# Nonparametric Probability Density Function Estimation Using the Pade Approximation
## 1 Introduction
Thus, the PDF will ultimately vary between adjacent bins because of fluctuations in the number of occurrences. By increasing the bin size, each bin tends to contain more samples [17]; however, many important details will be filtered out. This weakness causes the inferred PDF to be discontinuous, and so it cannot be viewed as an accurate method, especially if the derivatives of the histogram are required. These drawbacks led to the development of more advanced methods, such as Kernel Density Estimation (KDE), Adaptive KDE, Gaussian Mixture Model (GMM), Quantized Histogram-Based Smoothing (QHBS), and the one-point Pade approximation model [18]. KDE [19] is a common density estimator resulting in smooth and continuous densities. It estimates the PDF by deploying a kernel function at each data point and then computing the sum of these kernels to construct a nonparametric estimate of the PDF. The kernel is usually a symmetric, differentiable function, such as the Gaussian. The bandwidth hyperparameter, which defines the width of the kernel, is crucial in determining the smoothness of the resulting estimate. Adaptive KDE [20] is a form of KDE in which the bandwidth is not constant but instead changes according to the local density of the data. This approach allows for a more accurate depiction of the features of the distribution, especially in regions with dense data, while producing a smoother estimate where there are fewer data points. GMM [21; 22] is a parametric density estimation technique that assumes that the data have been produced by a weighted sum of Gaussian distributions. GMM models the mixture density by estimating the parameters of the (assumed) source Gaussian distributions. Each component of the mixture in the form of Gaussian has its own mean and covariance, while the overall density function is the weighted sum of these individual Gaussian densities. GMM offers a number of advantages over other models and can capture more complex and multi-modal distributions. It is, however, a parametric approach whose accuracy depends on the degree to which its underlying assumptions are met. QHBS [23] is an approach to address the challenges posed by conventional histograms. It approximates the PDF by smoothing a histogram in a manner that reduces the discontinuities and enhances the smoothness of the resulting density estimate. This approach splits the data into sets of bins and then employs a quantization procedure to soften the estimates and thus produce a smoother, continuous, and differentiable representation of the PDF. Each of these methods has its strengths and challenges. For instance, while KDE and Adaptive KDE offer smooth and continuous density estimates, selecting the appropriate kernel function and bandwidth can be difficult. On the other hand, methods like GMM and QHBS can address multimodal and discontinuous distributions effectively, but GMM requires knowledge of the number of component distributions, while QHBS is sensitive to the number of bins used in the underlying histogram. Despite these challenges, these methods are widely used in practice due to their ability to provide accurate and flexible density estimates in various applications. Although we have presented only a rather brief review of these schemes, it can serve as a backdrop to our current work. In this paper, we shall argue that, unlike the previous methods, the use of the _moments_ of the data can serve to yield an even more effective method to estimate the PDF, which also quite naturally leads us to the Pade transform, explained below. _Note that we will synonymously refer to this phenomenon as the Pade approximation and the Pade Transform._ [PERSON] and [PERSON] first proposed the use of the Pade approximation for estimating PDFs [24]. We include [PERSON]'s original method in our quantitative evaluations below, here referred to as the \"one-point Pade approximation model\". As originally formulated, this approach suffers from several challenges, which are mentioned in [24].