## 4 Results  
### Hypotheses on \(f\)  
In this paragraph, we define the set of hypotheses on \(f\) which could possibly be of use in our work. Discussion on several of these hypotheses can be found in Appendix E.  
In the remaining of this section, for legibility reasons, we replace \(g\) with \(g^{(k-1)}\). Let  
\[\Theta=\mathbb{R}^{d},\;\Theta^{D_{\phi}}=\left\{b\in\Theta\;|\;\; \int\varphi^{*}\left(\varphi^{\prime}\left(\frac{g(x)}{f(x)}\frac{f_{b}(b^{ \top}x)}{g_{b}(b^{\top}x)}\right)\right)d\mathbf{P}<\infty\right\}\] \[M(b,a,x)=\int\varphi^{\prime}\left(\frac{g(x)}{f(x)}\frac{f_{b}( b^{\top}x)}{g_{b}(b^{\top}x)}\right)g(x)\frac{f_{a}(a^{\top}x)}{g_{a}(a^{ \top}x)}dx-\;\varphi^{*}\left(\varphi^{\prime}\left(\frac{g(x)}{f(x)}\frac{f_{ b}(b^{\top}x)}{g_{b}(b^{\top}x)}\right)\right)\] \[\mathbb{P}_{n}M(b,a)=\int M(b,a,x)d\mathbb{P}_{n},\;\mathbf{P}M(b,a )=\int M(b,a,x)d\mathbf{P}\]  
where \(\mathbf{P}\) is the probability measure presenting \(f\) as density. Similarly as in chapter \(V\) of [PERSON] [22], let us define :  
(_A1_) : For all \(\varepsilon>0\), there is \(\eta>0\), such that for all \(c\in\Theta^{D_{\phi}}\) verifying \(\|c-a_{k}\|\geq\varepsilon,\)  
we have \(\mathbf{P}M(c,a)-\eta>\mathbf{P}M(a_{k},a),\;\text{with }a\in\Theta.\)  
(_A2_) : \(\exists\;Z<0\), \(n_{0}>0\) such that \((n\geq n_{0}\;\Rightarrow\;\sup_{a\in\Theta}\sup_{c\in\{\Theta^{D_{\phi}}\}^{ \varepsilon}}\mathbb{P}_{n}M(c,a)<Z)\)  
(_A3_) : There exists \(V\), a neighbourhood of \(a_{k}\), and \(H\), a positive function, such that, for all \(c\in V\),  
we have \(|M(c,a_{k},x)|\leq H(x)(\mathbf{P}-a.s.)\) with \(\mathbf{P}H<\infty,\)  
(_A4_) : There exists \(V\), a neighbourhood of \(a_{k}\), such that for all \(\varepsilon\), there exists a \(\eta\) such that for all \(c\in V\)  
and \(a\in\Theta\), verifying \(\|a-a_{k}\|\geq\varepsilon,\;\text{we have }\mathbf{P}M(c,a_{k})<\mathbf{P}M(c,a)-\eta.\)  
Putting \(I_{a_{k}}=\frac{\partial^{2}}{\partial a^{2}}D_{\phi}(g\frac{f_{a_{k}}}{g_{a_{ k}}},f)\), let us consider now four new hypotheses:  
(_A5_) : \(\mathbf{P}\|\frac{\partial}{\partial b}M(a_{k},a_{k})\|^{2}\) and \(\mathbf{P}\|\frac{\partial}{\partial a}M(a_{k},a_{k})\|^{2}\) are finite and the expressions \(\mathbf{P}\frac{\partial^{2}}{\partial b_{i}\partial b_{j}}M(a_{k},a_{k})\) and \(I_{a_{k}}\)  
exist and are invertible. (_A6_) : There exists \(k\) such that \(\mathbf{P}M(a_{k},a_{k})=0\). \begin{table}
\begin{tabular}{c l} \hline \hline \(0.\) & We define \(g\), a density with same mean and variance as \(f\) and we set \(g^{(0)}=g\). \\ \hline \(i-1.\) & Given \(\check{g}_{n}^{(i-1)}\), find \(\check{a}_{i}\) such that the index \\  & \(\check{D_{\phi}}\left(\check{g}_{n}^{(i-1)}\frac{f_{a_{n}}}{\check{g}_{a,n}^{(i -1)}},f_{n}\right)=\frac{1}{n}\Sigma_{i=1}^{n}\varphi\left(\frac{\check{g}_{n} ^{(i-1)}(X_{i})\check{f}_{a_{n}}(a^{\top}X_{i})}{f_{n}(X_{i})\check{g}_{a,n}^{(i -1)}(a^{\top}X_{i})}\right)\) \\  & is minimized, where \(f_{a,n}\) is a marginal density estimate based on \(a^{\top}X_{1}\), \(a^{\top}X_{2}\),...,\(a^{\top}X_{n}\), \\  & and where \(\check{g}_{a,n}^{(i-1)}\) is a density estimate based on the projection to \(a\) of a Monte Carlo random sample from \(\check{g}_{n}^{(i)}\). \\  & And we set \\  & \(\check{g}_{n}^{(i)}=\check{g}_{n}^{(i-1)}\frac{f_{a_{1},n}}{\check{g}_{a_{1},n}^ {(i-1)}}\) \\ \(i.\) & Then we replace \(\check{g}_{n}^{(i-1)}\) with \(\check{g}_{n}^{(i)}\) and go back to \(i-1\) until the criteria reaches the stopping rule of this procedure (see below). \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Stochastic outline of the algorithm. _. _(A7)_: \((Var_{\mathbf{P}}(M(a_{k},a_{k})))^{1/2}\) exists and is invertible. _(A0)_: \(f\) and \(g\) are assumed to be positive and bounded and such that \(K(g,f)\geq\int|f(x)-g(x)|dx\)  
where \(K\) is the Kullback-Leibler divergence.