Distribution percentile estimates

Rice S, Church M (1996) Sampling surficial fluvial gravels the precision of size distribution percentile estimates. J Sedimentary Res 66(3) 654—665... 

The line the data supports on a hazard plot determines engineering information relating to the distribution of time to failure. Fan failure data and simulated data are illustrated here to explain how the information is obtained. The methods provide estimates of distribution parameters, percentiles, and probabilities of failure. The methods that give estimates of distribution parameters differ slightly from the hazard paper of one theoretical distribution to another and are given separately for each distribution. The methods that give estimates of distribution percentiles and failure probabilities are the same for all papers and are given first. 

When a set of data does not plot as a straight line on any of the available papers, then one may wish to draw a smooth curve through the data points on one of the plotting papers, and use the curve to obtain estimates of distribution percentiles and probabilities of failure for various given times. With such a nonparametric fit to the data, it is usually unsatisfactory to extrapolate beyond the data because it is difficult to determine how to extend how to extend the curve. Nonparametric fitting is best used only if the data contain a reasonably large number of failures. 

The comparison of the T25 method with the LMS method showed a good correlation between the two methods (correlation coefficient of 0.85 in a log-log plot) for 33 substances identified in the US-EPA IRIS database. The ratios between the lifetime cancer risks calculated by the T25 method and the LMS method were in the range 0.5-2.0 for 30 out of the 33 substances (calculated for the 10 lifetime cancer risk). The distribution of the ratios was plotted and the parameters characterizing this distribution were estimated. The mean and the median were both 1.21, the 5 th and 95 th percentiles were 0.50 and 1.87, respectively, and the minimum and maximum values were 0.45 and 2.31, respectively. For 24 substances, the T25 method gave a higher result than the LMS method, and for the remaining 9 substances a lower result. 

Figure 14-10 DoD plot for comparison of two drug assays parametric analysis. A histogram shows the relative frequency of N = 65 differences with the estimated Gaussian density distribution. Parametrically estimated 2.5, and 97.5 percentiles are shown with 90% CIs.

Then estimate the 0,90 confidence interval of each reference limit from the distribution of the percentile estimates. 

It would appear reasonable to assume that the upside and downside information regarding demand implies that the probabilities are well-centered around the mean of 10,000. Therefore, let us assume that demand follows a normal distribution. We can interpret the statement from Marketing to mean that the cumulative probability of demand equaling 14,000 units is 0.90. Thus, we can use a normal look-up table (perhaps the one embedded in the popular Excel spreadsheet software) and use this 90th-percentile estimate from Marketing to compute the standard deviation of our demand distribution. Specifically,... 

D. I. Gibbons and L. C. Vance, M Simulation Study of Estimators for the Parameters and Percentiles in the Two-Parameter Weibull Distribution, General Motors Research Publication No. GMR-3041, General Motors, Detroit, Mich., 1979. 

The median particle diameter is the diameter which divides half of the measured quantity (mass, surface area, number), or divides the area under a frequency curve ia half The median for any distribution takes a different value depending on the measured quantity. The median, a useful measure of central tendency, can be easily estimated, especially when the data are presented ia cumulative form. In this case the median is the diameter corresponding to the fiftieth percentile of the distribution. 

Mosleh, Kazarians, and Gekler obtained a Bayesian estimate of the failure rate, Z, of a coolant recycle pump in llie hazard/risk study of a chemical plant. The estimate was based on evidence of no failures in 10 years of operation. Nuclear industry experience with pumps of similar types was used to establish tire prior distribution of Z. Tliis experience indicated tliat tire 5 and 95 percentiles of lire failure rate distribution developed for tliis category were 2.0 x 10" per hour (about one failure per 57 years of operation) and 98.3 x 10 per hour (about one failure per year). Extensive experience in other industries suggested the use of a log-nonnal distribution witli tlie 5 and 95 percentile values as llie prior distribution of Z, tlie failure rate of the coolant recycle pump. 

Suppose, for example, that an estimate based on a Wei-bull fit to the fan data is desired of the fifth percentile of the distribution of time to fan failure. Enter the Weibull plot. Figure 62.6, on the probability scale at the chosen percentage point, 5 per cent. Go vertically down to the fitted line and then horizontally to the time scale where the estimate of the percentile is read and is 14,000 hours. 

The sample size of the survey will depend primarily on the need to accurately estimate the upper percentiles of the national distribution and the number of subgroup populations for which frequency distributions are needed and the level of accuracy needed for these distributions (Cox et el., 1985). 

These estimates are for a national distribution and will increase depending upon the number of subpopulation groups for which frequency distribution data is desired. Determining the sample size, therefore, requires decisions regarding the percentile of interest and level of accuracy needed, and balancing these against the costs of the required sample. 

Cumulative distributions of the logarithms of NOELs were plotted separately for each of the stmcmral classes. The 5th percentile NOEL was estimated for each stmctural class and this was in mrn converted to a human exposure threshold by applying the conventional default safety factor of 100 (Section 5.2.1). The stmcmre-based, tiered TTC values established were 1800 p,g/person/ day (Class I), 540 pg/person/day (Class II), and 90 pg/person/day (Class III). Endpoints covered include systemic toxicity except mutagenicity and carcinogenicity. Later work increased the number of chemicals in the database from 613 to 900 without altering the cumulative distributions of NOELs (Barlow 2005). 

The distributions of two individual assessment factors can be combined forming a new distribution (with a new GM and GSD) characterizing the combination of the two individual factors a certain percentile can be chosen from this new distribution. Which percentile of a distribution to be chosen is a policy issue. When distributions of the assessment factors are not available, the point estimate of a particular factor could be used. Distributions and point estimates can be used in parallel and combined when necessary for example the chosen percentile from a distribution of one factor can be combined with the point estimate of another factor by multiplication. 

Predictive methods of exposure assessment often rely on single values for input parameters to the exposure model that represent one point on the distribution curve of all possible values for this parameter. This point value can range from a 50th percentile, mean, median, or typical value to a worst-case estimate. In the predictive exposure assessment, a number of parameters are integrated through an algorithm to produce an output such as the predicted environmental concentration (PEC). If many worst-case values are involved, this integration can result in a PEC that has a... 

The general strategy of equating parameters to statistics is of course not restricted to moments. Reliance on sample percentiles (e.g., sample median) can lead to estimators that are not excessively sensitive to outliers. In general, to fit a distribution with k parameters, k parameters must be equated to distinct sample statistics. 

When a parametric distribution is fitted, each datum contributes to the estimate of each parameter or percentile. Whether this is good or not depends on whether the distribution to be fitted is reasonable. If it is assumed that one can identify the true distribution, the data will be used in a way that is in some sense optimal. In the real world, where the best distribution is uncertain, it may happen that estimated frequencies for one tail of a distribution are sensitive to observations on the other tail, e.g., estimates of high concentration percentiles are sensitive to observed low concentrations. 

The Bayesian equivalent to the frequentist 90% confidence interval is delineated by the 5th and 95th percentiles of the posterior distribntion. Bayesian confidence intervals for SSD (Figures 5.4 to 5.5), 5th percentile, i.e., HC5 and fraction affected (Figures 5.4 to 5.6) were calculated from the posterior distribution. Thns, the nncer-tainties of both HC and FA are established in 1 consistent mathematical framework FA estimates at the logio HC lead to the intended protection percentage, i.e., M °(logio HCf) = p where p is a protection level. Further full distribution of HC and FA uncertainty can be very easily extracted from posterior distribntion for any level of protection and visualized (Figures 5.5 to 5.7). 

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