Empirical distribution functions

When enough data are available, the need to assume a specific parametric distribution can be avoided by using the empirical distribution. The empirical distribution based on n observations is the distribution that assigns equal probability (1/n) to each observed value. A particular focus of a workshop on distribution selection (USEPA 1998) was considerations for choosing between the use of parametric distribution functions. .. and empirical distribution functions. That report of the workshop emphasizes case-specific criteria. 

FIGURE 6.9 Empirical distribution function (gray, below) and p-box (black, below) corresponding to a data set (triangles, above) containing measurement error. 

The dispersion phenomenon has been quantitatively approached by three models. Initially, Albrecht s theory (Tang and Albrecht, 1970) was applied to the finite segments of the polymer. Then, in the case of materials such as trans-Vk, use of an empirical distribution function P N) for the conjugation length made it possible to exactly reproduce the line shapes and line intensities resulting from excitation with different laser lines ... 

Different results may be observed under conditions that are ostensibly the same. To keep track of this variation, we must maintain records or statistics. There are two general strategies that we may employ. First, we may simply store the results. That is, if we have a thousand observations, we can maintain access to all the individual values. The record may then be employed as an empirical distribution function, in which particular percentiles may be identified on demand. Second, we may use a mathematical model to summarize the distribution. There are two very different reasons for doing this. First, a statistical model may be used to provide a concise summary. The facility with which an analyst can store and retrieve data makes this motivation less compelling than it once was. Second, when a sparse data set is not considered representative of a large population, a model may also be used to infer or predict values that are not represented in the data set. 

Measurement of values used directly in risk assessment (e.g., in empirical distribution functions). 

The smoothed bootstrap has been proposed to deal with the discreteness of the empirical distribution function (F) when there are small sample sizes (A < 15). For this approach one must smooth the empirical distribution function and then bootstrap samples are drawn from the smoothed empirical distribution function, for example, from a kernel density estimate. However, it is evident that the proper selection of the smoothing parameter (h) is important so that oversmoothing or undersmoothing does not occur. It is difficult to know the most appropriate value for h and once the value for h is assigned it influences the variability and thus makes characterizing the variability terms of the model impossible. There are few studies where the smoothed bootstrap has been applied (21,27,28). In one such study the improvement in the correlation coefficient when compared to the standard non-parametric bootstrap was modest (21). Therefore, the value and behavior of the smoothed bootstrap are not clear. 

Ayer, M., Brtmk, H. D., Ewing, G. M. and Silverman, E. (1955). An empirical distribution function for sampling with incomplete information. Annals of Mathematical Statistics, 26,641-7. 

The spectrophotometric measurements in Table 2.1 are to be tested versus a normal distribution by means of Kolmogorov-Smirnov s test at a significance level of a = 0.05. In the first step, the empirical distribution function, F x), is evaluated as shown in Figure 2.10. For comparison of the hypothetical distribution function, the cumulative frequency by using the mean and the standard deviation of the data in dependence on the (standard normal) deviate z are plotted (cf. Eq. (2.28)). 

Figure 1. Empirical distribution function F s) of the km-dependent failures.

The results for the functions Hg and D illustrated in Figure 24.13 show that the empirical distribution functions Hg and D computed from real data (plotted as a black dashed Hne) are more or less within the confidence bands obtained from simulated data (gray solid lines). However, the estimated pair-correlation function g from real data does not match the confidence band of simulated data perfectly (see Figure 24.10). However, the main structural properties of g such as the hard[Pg.684]

Hanson and Larson used the equation of Hunt to determine runup heights in the southern Baltic Sea for the period 1982-2004, from which the runup levels were derived by taking into account the water levels. Empiric distribution functions were then fitted to the data to extrapolate the total water level (runup level) to high return periods. Figure 38.9 illustrates the empiric distribution function for the annual maximum runup level plotted with Gringorten s formula together with a fitted Gumbel distribution. 

Figure 2. Empirical distribution function of chloroform concentration in drinking water at network points.

Jammalamadaka, S.R., Taufer, E., Testing exponentiality by comparing the empirical distribution function of the normalized spacings with that of the original data. Journal of Nonparametric Statistics, 15(6), 2003, 719-729. 

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