Bounding a Distributed Variable

For certain distributions, the set of values for which the pdf is positive (the support) is unbounded. For example, the pdf of the log-normal distribution is positive for all positive real numbers. Ordinarily, there will be values too extreme to be reasonable, and so it is common to place bounds on the support. However, selecting precise values for the bounds may be a difficult decision. 

Supposing that one has decided on bounds for a variable, one can fit a distribution that has a bounded support, such as the beta distribution or Johnson SB distribution. Alternatively, in a Monte Carlo implementation, one may sample the unbounded distribution and discard values that fall beyond the bounds. However, then a source of some discomfort is that the parameters of the distribution truncated in this way may deviate from the specification of the distribution (e.g., the mean and variance will be modified by truncation). It seems reasonable for Monte Carlo software to report the percentage discarded, and report means and variances of the distributions as truncated, for comparison to means and variances specified. 

A fitted distribution should be evaluated using graphical methods as well as statistical goodness-of-flt (GoF) tests. Appropriate procedures are available in texts on environmental statistics and risk assessment (e.g., Gilbert 1987 Helsel and Hirsch 1992 Millard and Neerchal 2000). It is suggested that USEPA (1998) be consulted regarding a number of practical considerations. 

Some statistical tests are specific for evaluation of normality (log-normality, etc., normality of a transformed variable, etc.), while other tests are more broadly applicable. The most popular test of normality appears to be the Shapiro-Wilk test. Specialized tests of normality include outlier tests and tests for nonnormal skewness and nonnormal kurtosis. A chi-square test was formerly the conventional approach, but that approach may now be out of date. 

For graphical evaluation of distribution fit, probability (P-P) plots and quantile (Q-Q) plots are particularly helpful. Sometimes the statistic has been used to quantify the linearity of a P-P plot or Q-Q plot however, in practice it appears that there may be substantial deviation between the observed and expected frequencies, despite an R that would be viewed as large in many statistical contexts. 

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