Figure 2

Boxplots showing the distribution of estimated values for parameter \(\beta\) from each of the \(m=1000\) samples with \(n=500\). The X-data in the samples contains a dragon-king. The horizontal axis represents the value of Y that has been generated according to either a Poisson or a NB distribution with \(E[Y|X] = \alpha X ^{\beta ^*}\), where \(\alpha = 0.01\) and \(\beta ^*=0.5\). The rest of values of Y are generated according to a Poisson or a NB distribution with \(E[Y|X] = \alpha X ^{\beta }\), where \(\beta = 1.15\). The vertical axis represents the values of the estimated scaling exponents. The bottom and top of the box indicate the first and third quartiles, the middle line in the box is the median and the upper and lower limits correspond to the 5th and 95th percentiles, giving a 90% confidence interval. The horizontal dashed line represents the true value of \(\beta\), used to generate the synthetic data samples.