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Three-Parameter Kappa Distribution Maximum Likelihood Estimates and Likelihood Ratio Tests

PAUL W. MIELKE JR.Department of Statistics, Colorado State University, Fort Collins, Colo.

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EARL S. JOHNSONDepartment of Statistics, Mary Washington College, Fredericksburg, Va.

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Abstract

Methods are presented for obtaining maximum likelihood estimates and tests of hypotheses involving the three-parameter kappa distribution. The obtained methods are then applied by fitting this distribution to realized sets of precipitation and streamflow data and testing for seeding effect differences between realized seeded and nonseeded sets of precipitation data. The kappa distribution appears to fit precipitation data as well as either the gamma or log-normal distribution. As a consequence, the sensitivity of test procedures based on the kappa distribution compares favorably with that of previously used test procedures.

Since both the density and cumulative distribution functions of the kappa distribution are in closed form, the density and cumulative distribution functions associated with each order statistic are also in closed form. In contrast, the gamma and log-normal cumulative distribution functions are not in closed form. As a consequence, computations involving order statistics are far more convenient with the kappa distribution than either the gamma or log-normal distributions.

Abstract

Methods are presented for obtaining maximum likelihood estimates and tests of hypotheses involving the three-parameter kappa distribution. The obtained methods are then applied by fitting this distribution to realized sets of precipitation and streamflow data and testing for seeding effect differences between realized seeded and nonseeded sets of precipitation data. The kappa distribution appears to fit precipitation data as well as either the gamma or log-normal distribution. As a consequence, the sensitivity of test procedures based on the kappa distribution compares favorably with that of previously used test procedures.

Since both the density and cumulative distribution functions of the kappa distribution are in closed form, the density and cumulative distribution functions associated with each order statistic are also in closed form. In contrast, the gamma and log-normal cumulative distribution functions are not in closed form. As a consequence, computations involving order statistics are far more convenient with the kappa distribution than either the gamma or log-normal distributions.

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