Moment-Ratio Diagrams for Univariate Distributions

We present two moment-ratio diagrams along with guidance for their interpretation. The first moment-ratio diagram is a graph of skewness vs. kurtosis for common univariate probability distributions. The second moment-ratio diagram is a graph of coefficient of variation vs. skewness for common univariate probability distributions. Both of these diagrams, to our knowledge, are the most comprehensive to date. The diagrams serve four purposes: (1) they quantify the proximity between various univariate distributions based on their second, third, and fourth moments, (2) they illustrate the versatility of a particular distribution based on the range of values that the various moments can assume, (3) they can be used to create a short list of potential probability models based on a data set, and (4) they clarify the limiting relationships between various well-known distribution families. The use of the moment-ratio diagrams for choosing a distribution that models given data is illustrated.

[1]  D. Pfeffermann,et al.  Small area estimation , 2011 .

[2]  Simon French,et al.  Statistical Analysis of Reliability Data , 1992 .

[3]  Samuel Kotz,et al.  Encyclopedia of Statistical Sciences: Volume 5 , 1986 .

[4]  Pandu R. Tadikamalla,et al.  A Look at the Burr and Related Distributioni , 1980 .

[5]  J. Bert Keats,et al.  Statistical Methods for Reliability Data , 1999 .

[6]  C. Craig.  A New Exposition and Chart for the Pearson System of Frequency Curves , 1936 .

[7]  P. Good.  Resampling Methods , 1999, Birkhäuser Boston.

[8]  P. R. Nelson.  Continuous Univariate Distributions Volume 2 , 1996 .

[9]  J. Lieblein,et al.  Statistical Investigation of the Fatigue Life of Deep-Groove Ball Bearings , 1956 .

[10]  N. L. Johnson,et al.  Continuous Univariate Distributions. , 1995 .

[11]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[12]  F. Famoye.  Continuous Univariate Distributions, Volume 1 , 1994 .

[13]  R. Berk,et al.  Continuous Univariate Distributions, Volume 2 , 1995 .

[14]  P. Davies,et al.  Kendall's Advanced Theory of Statistics. Volume 1. Distribution Theory , 1988 .

[15]  W. Nelson.  Statistical Methods for Reliability Data , 1998 .

[16]  Robert N. Rodriguez.  A guide to the Burr type XII distributions , 1977 .

[17]  M. Kendall,et al.  Kendall's advanced theory of statistics , 1995 .

[18]  D. Cox,et al.  Analysis of Survival Data. , 1985 .

[19]  Lawrence Leemis,et al.  Parametric Model Discrimination for Heavily Censored Survival Data , 2008, IEEE Transactions on Reliability.

[20]  Lawrence M. Leemis,et al.  Reliability: Probabilistic Models and Statistical Methods , 1994 .

[21]  Lawrence M Leemis,et al.  Univariate Distribution Relationships , 2008 .

[22]  Martin Crowder,et al.  Statistical Analysis of Reliability Data , 1991 .

[23]  C. Caroni,et al.  The Correct “Ball Bearings” Data , 2002, Lifetime data analysis.

[24]  J. Norris.  Appendix: probability and measure , 1997 .