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- Author or Editor: Paul W. Mielke Jr. x
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Abstract
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Abstract
Rapidly converging iterative procedures for obtaining exact maximum likelihood estimates of the two-parameter gamma distribution scale and shape parameters are presented. These procedures yield estimates of parameters associated with a likelihood ratio test based on the two-parameter gamma distribution for investigating possible treatment-induced scale differences.
Abstract
Rapidly converging iterative procedures for obtaining exact maximum likelihood estimates of the two-parameter gamma distribution scale and shape parameters are presented. These procedures yield estimates of parameters associated with a likelihood ratio test based on the two-parameter gamma distribution for investigating possible treatment-induced scale differences.
Abstract
A simple reparameterization of the beta distribution is presented. This reparameterization provides a more intuitive description of the beta distribution than the usual version. As a consequence, easily interpreted beta distribution likelihood ratio tests based on this reparameterization are introduced. To facilitate the use of this approach, convenient iterative procedures are given for obtaining the required maximum likelihood estimates of the relevant parameters. In addition, numerical examples of these techniques are illustrated with recently acquired data from a hail suppression project.
Abstract
A simple reparameterization of the beta distribution is presented. This reparameterization provides a more intuitive description of the beta distribution than the usual version. As a consequence, easily interpreted beta distribution likelihood ratio tests based on this reparameterization are introduced. To facilitate the use of this approach, convenient iterative procedures are given for obtaining the required maximum likelihood estimates of the relevant parameters. In addition, numerical examples of these techniques are illustrated with recently acquired data from a hail suppression project.
Abstract
A positively skewed two-parameterer family of distributions is introduced as an alternative description of precipitation amounts. Each distribution of this family is associated with a simple two-sample non-parametric test which, for large samples, is optimum in detecting scale changes induced, say, by cloud seeding. Since the nonpararmetric test in question here is specified by the shape parameter of this family, two procedures are given for estimating parameters of this family from observed precipitation data. In addition, specific comparisons are made between this new family and corresponding gamma and one-sided t families.
Abstract
A positively skewed two-parameterer family of distributions is introduced as an alternative description of precipitation amounts. Each distribution of this family is associated with a simple two-sample non-parametric test which, for large samples, is optimum in detecting scale changes induced, say, by cloud seeding. Since the nonpararmetric test in question here is specified by the shape parameter of this family, two procedures are given for estimating parameters of this family from observed precipitation data. In addition, specific comparisons are made between this new family and corresponding gamma and one-sided t families.
Abstract
This paper is concerned. with the application of well-known statistical methods (e.g. matched-pairs t-test, two-sample t-test, one-way analysis of variance and significance test of Pearson's correlation coefficient) in the atmospheric sciences. This concern results from the fact that these statistical methods are based on a complex nonintuitive geometry which does not correspond with the perceived Euclidean geometry of the data intended to be analyzed. The real and artificial examples of this paper demonstrate how these commonly used statistical methods yield conclusions which may contradict rational interpretations by investigators. The geometric problem underlying these well-known statistical methods is their dependence on a peculiar distance measure defined between all pairs of measurements (this distance measure does not satisfy the triangle inequality condition of metric spaces, e.g., the familiar Euclidean space). Alternative statistical methods are suggested which overcome this geometric problem.
Abstract
This paper is concerned. with the application of well-known statistical methods (e.g. matched-pairs t-test, two-sample t-test, one-way analysis of variance and significance test of Pearson's correlation coefficient) in the atmospheric sciences. This concern results from the fact that these statistical methods are based on a complex nonintuitive geometry which does not correspond with the perceived Euclidean geometry of the data intended to be analyzed. The real and artificial examples of this paper demonstrate how these commonly used statistical methods yield conclusions which may contradict rational interpretations by investigators. The geometric problem underlying these well-known statistical methods is their dependence on a peculiar distance measure defined between all pairs of measurements (this distance measure does not satisfy the triangle inequality condition of metric spaces, e.g., the familiar Euclidean space). Alternative statistical methods are suggested which overcome this geometric problem.
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.
Abstract
A new covariate ratio procedure is presented for estimating a treatment-induced effect. The procedure 1) allows for uncontrolled natural variability, 2) adjusts for disproportionate allocation of non-treated and treated experimental units, 3) diminishes the influence in an objective manner of an individual value corresponding to any experimental unit, and 4) accounts for differential treatment effects, i.e., a simple location or scale change is not assumed. This procedure is applied to specific meteorologically defined partitions involving data of the Climax I and II experiments. Results based on the pooled data indicate a 32% precipitation increase for the −20 to − 11°C 500 mb temperature partition, a 49% increase for the 190 to 250° 700 mb wind direction partition, and a 13% increase for the total sample. Comparisons based on Monte Carlo simulations (re-randomization) indicate that this procedure yields estimates which are more stable (precise) than corresponding estimates based on the double ratio.
Abstract
A new covariate ratio procedure is presented for estimating a treatment-induced effect. The procedure 1) allows for uncontrolled natural variability, 2) adjusts for disproportionate allocation of non-treated and treated experimental units, 3) diminishes the influence in an objective manner of an individual value corresponding to any experimental unit, and 4) accounts for differential treatment effects, i.e., a simple location or scale change is not assumed. This procedure is applied to specific meteorologically defined partitions involving data of the Climax I and II experiments. Results based on the pooled data indicate a 32% precipitation increase for the −20 to − 11°C 500 mb temperature partition, a 49% increase for the 190 to 250° 700 mb wind direction partition, and a 13% increase for the total sample. Comparisons based on Monte Carlo simulations (re-randomization) indicate that this procedure yields estimates which are more stable (precise) than corresponding estimates based on the double ratio.
Abstract
This paper considers the examination of possible differences in monthly sea-level pressure patterns, The satisfactory examination of such differences requires appropriate multi-response parametric methods based on unknown multivariate distributions (i.e., an appropriate parametric technique is probably non-existent). In order to avoid the likely insurmountable difficulties involving parametric methods, the application of multi-response permutation procedures (MRPP) is suggested as an appropriate approach for the examination of such differences.
Abstract
This paper considers the examination of possible differences in monthly sea-level pressure patterns, The satisfactory examination of such differences requires appropriate multi-response parametric methods based on unknown multivariate distributions (i.e., an appropriate parametric technique is probably non-existent). In order to avoid the likely insurmountable difficulties involving parametric methods, the application of multi-response permutation procedures (MRPP) is suggested as an appropriate approach for the examination of such differences.
Abstract
The Climax I and II wintertime orographic cloud seeding experiments have recently been reanalyzed (Mielke et al., 1981c). The primary inference technique of this recent reanalysis involved (i) target-control residual data resulting from a simple linear model fitted by least squares and (ii) analyses based on classical linear rank statistics. The respective problems associated with this inference technique are (i) the residual data are highly dependent on a few very large values due to the model being fit by least squares and (ii) the complex non-Euclidean geometry underlying classical linear rank tests. The purpose of this article is to describe and apply a new inference technique which resolves both of these problems. This new inference technique involves residual data resulting from a median regression line and analyses based on recently developed rank tests associated with multi-response permutation procedures (MRPP). Application of this new inference to specific meteorological partitions of the Climax I and II experiments indicates that the evidence for a seeding effect is a little stronger (i.e., smaller P values) with the new technique than with the old technique for the warm Climax II 500 mb temperature partition and a Climax I 700 mb wind velocity partition, a little weaker for the warm Climax II 700 mb equivalent potential temperature partition, and about the same for the other meteorological partitions examined.
Abstract
The Climax I and II wintertime orographic cloud seeding experiments have recently been reanalyzed (Mielke et al., 1981c). The primary inference technique of this recent reanalysis involved (i) target-control residual data resulting from a simple linear model fitted by least squares and (ii) analyses based on classical linear rank statistics. The respective problems associated with this inference technique are (i) the residual data are highly dependent on a few very large values due to the model being fit by least squares and (ii) the complex non-Euclidean geometry underlying classical linear rank tests. The purpose of this article is to describe and apply a new inference technique which resolves both of these problems. This new inference technique involves residual data resulting from a median regression line and analyses based on recently developed rank tests associated with multi-response permutation procedures (MRPP). Application of this new inference to specific meteorological partitions of the Climax I and II experiments indicates that the evidence for a seeding effect is a little stronger (i.e., smaller P values) with the new technique than with the old technique for the warm Climax II 500 mb temperature partition and a Climax I 700 mb wind velocity partition, a little weaker for the warm Climax II 700 mb equivalent potential temperature partition, and about the same for the other meteorological partitions examined.
