum_{i=1}^{n_k} (x_i^k - x_{i-1}^k) (f^{1/2}(x_{i-1}^k) - f^{1/2}(x_{i}^k))^2 \\
&\leq  n_k \eps_k^2 \leq  K_0 M^{2/3} \epsilon^2 \\
\end{split}
\end{equation*} 

We then define $h = \frac{g^2}{\int g^2} $ and have 

\begin{align*}
 \int g^2 dx &= \int (g^2 - f)(x) dx + 1 \\ 
	     &= \int (g-\sqrt{f})(g+\sqrt{f})(x) dx +1\\
	     &= 1 + \O(\epsilon)
\end{align*}

and deduce that $(\int g^2)^{1/2} = 1 + \O(\eps)$. Let $H$ be the Hellinger distance  

\begin{align*}
H^2(f,h) &= H^2\left( f,\frac{g^2}{\int g^2} \right) \\ 
	 &\leq H^2(f,g^2) + H^2(g^2,\frac{g^2}{\int g^2}) \\
	 &\leq \epsilon^2 + \int (g - \frac{g}{(\int g^2)^{1/2}})^2(x) dx \leq \O(\epsilon^2) \\
\end{align*}

Since $|| h/f||_\infty = ||g/f||_\infty (\int g^2)^{-1} \leq (\int g^2)^{-1} $ together with the above bound on $H(f,h)$ and the following lemma 8 from \cite{GVdV07:Dirichlet}, we obtain the required result.

Let $P$ be a probability distribution defined by 
$$ 
P = \sum_{i=1}^{n_k} p_i \delta(x_i^k) \quad p_i = (h_{i-1} - h_{i})x_i^k \quad p_{n_k} = h_{n_k}x_{n_k}^k = h_{n_k}L
$$
thus $f_P = h$ and given the previous result, lemma \ref{lem:KL} is proved. 
\end{proof}

The proof of Theorem \ref{th1} is based on theorem 2.1 of \cite{ggv00}. It consists in obtaining a lower bound on the prior mass of Kullback Leiber neighbourhoods of any density in $\F$. An interesting feature of mixtures distributions having a parameter dependent support is that in many cases, the prior mass of sets the sets $\{f, KL(f_0,f) = + \infty \}$ is $1$ for most $f_0 \in \F$. Hence we cannot apply the result of \citet{ggv00}. We thus extended the approach used in \citet{KHAZAEI:2010:INRIA-00494692:1} to the convergence rate framework and get similar results as those presented in \citet{ggv00}. Thus we only have to bound from below, for a sequence $\theta_n$ such that $F_0(\theta_n)^n \in [1-\epsilon, 1-\epsilon/2]$ , the prior mass of sets 
\begin{equation}\label{eq:def_Sn}
S_n(\epsilon,\theta_n) = \left\{f, KL(f_n,f_{0,n}) \leq \epsilon^ 2, \int f_{0,n}(x) \log\left({f(x) \over f_0(x)} \right)^2dx \leq \epsilon^2 \right\}
\end{equation}
where 
\begin{equation} \label{eq:def_fn}
f_n(\cdot) = {f(\cdot) \I_{[0,\theta_n]}(\cdot) \over F(\theta_n) } ,\;  f_{0,n}(\cdot) = {f_0(\cdot) \I_{[0,\theta_n]}(\cdot) \over F_0(\theta_n) }
\end{equation}
% 
\begin{lemm}
Let $\Pi$ either a Type 1 or Type 2 prior on $\F$ satisfying \eqref{eq:cond_th1} and let $S_n(\epsilon,\theta_n)$ be a set as in \eqref{eq:def_Sn}, then 
\begin{equation}
\Pi(S_n(\epsilon, \theta_n) \gtrsim e^{C_1\epsilon^{-1}\log(\epsilon)} 
\end{equation}
\end{lemm}

Consider $\epsilon > 0$ and $\theta_n$ such that $F_0(\theta_n)^{n} \in [1-\epsilon, 1-\epsilon/2]$, thus $\theta_n \leq 1- \epsilon(2nf_0(0))^{-1}$ and define $f_{Pn}$ and $f_{0,n}$ as in \eqref{eq:def_fn}. Using lemma \ref{lem:KL} with $L = \theta_n$, we obtain that there exists a distribution $P=\sum_{i=1}^{n_k} \delta_{x_i} p_i$ such that for some constants $C_0, C'_0 > 0 $ such that  
%
$$
KL(f_{0,n},f_{Pn}) \leq C_0 \epsilon^2, \text{ and } \int f_{0,n} \log\left({f_{0,n}\over f_{Pn}} \right)^2 \leq C'_0 \epsilon^2
$$
%
Note that $f_P$ has support $[0, \theta_n]$ and put a positive mass on $\theta_n$. Now, set $m = n_k$ and consider $P'$ the mixing distribution associated with $\{m, x'_1,\dots,x'_m, p'_1 \dots, p'_m \}$ with $\sum_{i=1}^{m} p'_i = 1$. Define for $1 \leq j \leq m-1$ the set $U_i = [0\vee (x_i - \epsilon^3, x_i + \epsilon^3]$ and $U_m = (\theta_n, \theta_n + \epsilon(1-\theta_n) \wedge \epsilon^3]$. Construct $P'$ such that $x'_i \in U_i$ and $|P'(U_i) - p_i| \leq \epsilon^2 n^{-1}$ 
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