# Nonparametric Probability Density Function Estimation Using the Pade Approximation
## 5 Quantitative and Comparative Assessment
### Example 4: The Exponential Distribution  
The above example illustrates the nature of moment-based methods, which are flexible with a limited number of samples. The next distribution model that we consider is the Exponential distribution. The Exponential distribution is commonly used to model the time between events in a Poisson process and is defined by a rate parameter, \(\lambda\). The formula for the Probability Density Function (PDF) of the Exponential distribution is shown in Equation (40). In our experiments, \(\lambda\) was set to 1. The results of the simulations are given in Figure 9.  
\[f(x;\lambda)=\lambda e^{-\lambda x}\quad\text{for}\quad x\geq 0 \tag{40}\]  
Upon reviewing the data shown in Figure 7 and Tables 10-12, it is noticeable that the effectiveness of all techniques has decreased. The main cause for this decline is attributed to the fact that the Exponential distribution presents difficulties in capturing its characteristics near zero, which poses a challenge for the techniques to accurately depict its traits in this region. In some instances, it seems that the One-Point Pade approximation demonstrates better performance. The reason for this could be that the One-Point Pade approximation introduces a distortion near zero, which in this scenario appears advantageous at first glance. The distortion aids in approximating the behaviour near zero to match the characteristics \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Method** & \(\downarrow\)**WD** & \(\downarrow\)**BD** & \(\uparrow\)**CC** & \(\downarrow\)**KLD** & \(\downarrow\)**L1** & \(\downarrow\)**L2** \\ \hline KDE & 19.4452 & 4.4273 & 0.9888 & 0.00025 & 0.1274 & 0.0053 \\ \hline Adaptive KDE & 15.8213 & 4.4260 & 0.9840 & 0.00027 & 0.1285 & 0.0051 \\ \hline GMM & 14.5563 & 4.4266 & 0.9680 & 0.00033 & 0.1496 & 0.0065 \\ \hline QHBS & **9.7451** & 4.4308 & 0.9265 & 0.00031 & 0.1876 & 0.0101 \\ \hline One-Point Pade & 13.4854 & 4.4250 & 0.9926 & 0.00024 & 0.0899 & 0.0037 \\ \hline Proposed Method & 13.9661 & **4.4245** & **0.9950** & **0.00200** & **0.0826** & **0.0032** \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison of PDF approximation methods for the Gamma distribution using 300 samples. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Method** & \(\downarrow\)**WD** & \(\downarrow\)**BD** & \(\uparrow\)**CC** & \(\downarrow\)**KLD** & \(\downarrow\)**L1** & \(\downarrow\)**L2** \\ \hline KDE & 18.6965 & 4.4309 & 0.9822 & 0.00265 & 0.1552 & 0.0065 \\ \hline Adaptive KDE & 15.2305 & 4.4316 & 0.9506 & 0.00285 & 0.1996 & 0.0081 \\ \hline GMM & 9.8272 & 4.4266 & 0.9519 & 0.00310 & 0.1474 & 0.0079 \\ \hline QHBS & 8.0250 & 4.4290 & 0.9394 & 0.00360 & 0.1900 & 0.0090 \\ \hline One-Point Pade & 9.4705 & 4.4255 & 0.9772 & 0.00235 & 0.1226 & 0.0057 \\ \hline Proposed Method & **7.3515** & **4.4245** & **0.9928** & **0.00205** & **0.0794** & **0.0035** \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Performance metrics for different methods for the Gamma distribution using 100 Samples. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Method** & \(\downarrow\)**WD** & \(\downarrow\)**BD** & \(\uparrow\)**CC** & \(\downarrow\)**KLD** & \(\downarrow\)**L1** & \(\downarrow\)**L2** \\ \hline KDE & 12.0644 & 4.4329 & 0.9810 & 0.0250 & 0.1355 & 0.0058 \\ \hline Adaptive KDE & 12.8983 & 4.4406 & 0.9301 & 0.0300 & 0.2514 & 0.0096 \\ \hline GMM & 11.0408 & 4.4308 & 0.9749 & 0.0275 & 0.1451 & 0.0058 \\ \hline QHBS & 15.4045 & 4.4440 & 0.8379 & 0.0350 & 0.3498 & 0.0163 \\ \hline One-Point Pade & 11.4033 & 4.4283 & 0.9657 & 0.0225 & 0.1563 & 0.0067 \\ \hline Proposed Method & **10.5327** & **4.4272** & **0.9872** & **0.0200** & **0.1052** & **0.0041** \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Performance metrics for different methods for the Gamma distribution using 50 samples. of the Exponential function in that region.