# Nonparametric Probability Density Function Estimation Using the Pade Approximation
## 1 Introduction  
Most natural events are stochastic [1] because they involve uncertain interactions of random internal \"environments\". To study these events, researchers have applied probability theory, which is a tool encompassing a branch of mathematics that aids in the analysis of the behavior of stochastic events. Probability theory, in particular, and stochastic processes, in general, are among the most fundamental fields of mathematics. They have applications in signal processing, pattern recognition, classification, forecasting, medical diagnosis, and many other applications. In practice, probability distributions find application in diverse fields such as electron emission, radar detection, weather prediction, remote sensing, economics, noise modelling, etc. [2; 3; 4; 5]. Once the probabilistic model is well-defined, subsequent analysis often involves the theory of \"estimation\" (synonymously, learning, or training, depending on the application domain). Such estimation is always valuable, but it is nonetheless challenging [6], especially when the available data are limited [7; 8]. In this context, the PDF of a random variable is central to various aspects of the analysis and characterization of the data and the model. Many studies have emphasized the need for estimating the PDF [9], which is challenging due to the absence of efficient construction methods. Suppose that \(X_{N}\) is a set of observed data with \(N\) samples randomly distributed with an unknown PDF. The given \(X_{N}\) is characterized by its PDF (\(f(x)\)) of the underlying random variable, which satisfies the fundamental properties of a PDF:  
\[ \begin{split}\int_{-\infty}^{+\infty}f(x)\,dx&=1, \\ f(x)&\geq 0.\end{split}  \tag{1}\]  
Once the PDF has been estimated based on a well-defined criterion, one also needs an assessment method to compare \(f(x)\) with the set of observed data to identify the accuracy of our estimation. This PDF can be used to study the statistical properties of the dataset, estimate future outcomes, generate more samples, and so on. Numerous methods have been proposed for estimating the PDF, and these can be classified into two approaches, namely, using parametric or nonparametric models  
Unlike parametric models, which make assumptions about the parametric form of the PDF, nonparametric density estimation does not assume the form of the distribution from which the samples are drawn [10]. In parametric models, one develops a model to fit a probability distribution for the observed data, followed by the determination of the parameters of this distribution. The procedure is summarized into four steps:  
1. Select some known candidate probability distribution models based on the physical characteristics of the process and known characteristics of the distribution of the available samples. 2. Determine the parameters of the candidate models. 3. Assess the goodness-of-fit of the fitted generative model using established statistical tests and graphical analysis. 4. Choose the model that yields the best results [2; 11; 12]. However, these methods cannot cover all data because some datasets do not follow specific distributional shapes. As a result, these methods may fail to accurately approximate the true but unknown PDF [3]. Nevertheless, they contribute to solving many applications. In nonparametric density estimation, we resort to techniques to find a model to fit the idealistic probability distribution of the observed data [13]. For example, a histogram approximation is one of the earliest density estimation methods, introduced by [PERSON] [14]. Here, the entire range of values is divided into a series of intervals, referred to as \"bins\", followed by adding up the number of samples falling into each bin. Although the so-called bin width is an arbitrary value, extensive research has been done to find a robust way to indicate the effective bin width [15; 16]. Moreover, some bins will be empty, while others may have few occurrences.