GAUSSian quantiles

Because (a) is the area in the lower tail of the normal distribution, Za is called the ath quantile of the standard normal distribution, (or the (100)(a)th percentile). A useful identity follows directly from the symmetry of the Gaussian distribution in Equation A-2 (4). 

Table 2.6 Quantile of the one-sided Student s f distribution for three significance levels a and different degrees of freedom f. Note how the distribution approaches the Gaussian distribution if the degrees of freedom tend to infinity (cf. Table 2.5).

For example, we can build B = 400 parametric bootstrap samples, each by adding noise to the model for predicting y from the last example. We assume the noise, denoted s has a Gaussian distribution with mean zero and variance = YH=i( yi y<9 In, written as a,- A1(0, ri). Then we can write y = a -1-/3/fc > 0. 5) -b Si, where a and /3 are the maximum-UkeUhood estimates of the intercept and slope, respectively. As B increases, the parametric bootstrap samples capture the variability of the estimates for a and a -b /3. For example, as with the nonparametric bootstrap, with B = 400, we can extract a 95% confidence interval for the estimates by selecting the 0.025 and 0.975 quantiles of the bootstrap estimates for each z-value. 

A. Each epistemic variable is represented by its median and a range We have supposed that the true and unknown distributions of the epistemic variables are Gaussian. The results will depend in the capacity of the expert to give quantiles and interval close to the actual values. We have supposed first that the expert gives the true median and the true 95% interval for the epistemic variables (Figure 3). Belief and plausibility functions of the response Pf are described here by 256 focal intervals indeed, the 8 epistemic variables are each described by two intervals, that means 2 (= 256) hypercubes to be evaluated in the space of epistemic variables. For each hypercube, the evaluation of the minimum and maximum values of the response provides the corresponding... 

Fig. 3. The figure examines graphically the Ganssianity of 5000 attribute vectors afc = (ofci, 0 2), at randomly selected locations k within a seismic cube. The upper plots show quantile-quantile plots of Oki and ak2, comparing their empirical quantiles to the theoretical quantiles of a Gaussian distribution (dots). The lines join the first and third quartiles of the distributions. The lower left figure shows histograms of Bfei and ak2, with fitted Gaussian distributions superimposed for aki (dotted line) and ak2 (solid line). The lines are almost identical due to normalization. The lower right figure illustrates a joint frequency plot of aki and 0 2-...

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