Summary
Heaviness of tail of distributions is compared each other analytically and systematically. Distributions under the study are the lognormal, loglogistic, Weibull, gamma, exponential-polynomial distributions. The beta loglogistic, which covers some of these distributions as its limits, is also discussed.
Heaviness of tail is an important notion in life-test, robust estimation and rank test. Here the notion is studied to examine models for estimating safe doses. Some results about heaviness of tail are given, and a new notion of heaviness of tail at the origin is defined and discussed.
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References
Armitage, P. and Doll, R. (1961). Stochastic models for carcinogens,Proc. 4th Berkeley Symp. Math. Statist. Prob.,4, 19–38.
Barlow, R. E. and Proshan, F. (1965).Mathematical Theory of Reliability, John Wiley & Sons, Inc., New York.
Barlow, R. E., Bartholomew, D. J., Bremner, J. M. and Brunk, H. D. (1972).Statistical Inference under Order Restrictions, John Wiley & Sons, Inc., New York.
Chand, N. and Hoel, D. G. (1974). A comparison of models for determining safe levels of environmental agents,Reliability and Biometry, SIAM, 681–700.
Crump, K. S., Guess, H. A. and Deal, K. L. (1977). Confidence intervals and test of hypotheses concerning dose-response relations inferred from animal carcinogenicity data,Biometrics,33, 437–451.
Doksum, K. (1969). Starshaped transformations and the power of rank tests,Ann. Math. Statist.,40, 1167–1176.
Food and Drug Administration. Advisory Committee on Protocols for Safety Evaluation (1971). Panel on carcinogenesis report on cancer testing in the safety evaluation of food additive and pesticides,Toxicol. Appl. Pharmacol.,20, 419–438.
Fraser, D. A. S. (1957).Nonparametric Methods in Statistics, John Wiley & Sons, Inc., New York.
Gastwirth, J. L. (1970). On robust tests,Nonparametric Techniques in Statistical Inference, (ed. M. L. Puri) University Press, Cambridge, 89–101.
Gehan, E. (1969). Estimating survival functions from the life table.J. Chronic Diseases,21, 629–644.
Hartley, H. O. and Sielken, R. L., Jr. (1977). Estimation of “safe doses” in carcinogenic experiments,Biometrics,33, 1–30.
Iversen, S. and Arley, N. (1950). On the mechanism of experimental carcinogenesis,Acta Path. Microbiol. Scand.,27, 773–803.
Mantel, N. and Bryan, W. R. (1961). “Safety” testing of carcinogenic agents,J. Nat. Cancer Inst.,27, 455–470.
Mantel, N., Bohidar, N. R., Brown, C. C., Ciminera, J. L. and Tukey, J. W. (1975). An improved “Mantel-Bryan” procedure for “safety” testing of carcinogens,Cancer Res.,35, 865–872.
Peto, R. and Lee, P. (1973). Weibull distributions for continuous carcinogenesis experiments,Biometrics,29, 457–470.
Prentice, R. L. (1975). Discrimination among some parametric models,Biometrika,62, 607–614.
Prentice, R. L. (1976). A generalization of the probit and logit methods for dose response curves,Biometrics,32, 761–768.
Rai, K. and van Ryzin, J. (1979). Risk assessment of toxic environmental substances using a generalized multi-hit dose response model,Energy and Health, SIMS Conf., SIAM Press, Philadelphia.
Sibuya, M. and Yanagimoto, T. Notions of heaviness of tail and comparison of distributions (in preparation).
Takeuchi, K. (1972). On robust estimation—II,Jap. J. Appl. Statist.,2, 69–93 (in Japanese).
Yanagimoto, T. and Sibuya, M. (1976). Isotonic tests for spread and tail,Ann. Inst. Statist. Math.,28, A, 329–342.
van Zwet, W. R. (1964).Convex Transformation of Random Variables, Math. Centrum, Amsterdam.
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Yanagimoto, T., Sibuya, M. Comparison of tails of distributions in models for estimating safe doses. Ann Inst Stat Math 32, 325–340 (1980). https://doi.org/10.1007/BF02480337
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DOI: https://doi.org/10.1007/BF02480337