## 1 Introduction  
Consider a family \(f(x;\,r)\) of functions such that for every \(r\in[R_{1},\,R_{2}]\subset\mathbb{R}\,f(\cdot;\,r):\mathbb{R}\to\mathbb{R}\) is a probability density function (p.d.f.) on real line \(\mathbb{R}\), while non-random parameter \(r\) takes values from a finite interval \([R_{1},\,R_{2}]\). Assume that we have observations \((\kappa_{l},\,r_{l})\), \(l=1,\,2,\,\ldots d\) at our disposal, where for \(r_{l}\in[R_{1},\,R_{2}]\) observation \(\kappa_{l}\) is drawn at random according to p.d.f. \(f(x;\,r_{l})\), \(x\in\mathbb{R}\). Our aim is to propose, under mild assumptions, a nonparametric estimator of the whole continuum set of p.d.f. 's \(\mathcal{F}=\{f(\cdot;\,r),\,\,r\in[R_{1},\,R_{2}]\}\), assuming that the number of data \(d\to\infty\) and that \(r_{l}\)'s cover \([R_{1},\,R_{2}]\) more and more densely as \(d\to\infty\). Later on, we shall refer to the above stated problem as the \(\mathcal{F}\)-estimation problem. In this paper, we concentrate mainly on the algorithmic aspects of the \(\mathcal{F}\)-estimation problem, since it is computationally demanding. However, we also provide an outline of the proof that the proposed learning algorithm is convergent in the integrated mean squared error (IMSE) sense. Before providing motivating examples, we briefly indicate similarities, differences, and a generality of this problem among other nonparametric estimation tasks that were considered from the 1950s [1; 2; 3; 4]:  
1. The \(\mathcal{F}\)-estimation problem has certain similarities to the nonparametric estimation problem of a bivariate p.d.f. Notice, however, the important difference, namely in our case \(r\) is a non-random parameter. In other words, our sub-task is to select \(r_{l}\)'s in an appropriate way (or to confine ourselves to such an interval \([R_{1},\,R_{2}]\) which is covered densely by passive observations of pairs \((\kappa_{l},\,r_{l})\), \(l=1,\,2,\,\ldots d\)). 2. One can also notice a formal similarity of our problem and the problem of nonparametric estimation of a non-stationary p.d.f. \(f(x;\,t)\), say that was introduced to the statistical literature by [PERSON] (see [5; 6; 7] ) and continued in the papers on the concept drift tracking (see [8; 9; 10]). There is, however, a fundamental difference between the time \(t\) and parameter \(r\). Namely, time is not reversible and we usually do not have an opportunity to repeat observations at instants preceding present \(t\). On the contrary, in the \(\mathcal{F}\)-estimation problem, we allow the next observation to be done at \(r_{l+1}<r_{l}\). Furthermore, we allow also for repeated observations for the same value of \(r\). 3. The estimation of several p.d.f. 's was considered in [11], where it was pointed out that it is a computationally demanding problem, because all of these densities should be estimated simultaneously. However, the goal of this paper is quite different than ours. Namely, in [11], the goal was to compare several densities that are not necessarily indexed by the same additional parameter, which does not arise as a parameter of an estimator. 4. Denote by \(\hat{f}(. ;\,r)\) nonparametric estimators of \(f(.,\,r)\), \(r\in[R_{1},\,R_{2}]\). Having them at our disposal, we immediately obtain nonparametric estimators of a regression function: \(\int_{-\infty}^{\infty}x\,\hat{f}(x;\,r)\,dx\) with \(r\) as the input variable. 5. Similarly, calculating the median of \(\hat{f}(. ;\,r)\), we obtain an estimator of the median regression on \(r\). Analogously, estimators of other quantile regression functions can be obtained. 6. When we allow that \(x\) and/or \(r\) if \(f(x;\,r)\) are vectors, then the \(\mathcal{F}\)-estimation problem covers also multivariate density and regression estimation problems. In this paper, we do not follow these generalizations, since even for univariate \(x\) and \(r\) we obtain computationally and data demanding problem. On the other hand, we propose double orthogonal expansion as the base for solving the \(\mathcal{F}\)-estimation problem.