# Nonparametric Probability Density Function Estimation Using the Pade Approximation
## 6 Discussion and Conclusions  
In this paper, we have considered the problem of estimating the PDF of observed data. Our novel scheme uses two powerful mathematical tools: the concept of moments and the relatively little-known Pade approximation. While moments incorporate crucial information which is central to both the time and frequency domains, the Pade approximation provides an effective means of obtaining convergent series from the data. Both of these phenomena are used in our new method to estimate the PDF in an inter-twined manner. The method we propose is _nonparametric_. It invokes the concept of _matching_ the moments of the original function and its learned PDF, and this, in turn is achieved by using the Pade approximation. Apart from the theoretical analysis, we have also experimentally evaluated the validity and efficiency of our scheme. Although the Pade approximation is asymmetric, we have taken advantage of this asymmetry to our advantage. We have done this by working with two \"mirrored\" versions of the data so as to obtains different versions of the PDF. We have then effectively \"superimposed\" (or coalesced) them together to yield the final composite PDF. We are not aware of any other research that utilizes such a composite strategy, in any signal processing domain. To evaluate the performance of the proposed method, we have employed synthetic samples obtained from various well-known distributions, including mixture densities. The accuracy of the proposed method has also been compared with that obtained by other approaches representative of the families of time- and frequency-domain methods available in the literature. Our method has shown promising results across the distributions tested. However, we recognize the potential for further exploration. While our chosen examples demonstrate the method's efficacy, future work could involve testing on a broader range of distributions, including heavy-tailed and multimodal distributions. This extended testing could provide additional insights into the method's performance under diverse scenarios and potentially uncover new areas for refinement or application. Moreover, in future research, we intend to explore methods that could enhance the robustness of the proposed approach, particularly in handling outliers. The current method, like other nonparametric density estimation techniques, may be sensitive to extreme values, as sample moments, which can be influenced by outliers. Although this issue can be mitigated through outlier detection and censoring prior to density estimation, we anticipate a potential for improvement. One promising direction would be to investigate the integration of more robust estimators for sample moments into the density estimation method. This could potentially increase the method's resilience to outliers, without compromising its performance on \"clean\" data, thereby expanding its applicability to datasets possessing more challenging characteristics. The results underscore the robustness and effectiveness of our method, particularly in scenarios when the sample sizes are considerably reduced. Thus, this research confirms \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Method** & \(\downarrow\)**WD** & \(\downarrow\)**BD** & \(\uparrow\)**CC** & \(\downarrow\)**KLD** & \(\downarrow\)**L1** & \(\downarrow\)**L2** \\ \hline KDE & 21.9270 & 5.3114 & 0.9694 & 0.0017 & 0.1903 & 0.0157 \\ \hline Adaptive KDE & 14.0566 & 5.3204 & 0.9579 & 0.0106 & 0.2266 & 0.0195 \\ \hline GMM & 20.7139 & 5.3246 & 0.9112 & 0.0009 & 0.3420 & 0.0334 \\ \hline QHBS & 20.3703 & 5.3223 & 0.9123 & 0.0006 & 0.3835 & 0.0339 \\ \hline One-Point Padé & **13.4794** & 5.3203 & 0.9296 & 0.0017 & 0.2398 & 0.0306 \\ \hline Proposed Method & 16.5618 & **5.3111** & **0.9897** & **0.0003** & **0.1702** & **0.0102** \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Performance metrics for different methods for the Exponential distribution using 50 samples.