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Some Useful Distributions

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Statistical Theory and Inference

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Abstract

This chapter contains many useful examples of parametric distributions, one- and two-parameter exponential families, location–scale families, maximum likelihood estimators, method of moment estimators, transformations t(Y ), E(Y ), V (Y ), moment generating functions, and confidence intervals. Many of the distributions can be used to create exam questions on the above topics as well as the kernel method, MSE, and hypothesis testing. Using the population median and median absolute deviation, robust estimators of parameters can often be found using the sample median and median absolute deviation.

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References

  • Abuhassan, H., and Olive, D.J. (2008), “Inference for the Pareto, Half Normal and Related Distributions,” unpublished manuscript, (http://lagrange.math.siu.edu/Olive/pppar.pdf).

  • Adell, J.A., and Jodrá, P. (2005), “Sharp Estimates for the Median of the Γ(n + 1, 1) Distribution,” Statistics & Probability Letters, 71, 185–191.

    Article  MATH  MathSciNet  Google Scholar 

  • Al-Mutairi, D.K., Ghitany, M.E., and Kundu, D. (2013), “Inference on Stress-Strength Reliability from Lindley Distributions,” Communications in Statistics: Theory and Methods, 42, 1443–1463.

    Article  MATH  MathSciNet  Google Scholar 

  • Barndorff-Nielsen, O. (1978), Information and Exponential Families in Statistical Theory, Wiley, New York, NY.

    MATH  Google Scholar 

  • Besbeas, P., and Morgan, B.J.T. (2004), “Efficient and Robust Estimation for the One-Sided Stable Distribution of Index 1∕2,” Statistics & Probability Letters, 66, 251–257.

    Article  MATH  MathSciNet  Google Scholar 

  • Bickel, P.J., and Doksum, K.A. (2007), Mathematical Statistics: Basic Ideas and Selected Topics, Vol. 1., 2nd ed., Updated Printing, Pearson Prentice Hall, Upper Saddle River, NJ.

    Google Scholar 

  • Bowman, K.O., and Shenton, L.R. (1988), Properties of Estimators for the Gamma Distribution, Marcel Dekker, New York, NY.

    MATH  Google Scholar 

  • Brownstein, N., and Pensky, M. (2008), “Application of Transformations in Parametric Inference,” Journal of Statistical Education, 16 (online).

    Google Scholar 

  • Casella, G., and Berger, R.L. (2002), Statistical Inference, 2nd ed., Duxbury, Belmont, CA.

    Google Scholar 

  • Castillo, E. (1988), Extreme Value Theory in Engineering, Academic Press, Boston, MA.

    MATH  Google Scholar 

  • Chen, J., and Rubin, H. (1986), “Bounds for the Difference Between Median and Mean of Gamma and Poisson Distributions,” Statistics & Probability Letters, 4, 281–283.

    Article  MATH  MathSciNet  Google Scholar 

  • Cohen, A.C., and Whitten, B.J. (1988), Parameter Estimation in Reliability and Life Span Models, Marcel Dekker, New York, NY.

    MATH  Google Scholar 

  • Cooke, D., Craven, A.H., and Clarke, G.M. (1982), Basic Statistical Computing, Edward Arnold Publishers, London, UK.

    MATH  Google Scholar 

  • Cramér, H. (1946), Mathematical Methods of Statistics, Princeton University Press, Princeton, NJ.

    MATH  Google Scholar 

  • Datta, G.S. (2005), “An Alternative Derivation of the Distributions of the Maximum Likelihood Estimators of the Parameters in an Inverse Gaussian Distribution,” Biometrika, 92, 975–977.

    Article  MATH  MathSciNet  Google Scholar 

  • DeGroot, M.H., and Schervish, M.J. (2012), Probability and Statistics, 4th ed., Pearson Education, Boston, MA.

    Google Scholar 

  • Feller, W. (1971), An Introduction to Probability Theory and Its Applications, Vol. II, 2nd ed., Wiley, New York, NY.

    Google Scholar 

  • Ferguson, T.S. (1967), Mathematical Statistics: a Decision Theoretic Approach, Academic Press, New York, NY.

    MATH  Google Scholar 

  • Forbes, C., Evans, M., Hastings, N., and Peacock, B. (2011), Statistical Distributions, 4th ed., Wiley, Hoboken, NJ.

    MATH  Google Scholar 

  • Gabel, R.A., and Roberts, R.A. (1980), Signals and Linear Systems, Wiley, New York, NY.

    Google Scholar 

  • Greenwood, J.A., and Durand, D. (1960), “Aids for Fitting the Gamma Distribution by Maximum Likelihood,” Technometrics, 2, 55–56.

    Article  MATH  MathSciNet  Google Scholar 

  • Hamza, K. (1995), “The Smallest Uniform Upper Bound on the Distance Between the Mean and the Median of the Binomial and Poisson Distributions,” Statistics & Probability Letters, 23, 21–25.

    Article  MATH  MathSciNet  Google Scholar 

  • Headrick, T.C., Pant, M.D., and Sheng, Y. (2010), “On Simulating Univariate and Multivariate Burr Type III and Type XII Distributions,” Applied Mathematical Sciences, 4, 2207–2240.

    MATH  MathSciNet  Google Scholar 

  • Johnson, N.L., and Kotz, S. (1970ab), Distributions in Statistics: Continuous Univariate Distributions, Vol. 1–2, Houghton Mifflin Company, Boston, MA.

    Google Scholar 

  • Johnson, N.L., Kotz, S., and Kemp, A.K. (1992), Distributions in Statistics: Univariate Discrete Distributions, 2nd ed., Wiley, New York, NY.

    Google Scholar 

  • Kalbfleisch, J.D., and Prentice, R.L. (1980), The Statistical Analysis of Failure Time Data, Wiley, New York, NY.

    MATH  Google Scholar 

  • Kennedy, W.J., and Gentle, J.E. (1980), Statistical Computing, Marcel Dekker, New York, NY.

    MATH  Google Scholar 

  • Kotz, S., and Johnson, N.L. (editors) (1982ab), Encyclopedia of Statistical Sciences, Vol. 1–2, Wiley, New York, NY.

    Google Scholar 

  • Kotz, S., and Johnson, N.L. (editors) (1983ab), Encyclopedia of Statistical Sciences, Vol. 3–4, Wiley, New York, NY.

    Google Scholar 

  • Kotz, S., and Johnson, N.L. (editors) (1985ab), Encyclopedia of Statistical Sciences, Vol. 5–6, Wiley, New York, NY.

    Google Scholar 

  • Kotz, S., and Johnson, N.L. (editors) (1986), Encyclopedia of Statistical Sciences, Vol. 7, Wiley, New York, NY.

    MATH  Google Scholar 

  • Kotz, S., and Johnson, N.L. (editors) (1988ab), Encyclopedia of Statistical Sciences, Vol. 8–9, Wiley, New York, NY.

    Google Scholar 

  • Kotz, S., and van Dorp, J.R. (2004), Beyond Beta: Other Continuous Families of Distributions with Bounded Support and Applications, World Scientific, Singapore.

    Book  Google Scholar 

  • Leemis, L.M., and McQueston, J.T. (2008), “Univariate Distribution Relationships,” The American Statistician, 62, 45–53.

    Article  MathSciNet  Google Scholar 

  • Lehmann, E.L. (1983), Theory of Point Estimation, Wiley, New York, NY.

    Book  MATH  Google Scholar 

  • Lehmann, E.L. (1999), Elements of Large–Sample Theory, Springer, New York, NY.

    MATH  Google Scholar 

  • Lindsey, J.K. (2004), Introduction to Applied Statistics: a Modelling Approach, 2nd ed., Oxford University Press, Oxford, UK.

    Google Scholar 

  • Mahmoud, M.A.W., Sultan, K.S., and Amer, S.M. (2003), “Order Statistics for Inverse Weibull Distributions and Characterizations,” Metron, LXI, 389–402.

    Google Scholar 

  • Mann, N.R., Schafer, R.E., and Singpurwalla, N.D. (1974), Methods for Statistical Analysis of Reliability and Life Data, Wiley, New York, NY.

    MATH  Google Scholar 

  • Marshall, A.W., and Olkin, I. (2007), Life Distributions, Springer, New York, NY.

    MATH  Google Scholar 

  • Meeker, W.Q., and Escobar, L.A. (1998), Statistical Methods for Reliability Data, Wiley, New York, NY.

    MATH  Google Scholar 

  • Patel, J.K., Kapadia C.H., and Owen, D.B. (1976), Handbook of Statistical Distributions, Marcel Dekker, New York, NY.

    MATH  Google Scholar 

  • Pourahmadi, M. (1995), “Ratio of Successive Probabilities, Moments and Convergence of (Negative) Binomial to Poisson Distribution,” Unpublished Manuscript.

    Google Scholar 

  • Pratt, J.W. (1968), “A Normal Approximation for Binomial, F, Beta, and Other Common, Related Tail Probabilities, II,” Journal of the American Statistical Association, 63, 1457–1483.

    MATH  MathSciNet  Google Scholar 

  • Rousseeuw, P.J., and Croux, C. (1993), “Alternatives to the Median Absolute Deviation,” Journal of the American Statistical Association, 88, 1273–1283.

    Article  MATH  MathSciNet  Google Scholar 

  • Schwarz, C.J., and Samanta, M. (1991), “An Inductive Proof of the Sampling Distributions for the MLE’s of the Parameters in an Inverse Gaussian Distribution,” The American Statistician, 45, 223–225.

    MathSciNet  Google Scholar 

  • Stein, C. (1981), “Estimation of the Mean of a Multivariate Normal Distribution,” The Annals of Statistics, 9, 1135–1151.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Southern Illinois University, Carbondale, IL, USA

    David J. Olive

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  1. David J. Olive
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Olive, D.J. (2014). Some Useful Distributions. In: Statistical Theory and Inference. Springer, Cham. https://doi.org/10.1007/978-3-319-04972-4_10

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