# Nonparametric Probability Density Function Estimation Using the Pade Approximation
## 5 Quantitative and Comparative Assessment
### Example 2: The Beta Distribution  
The next distribution model that we consider is the Beta distribution. The Beta distribution is a continuous probability distribution defined on the interval [0, 1], making it suitable for modelling probabilities and proportions. It is governed by two shape parameters, \(\alpha\) and \(\beta\), which influence the distribution's shape, allowing it to be symmetric, skewed, or U-shaped. The Beta distribution is extensively utilized in Bayesian statistics as a conjugate prior for binomial and Bernoulli processes, enabling efficient updates of beliefs based on observed data. In our experiment, the parameters were chosen as \(\alpha=2\) and \(\beta=5\) to evaluate the performance of our method in comparison to other methods. The formula for the PDF of the Beta distribution is shown in Equation (38). The results of the simulations are given in Figure 7.  
\[f(x;\alpha,\beta)=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{ \alpha-1}(1-x)^{\beta-1}\quad\text{for}\quad 0\leq x\leq 1 \tag{38}\]  
The results (see Tables 4-6) demonstrate that our method outperformed competing approaches and secured first place in the competition. For a sample size of 50, the results \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 & 32.4425 & 2.3137 & 0.9324 & 0.0045 & 0.2329 & 0.0084 \\ \hline Adaptive KDE & 28.8019 & 2.3091 & 0.9369 & 0.0029 & 0.1939 & 0.0074 \\ \hline GMM & **28.2692** & **2.3050** & 0.9672 & **0.0013** & 0.1312 & 0.0053 \\ \hline QHBS & 30.1941 & 2.3082 & 0.9272 & 0.0026 & 0.1877 & 0.0087 \\ \hline One-Point Padé & 30.4723 & 2.3088 & 0.9496 & 0.0032 & 0.1605 & 0.0065 \\ \hline Proposed Method & 31.4479 & 2.3057 & **0.9712** & 0.0016 & **0.1163** & **0.0049** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Performance metrics for different methods for the Gaussian mixture using 300 samples. Here and elsewhere, bold is used to indicate the best results. \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 & 30.5430 & 2.3143 & 0.9195 & 0.0048 & 0.2430 & 0.0087 \\ \hline Adaptive KDE & 28.4085 & 2.3094 & 0.9494 & 0.0031 & 0.1782 & 0.0065 \\ \hline GMM & **28.2237** & 2.3105 & 0.9501 & 0.0038 & 0.1590 & 0.0068 \\ \hline QHBS & 28.9642 & 2.3102 & 0.9237 & 0.0034 & 0.2033 & 0.0092 \\ \hline One-Point Padé & 28.7078 & 2.3088 & 0.9572 & 0.0036 & 0.1462 & 0.0061 \\ \hline Proposed Method & 29.5758 & **2.3049** & **0.9742** & **0.0013** & **0.1185** & **0.0047** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Performance metrics for different methods for the Gaussian mixture 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 & 39.6778 & 2.3137 & 0.9191 & 0.0046 & 0.2352 & 0.0083 \\ \hline Adaptive KDE & 38.3835 & 2.3138 & 0.9243 & 0.0052 & 0.2331 & 0.0086 \\ \hline GMM & 39.8234 & 2.3127 & 0.9452 & 0.0049 & 0.2189 & 0.0095 \\ \hline QHBS & 40.0201 & 2.3255 & 0.8182 & 0.0095 & 0.3795 & 0.0170 \\ \hline One-Point Padé & 40.4137 & 2.3102 & 0.9659 & 0.0045 & 0.1729 & 0.0062 \\ \hline Proposed Method & **38.8917** & **2.3074** & **0.9672** & **0.0025** & **0.1567** & **0.0059** \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Performance metrics for different methods for the Gaussian mixture using 50 samples. indicate that KDE achieved superior performance in terms of the Correlation Coefficient, showing an advantage in this specific criterion. However, across other evaluation metrics our method consistently delivered better performance.