Cumulant Estimators

Instead of estimating -/ 1 ln(exp(-/W(f)) directly using (5.44), one can use cumulant expansion approaches, as in regular free energy perturbation theory (see e.g., [20, 39] for combining cumulant expansions about the initial and final states). Unbiased estimators for cumulants can be used. Probably the most useful relations involve averages and variances of the work  

Results from forward and backward averages can also be combined [20] into symmetric perturbation formulas  

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In the next several sections, the theoretical distributions and tests of significance will be examined beginning with Student s distribution or t test. If the data contained only random (or chance) errors, the cumulative estimates x and 5- would gradually approach the limits p and cr. The distribution of results would be normally distributed with mean p and standard deviation cr. Were the true mean of the infinite population known, it would also have some symmetrical type of distribution centered around p. However, it would be expected that the dispersion or spread of this dispersion about the mean would depend on the sample size. 

To provide a sense of the value of some of these deviations from neutral taxation. Table 2 reports cumulative estimates over five years of selected energy-related tax expenditures. The Congressional Budget and Impoundment Control Act defines a tax expenditure as a revenue loss attributable to a special inclusion, exemption, or deduction from gross income or a special credit, preferential tax rate, or deferral of tax. 

At the end of 2003, an estimated 1,039,000 to 1,185,000 persons were living with HIV/AIDS in the United States, with 24% to 27% undiagnosed and unaware of their HIV infection (who may be unknowingly transmitting the virus to others). Of the total number of HIV-infected patients, approximately 4% (42,514) were diagnosed with AIDS in 2004, and 2% (18,017) died with AIDS. The cumulative estimated number of diagnoses of AIDS through 2004 was 944,305, one-half of whom (529,113) had died. 

Analyze the work distributions. Use the histogram analysis, optimal and cumulant estimators of Sect. 5.7 to determine the free energy difference. 

Fig. 5.3. Comparison of different free energy estimators. Plotted are distributions of estimated free energies using sample sizes (i.e., number of independent simulation runs) of N = 100 simulations (solid lines), as well as N = 1, 000 (long dashed) and N = 10,000 simulations short dashed lines), (a) Exponential estimator, (5.44). (b) Cumulant estimator using averages from forward and backward paths, (5.47). (c) Cumulant estimator using averages and variances from forward and backward paths, (5.48). (d) Bennett s optimal estimator, (5.50)...

In summary, using work collected from forward and backward paths greatly improves the accuracy of the estimates, and for the symmetric system studied here eliminates the bias. In our particular example, the cumulant estimators using forward and backward work data produce the most precise free energy estimates, followed by Bennett s optimal estimator. However, this somewhat poorer performance of the optimal estimator is caused in part by the high degree of symmetry of the system studied. 

Table 2.7 Toxicology testing recommendations for food contact substances based on dietary concentration (EXT) and corresponding estimated daily intake (EDI) values. Note that the cumulative exposures are based on non-biocidal chemicals biocidal tiers are one-fifth the cumulative dietary concentration (CDC) and cumulative estimated daily intake (CEDI) values expressed. DC and CDC values are in units of mass of migrant per mass food consumed. EDI and CEDI values are in units of mass of migrant per person per day. Abbreviations are as follows p,g (microgram), kg (kilogram), mg (milligram), < (less than or equal to), < (less than), > (greater than), and > (greater than or equal to)...

Table 7.1 Toxicology testing recommendations for food contact substances based on dietary concentration (DC) and corresponding estimated daily intake (EDI) values. Note that the cumulative exposures are based on non-biocidal chemicals biocidal tiers are one-fifth the cumulative dietary concentration (CDC) and cumulative estimated daily intake (CEDI) values expressed...

Gerin (Canada) 1993 Application of a welding process exposure matrix to welding history of 11,092 welders Welding fumes, total chromium, chromium (VI), nickel Cumulative estimates (mgyr/m ) Average concentrations (mg/m )... 

CEDI/ADI Database US PDA/CPSAN Cumulative Estimated Daily Intake/Acceptable Daily Intake Database of food compounds http //www.fda.gov/Food/IngredientsPackag-ingLabeling/Packagi ngF C S/C EDI/default.htm... 

Assessment of whether the level of NP migrating from FCMs should be of concern has not reached a consensus in regulatory agencies, partly because NP is predominantly present as an impurity or degradation product as a NIAS. The U.S. Food and Drug Administration (FDA) includes nonylphenol, nonylphenol,p—, octylphenol, octylphenol,p—, and tris(nonylphenyl)phosphite (see Table 5.1) in their List of Indirect Additives Used in Food Contact Substances (FDA 2014), but NP and OP are not approved by the FDA for use as direct food additives. Neither NP nor OP are listed in FDA s Cumulative Estimated Daily Intake database for food contact substances (FDA 2013). 

Food and Drug Administration (FDA) (2012) Cumulated Estimated Daily Intake, antimony trioxide. U.S. Food and Drug Administration, http //www.accessdata.fda.gov/scripts/sda/ sdNavigation.cfm sd=edisrev displayAll=false page=7. Accessed 4 Sept 2013... 

No specific regulatory limits or listings of cadmium compounds, including cadmium pigments, were located in FDA packaging and food contact substances databases at http //www.fda.gov/Food/IngredientsPackagingLabeIing/PackagingPCS/ default.htm (e.g., EAFUS, Inventory of Effective Food Contact Substance Notifications, List of Indirect Additives Used in Food Contact Substances, and Cumulative Estimated Dietary Intake databases). 

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