Fiducial confidence limits and prediction limits for a gamma distribution: Censored and uncensored cases

The problems of finding confidence limits for the mean and an upper percentile, and upper prediction limits for the mean of a future sample from a gamma distribution are considered. Simple methods based on cube root transformation and fiducial approach are proposed for constructing confidence limits and prediction limits when samples are uncensored or censored. Monte Carlo simulation studies indicate that the methods are accurate for estimating the mean and percentile and for predicting the mean of a future sample as long as the percentage of nondetects is not too large. Algorithms for computing confidence limits and prediction limits are provided. Necessary R programs for calculating confidence limits and prediction limits are also provided as a supplementary file. The methods are illustrated using some real uncesnored/censored environmental data sets.

[1]  T. E. Sterne,et al.  Inverse Probability , 1930, Nature.

[2]  Bradley Efron,et al.  R. A. Fisher in the 21st century (Invited paper presented at the 1996 R. A. Fisher Lecture) , 1998 .

[3]  Douglas M. Hawkins,et al.  A Note on the Transformation of Chi-Squared Variables to Normality , 1986 .

[4]  Dulal K. Bhaumik,et al.  Statistical Methods for Groundwater Monitoring , 1994 .

[5]  L. León-Novelo,et al.  Small sample inference for gamma parameters: one‐sample and two‐sample problems , 2014 .

[6]  D. G. Chapman,et al.  On tests and estimates for the ratio of poisson means , 1952 .

[7]  David L. Daniel.  A Case Study Perspective on Working with ProUCL and a State Environmental Agency in Determining Background Threshold Values , 2015, International journal of environmental research and public health.

[8]  B. Bolstad.  Comparing some iterative methods of parameter estimation for censored gamma data , 1998 .

[9]  F. Garwood,et al.  i) Fiducial Limits for the Poisson Distribution , 1936 .

[10]  E. S. Pearson,et al.  THE USE OF CONFIDENCE OR FIDUCIAL LIMITS ILLUSTRATED IN THE CASE OF THE BINOMIAL , 1934 .

[11]  Dulal K. Bhaumik,et al.  Testing Parameters of a Gamma Distribution for Small Samples , 2009, Technometrics.

[12]  E. B. Wilson,et al.  The Distribution of Chi-Square. , 1931, Proceedings of the National Academy of Sciences of the United States of America.

[13]  K. Krishnamoorthy,et al.  Normal-Based Methods for a Gamma Distribution , 2008, Technometrics.

[14]  K. Krishnamoorthy,et al.  Confidence limits for lognormal percentiles and for lognormal mean based on samples with multiple detection limits. , 2011, The Annals of occupational hygiene.

[15]  T. Ogden.  Handling results below the level of detection. , 2010, The Annals of occupational hygiene.

[16]  S. L. Zabell,et al.  R. A. Fisher and Fiducial Argument , 1992 .

[17]  Robert D. Gibbons,et al.  Statistical Methods for Groundwater Monitoring: Second Edition , 2009 .

[19]  N. Reid,et al.  Simple and accurate inference for the mean of the gamma model , 1997 .

[20]  K. Krishnamoorthy,et al.  Likelihood ratio tests for comparing several gamma distributions , 2015 .

[21]  Anja Vogler,et al.  An Introduction to Multivariate Statistical Analysis , 2004 .

[22]  K. Krishnamoorthy,et al.  Handbook of Statistical Distributions with Applications, Second Edition , 2015 .

[23]  Gregory William Bennett,et al.  The fiducial argument in statistical inference , 1965 .

[24]  Dulal K. Bhaumik,et al.  One-Sided Approximate Prediction Intervals for at Least p of m Observations From a Gamma Population at Each of r Locations , 2006, Technometrics.

[25]  A. P. Dawid,et al.  The Functional-Model Basis of Fiducial Inference , 1982 .

[26]  K. Fent,et al.  PHENYLTINS IN WATER, SEDIMENT, AND BIOTA OF FRESHWATER MARINAS , 1991 .

[27]  Dulal K. Bhaumik,et al.  An Upper Prediction Limit for the Arithmetic Mean of a Lognormal Random Variable , 2004, Technometrics.

[28]  S. E. Ahmed,et al.  Handbook of Statistical Distributions with Applications , 2007, Technometrics.

[29]  D. Wilks.  Maximum Likelihood Estimation for the Gamma Distribution Using Data Containing Zeros , 1990 .