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Paul W. Mielke Jr.

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Paul W. Mielke Jr.

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Paul W. Mielke Jr.

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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.

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Paul W. Mielke Jr.

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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.

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Paul W. Mielke Jr.

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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.

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Paul W. Mielke Jr.

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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.

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Paul W. Mielke Jr. and Jonnie G. Medina

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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.

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PAUL W. MIELKE JR. and EARL S. JOHNSON

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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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Paul W. Mielke Jr., Kenneth J. Berry, and Glenn W. Brier

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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.

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Paul W. Mielke Jr., Lewis O. Grant, and Charles F. Chappell

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This study is concerned with the elevation and spatial variation effects of wintertime orographic cloud seeding over an area encompassing Fremont, Hoosier and Vail mountain passes in the central Colorado mountains during a period from 1960 to 1965. The observation network consisted of 65 precipitation stations distributed over the three passes. Depending on the grouping of precipitation stations used to represent the prime target area of the study, the average daily precipitation for all 120 seeded days was from 6 to 11% greater than the average daily precipitation for all 131 non-seeded days. There is a high probability that these differences could have occurred by chance alone.

Analyses have also been made according to physically defined stratifications based on a model which describes the seeding effects ascribed to the various strata. Statistically significant increases (decreases) were observed over much of the area for the seeded periods in comparison with the non-seeded periods when 500 mb temperatures were −20C and warmer (−27C and colder). Little or no effects were noted in the intermediate temperature range. When 500-mb wind velocities were from 22–28 m sec−1, statistically significant increases were observed during the seeded period in comparison with the non-seeded period throughout the area.

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