# Nonparametric Probability Density Function Estimation Using the Pade Approximation
## 5 Quantitative and Comparative Assessment  
In this section, we report the results obtained by conducting several experiments to assess the performance of the proposed method. The experiments can be summarized as follows. Firstly, the proposed method was applied to datasets generated from well-known probability distributions. The results were then compared with the ideal PDF. Subsequently, the proposed method was compared with other state-of-the-art methods. We aimed to show that our new approach could produce favourable outcomes when working with fewer samples, i.e., in scenarios where data are limited, which is a common occurrence in real-world situations. In addition, we have made the MATLAB code, implementing all the proposed methods, publicly available in the following GitHub repository: [[https://github.com/hamidddds/Twosited_PadeAproximation](https://github.com/hamidddds/Twosited_PadeAproximation)]([https://github.com/hamidddds/Twosited_PadeAproximation](https://github.com/hamidddds/Twosited_PadeAproximation)) (accessed on 23 January 2025). In our research project, we conducted a series of tests to thoroughly investigate the problem using three datasets of varying sizes. The first dataset consisted of 300 samples, which we categorized as a _large_ dataset, providing a robust representation for comprehensive analysis. The second dataset contained 100 samples, classified as a _medium-sized_ dataset, offering a sufficient number of samples for a balanced evaluation. Lastly, the third dataset comprised only 50 samples, representing a _very small_ dataset specifically designed to simulate challenging scenarios in which access to extensive data is limited. This enabled us to assess how well the performance varied across the sample sizes. In real-world applications, sample sizes of 50 are often insufficient to accurately calculate all data properties in specific ranges. In many cases, scientists address this limitation by making assumptions about the underlying structure of the density and then using parametric methods to estimate the parameters, although the overall accuracy of the results tends to decrease, particularly when incorrect assumptions are made regarding the density. Despite these challenges, we included these small sample sizes to demonstrate that, even in harsh conditions, our method can capture some characteristics of the data and perform reasonably well. However, it is important to note that such small sample sizes are highly sensitive to data sparsity, which can significantly impact performance. Clearly, it is important to mention that the accuracy of all estimations improved with increasing numbers of samples. Figure 5: Estimation of Gaussian density function by considering the negative values. The blue and red lines show the estimated and ideal density functions, respectively. In this article, we employ five techniques for comparison with our proposed approach: Kernel Density Estimation (KDE), Adaptive KDE, Gaussian Mixture Model (GMM), Quantized Histogram-Based Smoothing (QHBS), and the one-point approximation model. Since our method is based on the one-point Pade approximation model, we have included it as a reference point for evaluating our approach. The KDE is a nonparametric technique that smooths data points to estimate the underlying PDF without assuming a specific distribution. It employs a kernel function, such as the Gaussian, and a bandwidth parameter to control the level of smoothing. In this study, a Gaussian kernel was used, with the bandwidth calculated using [PERSON]'s rule of thumb [32]. The Adaptive KDE enhances the standard KDE by dynamically adjusting the bandwidth based on the density of data points. This adaptability enables it to handle varying data densities efficiently, capturing intricate details in densely populated regions while ensuring uniformity in sparser areas. The Adaptive KDE utilized the same Gaussian kernel but varied the bandwidth locally to reflect data density.