# Efficient Estimation of Generative Models Using Tukey Depth
## 5 Synthetic Experiments
### Experiments and Results  
Given observations \(\overline{y}_{dt},\ d=1,\ldots,D,\ t=1,\ldots,T\), we consider the problem of estimating the parameters \(\phi\), \(\beta\) and \(\lambda\) of the beta and exponential distributions formulated  
Figure 6: Example plot of the synthetic data \(\overline{y}_{1t}\). Figure 7: **Left panel**: Beta distribution with shape parameters \(\phi=6\) and \(\beta=4\). **Right panel**: Exponential distribution with rate \(\lambda=10\). above. In [PERSON] et al. (2018), [PERSON] et al. presented the resulting likelihood functions and explained why it is not possible to compute them. We, therefore, resort to ABC to estimate the parameters, and below we explain the summary statistics used. For each day \(d\), we suggest the following three statistics  
\[S_{1d}\coloneqq\frac{1}{T}\sum_{t=1}^{T}\overline{y}_{dt}, \tag{17}\]  
\[S_{2d}\coloneqq\sum_{t=1}^{T}t\cdot\left(\frac{\overline{y}_{dt}}{\sum_{t=1}^{ T}\overline{y}_{dt}}\right), \tag{18}\]  
\[S_{3d}\coloneqq\sum_{t=1}^{T}t^{2}\left(\frac{\overline{y}_{dt}}{\sum_{t=1}^{ T}\overline{y}_{dt}}\right)-\left[\sum_{t=1}^{T}t\cdot\left(\frac{\overline{y}_{dt}}{ \sum_{t=1}^{T}\overline{y}_{dt}}\right)\right]^{2}. \tag{19}\]  
The reasons for these choices of statistics are as follows. \(S_{1d}\) is the mean CPU consumption on day \(d\), and is directly related to the expected sizes of the tasks, \(\lambda\), since the number of tasks per day is assumed to be known. Next, \(S_{2d}\) is the mean time of the day when tasks are received, that is, are the tasks mainly received early on the day or later in the day. This is directly related to the expectation of the Beta distribution in Equation (13). Finally, \(S_{3d}\) is the variance in when the tasks are received, i.e., whether they are all received in a small time interval or more evenly spread throughout the day. This statistic is directly related to the variance of the Beta distribution. In summary, these three statistics should be able to capture the main properties of the data to efficiently estimate the three unknown parameters \(\phi\), \(\beta\), and \(\lambda\). Computing the three statistics for \(D\) days results in a total of \(3D\) statistics. \[\mathbf{S}_{1}\coloneqq(S_{11},\dots,S_{1D}),\quad\mathbf{S}_{2}\coloneqq(S_{ 21},\dots,S_{2D}),\quad\mathbf{S}_{3}\coloneqq(S_{31},\dots,S_{3D}). \tag{20}\]  
Recall from the last passage of Section 4 that the Tukey depth contours are especially useful to reduce the high dimensionality of the set of summary statistics. Therefore, to sufficiently extract the main properties of the \(3D\)-dimensional distribution of \((\mathbf{S}_{1},\mathbf{S}_{2},\mathbf{S}_{3})\), we estimate the Tukey depth contours of the distribution of the statistics over the \(D\) days. In other words, we use \((\mathbf{S}_{1},\mathbf{S}_{2},\mathbf{S}_{3})\) as \(\mathbf{X}\) in the method in Section 4. As an alternative, we could simply compute the average of the statistics over all the \(D=40\) each day, but given that the statistics are not sufficient ([PERSON] et al., 2018) for the unknown parameters, computing the Tukey depth contours will be able to provide us with more information, which again will improve the estimation of the unknown parameters. We assumed that the CPU usage times were observed for \(D=40\) days and that the observations were the average CPU usage in \(T=144\) disjoint time intervals per day, i.e., \(10\) min intervals. Finally, we assumed that the number of tasks per day was \(N_{d}=20\) for \(d=1,\dots,40\).