Ratio estimators in agricultural research : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Statistics at Massey University, Palmerston North, New Zealand

This thesis addresses the problem of estimating the ratio of quantitative variables from several independent samples in agricultural research. The first part is concerned with estimating a binomial proportion, the ratio of discrete counts, from several independent samples under the assumption that there is a single underlying binomial proportion p in the population of interest. The distributions and properties of two linear estimators, a weighted average and an arithmetic average, are derived and merits of the approaches discussed. They are both unbiased estimators of the population proportion, with the weighted average having lower variability than the arithmetic average. These findings are obtained through a first principles analysis, with a geometrical interpretation presented. This variability result is also a consequence of the Rao-Blackwell theorem, a well-known result in the theory of statistical inference. Both estimators are used in the literature but we conclude that the weighted average estimate should always be used when the sample sizes are unequal. These results are illustrated by a simulation experiment and are validated using survey data in the study of lodging percentage of sunflower cultivar, Improved Peredovic, in Jilin Province, China in 1994. The second part of the research addresses the problem of estimating the ratio μͯ / μ, of the means of continuous variables in agricultural research. The distributional properties of the ratio X/Y of independent normal variables are examined, both theoretically and using simulation. The results show that the moments of the ratio do not exist in general. The moments exist, however, for a punctured normal distribution of the denominator variable if we only sample points for which | Y |>ε, ε being a small positive quantity. We draw out the practical rule-of-thumb that the ratio of two independent normal variables can be used to estimate μͯ / μ, when the coefficient of variation of the denominator variable is sufficiently small (less than or equal to 0.2). Lastly the thesis evaluates the relative merits of two common estimators of the ratio of the means of continuous variables in agricultural research, an arithmetic average and a weighted average, via simulation experiments using normal distributions. In the first simulation, the ratio and common coefficient of variation are changed while the sample size is kept moderately large. In the second simulation, the ratio and sample size are changed while the coefficient of variation is held constant. Results show that the weighted average always provides a better estimate of the true ratio and has lower variability than the arithmetic average. It is recommended that the weighted average be used for estimating the ratio from several pairs of observations. These results are tested using research data from rice breeding multi-environment trials in Jilin Province, China in 1995 and 1996. These data are used to demonstrate the diagnostic approach developed for assessing the 'safety' use of the arithmetic and the weighted average methods for estimating the ratio of the means of independent normal variables

Systematic analysis of the incoming quark energy loss in cold nuclear matter

The investigation into the fast parton energy loss in cold nuclear matter is crucial for a good understanding of the parton propagation in hot-dense medium. By means of four typical sets of nuclear parton distributions and three parametrizations of quark energy loss, the parameter values in quark energy loss expressions are determined from a leading order statistical analysis of the existing experimental data on nuclear Drell-Yan differential cross section ratio as a function of the quark momentum fraction. It is found that with independence on the nuclear modification of parton distributions, the available experimental data from lower incident beam energy rule out the incident-parton momentum fraction quark energy loss. Whether the quark energy loss is linear or quadratic with the path length is not discriminated. The global fit of all selected data gives the quark energy loss per unit path length {\alpha} = 1.21\pm0.09 GeV/fm by using nuclear parton distribution functions determined only by means of the world data on nuclear structure function. Our result does not support the theoretical prediction: the energy loss of an outgoing quark is three times larger than that of an incoming quark approaching the nuclear medium. It is desirable that the present work can provide useful reference for the Fermilab E906/SeaQuest experiment

Quark energy loss and shadowing in nuclear Drell-Yan process

The energy loss effect in nuclear matter is another nuclear effect apart from the nuclear effects on the parton distribution as in deep inelastic scattering process. The quark energy loss can be measured best by the nuclear dependence of the high energy nuclear Drell-Yan process. By means of three kinds of quark energy loss parameterizations given in literature and the nuclear parton distribution extracted only with lepton-nucleus deep inelastic scattering experimental data, measured Drell-Yan production cross sections are analyzed for 800GeV proton incident on a variety of nuclear targets from FNAL E866. It is shown that our results with considering the energy loss effect are much different from these of the FNAL E866 who analysis the experimental data with the nuclear parton distribution functions obtained by using the deep inelastic lA collisions and pA nuclear Drell-Yan data . Considering the existence of energy loss effect in Drell-Yan lepton pairs production,we suggest that the extraction of nuclear parton distribution functions should not include Drell-Yan experimental data.Comment: 12 page

Nuclear geometry effect and transport coefficient in semi-inclusive lepton-production of hadrons off nuclei

Hadron production in semi-inclusive deep-inelastic scattering of leptons from nuclei is an ideal tool to determine and constrain the transport coefficient in cold nuclear matter. The leading-order computations for hadron multiplicity ratios are performed by means of the SW quenching weights and the analytic parameterizations of quenching weights based on BDMPS formalism. The theoretical results are compared to the HERMES positively charged pions production data with the quarks hadronization occurring outside the nucleus. With considering the nuclear geometry effect on hadron production, our predictions are in good agreement with the experimental measurements. The extracted transport parameter from the global fit is shown to be $\hat{q} = 0.74\pm0.03 GeV^2/fm$ for the SW quenching weight without the finite energy corrections. As for the analytic parameterization of BDMPS quenching weight without the quark energy E dependence, the computed transport coefficient is $\hat{q} = 0.20\pm0.02 GeV^2/fm$. It is found that the nuclear geometry effect has a significant impact on the transport coefficient in cold nuclear matter. It is necessary to consider the detailed nuclear geometry in studying the semi-inclusive hadron production in deep inelastic scattering on nuclear targets.Comment: 14 pages, 3 figures. arXiv admin note: text overlap with arXiv:1310.569

Fuzzy bases and the fuzzy dimension of fuzzy vector spaces

In this paper, new definitions of a fuzzy basis and a fuzzy dimension for a fuzzy vector space are presented. A fuzzy basis for a fuzzy vector space $(E,mu)$ is a fuzzy set $beta$ on $E$. The cardinality of a fuzzy basis $beta$ is called the fuzzy dimension of $(E,mu)$. The fuzzy dimension of a finite dimensional fuzzy vector space is a fuzzy natural number. For a fuzzy vector space, any two fuzzy bases have the same cardinality. If $widetilde{E}_1$ and $widetilde{E}_2$ are two fuzzy vector spaces, then $dim(widetilde{E}_1+widetilde{E}_2)+dim(widetilde{E}_1cap widetilde{E}_2)=dim(widetilde{E}_1) +dim(widetilde{E}_2)$ and $dim({widetilde{rm{ker }}f})+dim({widetilde{rm{im }}f})=dim(widetilde{E})$ hold without any restricted conditions. end{abstract