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Figure 6. Normal probability plots showing the impact of decreasing and increasing variability

Figure 6 Normal probability plots showing the impact of decreasing and increasing variability

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Figure 1. Portrayal of (a) positive and (b) negative skewness Although the estimators for sample skewness and kurtosis are biased and therefore tend to underestimate the true values for the population, the consideration of these moments is necessary, as no probability distribution could technically be described as Gaussian normal unless both equal 0. Figure 2(a) depicts several examples of distributions exhibiting varying degrees of kurtosis where J, N, and U denote Johnson, nor- mal, and uniform distributions, respectively. It is important to note that, in each case, the distributions with positive or negative kur- tosis cross the normal curve twice on each side of the mean, which is a common trait of distributions with excess kurtosis. For distributions with excess positive kurtosis, such as the Johnson (0,1,0,10), this results in a pattern of higher-lower-higher probabilities on each side relative to each crossing point. For example, consider the J(0,1,0,10) relative to the N(0,1) in Figure 2(a). For values near the mean, the probabilities are higher for the J(0,1,0,10) distribution. Moving left or right from the mean, the densities cross and the relationship shifts to lower probabilities for the J(0,1,0,10) and then shifts back to higher probabilities, moving further to the left or right. For distributions with negative excess kurtosis, such as the uniform, the pattern becomes lower-higher-lower. This is a very important distinction, as when they cross only once, this is indicative of changes in variability vice kurtosis. The example of the normal distributions with different variances in Figure 2(b) depicts this.
Figure 1. Portrayal of (a) positive and (b) negative skewness Although the estimators for sample skewness and kurtosis are biased and therefore tend to underestimate the true values for the population, the consideration of these moments is necessary, as no probability distribution could technically be described as Gaussian normal unless both equal 0. Figure 2(a) depicts several examples of distributions exhibiting varying degrees of kurtosis where J, N, and U denote Johnson, nor- mal, and uniform distributions, respectively. It is important to note that, in each case, the distributions with positive or negative kur- tosis cross the normal curve twice on each side of the mean, which is a common trait of distributions with excess kurtosis. For distributions with excess positive kurtosis, such as the Johnson (0,1,0,10), this results in a pattern of higher-lower-higher probabilities on each side relative to each crossing point. For example, consider the J(0,1,0,10) relative to the N(0,1) in Figure 2(a). For values near the mean, the probabilities are higher for the J(0,1,0,10) distribution. Moving left or right from the mean, the densities cross and the relationship shifts to lower probabilities for the J(0,1,0,10) and then shifts back to higher probabilities, moving further to the left or right. For distributions with negative excess kurtosis, such as the uniform, the pattern becomes lower-higher-lower. This is a very important distinction, as when they cross only once, this is indicative of changes in variability vice kurtosis. The example of the normal distributions with different variances in Figure 2(b) depicts this.
Figure 2. Graphical representations of (a) distributions possessing varying degrees of excess kurtosis relative to a N(0,1) and (b) three normal distributions with differen’ variances and zero excess kurtosis
Figure 2. Graphical representations of (a) distributions possessing varying degrees of excess kurtosis relative to a N(0,1) and (b) three normal distributions with differen’ variances and zero excess kurtosis
Figure 3. Normal probability plot for the base case of data sampled from a SN(5, 2, 0) The plots in Figure 6 impart several interesting observations. As expected, the positive skew incorporated in the base case in Figure 6(a) evinces the characteristic concavity previously described in Section 3.2.1, along with a shortening and thinning of the left tail coupled with a corresponding elongation of the right tail. Beyond this, the plots clearly show that changes in variability have a much
Figure 3. Normal probability plot for the base case of data sampled from a SN(5, 2, 0) The plots in Figure 6 impart several interesting observations. As expected, the positive skew incorporated in the base case in Figure 6(a) evinces the characteristic concavity previously described in Section 3.2.1, along with a shortening and thinning of the left tail coupled with a corresponding elongation of the right tail. Beyond this, the plots clearly show that changes in variability have a much
Figure 4. Normal probability plots showing the effect of varying degrees of positive skewness
Figure 4. Normal probability plots showing the effect of varying degrees of positive skewness
Figure 5. Normal probability plots showing the effect of varying degrees of negative skewness Figures 6(b), (c), and (d) depict plots in which we sequentially reduce the variance from 2 to 0.5, 0.4, and finally to 0.125. In contrast to the examples with increasing variability, here we observe that the distribution shortens in both tails as the bulk of the values push in toward the mean of 5. We also note that the variance reduction begins to offset the skewness, inducing a degree of symmetry
Figure 5. Normal probability plots showing the effect of varying degrees of negative skewness Figures 6(b), (c), and (d) depict plots in which we sequentially reduce the variance from 2 to 0.5, 0.4, and finally to 0.125. In contrast to the examples with increasing variability, here we observe that the distribution shortens in both tails as the bulk of the values push in toward the mean of 5. We also note that the variance reduction begins to offset the skewness, inducing a degree of symmetry
about the mean. Obviously, as the variance approaches 0, we would expect to observe a straight line, perpendicular to the x axis, situ- ated at the mean. The plots are clearly trending this way, as shown by Figure 6(f). Despite the evolving symmetry, reducing the var- iance exacerbates the departure from normality, as demonstrated by the density plot in Figure 7, which is associated with the probability plot in Figure 6(f).
about the mean. Obviously, as the variance approaches 0, we would expect to observe a straight line, perpendicular to the x axis, situ- ated at the mean. The plots are clearly trending this way, as shown by Figure 6(f). Despite the evolving symmetry, reducing the var- iance exacerbates the departure from normality, as demonstrated by the density plot in Figure 7, which is associated with the probability plot in Figure 6(f).
Table IV. Simulation results showing the effects of decreasing variability on kurtosis
Table IV. Simulation results showing the effects of decreasing variability on kurtosis
describe kurtosis presented in Section 2.2 is scaled with respect to the variance and therefore is not affected by it. That is, kurtosis serves to describe the shape of a distribution irrespective of the variance. Despite this, the plots portray a clear relationship. Thus, the deeper implication appears to be that the kurtosis existed from the start but was masked by the variability.
describe kurtosis presented in Section 2.2 is scaled with respect to the variance and therefore is not affected by it. That is, kurtosis serves to describe the shape of a distribution irrespective of the variance. Despite this, the plots portray a clear relationship. Thus, the deeper implication appears to be that the kurtosis existed from the start but was masked by the variability.
Figure 8. Normal probability plots showing the effects of decreased and increased levels of excess kurtosis
Figure 8. Normal probability plots showing the effects of decreased and increased levels of excess kurtosis
demonstration of these effects, showing both the narrowing of the peak in the distribution coupled with a nearly twofold increase in its height. The sample statistics for Figures 12(c) and (d) certainly show the effects on excess kurtosis, but they also show a clear impact on the asymmetry, as the negative skewness increases by a factor of 3 in each case. This is further evidenced by the continued albeit slight leftward shift in the mean.
demonstration of these effects, showing both the narrowing of the peak in the distribution coupled with a nearly twofold increase in its height. The sample statistics for Figures 12(c) and (d) certainly show the effects on excess kurtosis, but they also show a clear impact on the asymmetry, as the negative skewness increases by a factor of 3 in each case. This is further evidenced by the continued albeit slight leftward shift in the mean.
Figure 10. Effects of changing variability on negative excess kurtosis
Figure 10. Effects of changing variability on negative excess kurtosis
Figure 11. Overlays of the density plots from Figures 10(b) and (c), revealing (a) increased peakedness and(b) heaviness in the tails typically indicate the presence of either positive or negative asymmetry or skewness, respectively. Moreover, the nature of the tails provides a hint of how variability affects the data. That is, whereas the nature of the curve can impart knowledge as to skewness about the mean, the nature of the tails can impart knowledge about the degree of variability present in the data. The plots in Figures 6(c), (e), and (g) serve as a good example of this, showing a clear elongation of the right tail in the face of increasing variability. However, the nature of the concave or convex curve may also reveal the presence of kurtosis in the data, suggesting a combined effect of dis- tribution characteristics at work within the data. The probability plot in Figure 11(e) provides a stark example of this, depicting the convex curve illustrative of negative asymmetry about the mean, but also a significant degree of positive excess kurtosis suggested by the dense, near-vertical curve in the immediate proximity of the mean.
Figure 11. Overlays of the density plots from Figures 10(b) and (c), revealing (a) increased peakedness and(b) heaviness in the tails typically indicate the presence of either positive or negative asymmetry or skewness, respectively. Moreover, the nature of the tails provides a hint of how variability affects the data. That is, whereas the nature of the curve can impart knowledge as to skewness about the mean, the nature of the tails can impart knowledge about the degree of variability present in the data. The plots in Figures 6(c), (e), and (g) serve as a good example of this, showing a clear elongation of the right tail in the face of increasing variability. However, the nature of the concave or convex curve may also reveal the presence of kurtosis in the data, suggesting a combined effect of dis- tribution characteristics at work within the data. The probability plot in Figure 11(e) provides a stark example of this, depicting the convex curve illustrative of negative asymmetry about the mean, but also a significant degree of positive excess kurtosis suggested by the dense, near-vertical curve in the immediate proximity of the mean.
Figure 12. Normal probability and density plots portraying the effects of decreasing variability in the presence of non-zero skewness and kurtosi:
Figure 12. Normal probability and density plots portraying the effects of decreasing variability in the presence of non-zero skewness and kurtosi:
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