Discrete data such as count data or data in the form of proportions arise in biological investigations and other similar fields. These data often show variation greater or smaller than predicted by the simple probability models such as the Poisson or the binomial model. Several discrete models have been used for modeling the counts or proportions by many authors (see, for example, Byers et al. 2003; Consul and Jain, 1973; Efron, 1986; Gibson and Austin, 1996; Kupper and Haseman, 1978; and Saha and Paul 2005). This article reviews briefly several aspects and the properties for some of the most commonly used discrete models for modeling counts or proportions, namely the negative binomial, the generalized Poisson, the double Poisson, the generalized negative binomial, the beta-binomial, the correlated binomial, the multiplicative binomial, and the double binomial models. The maximum likelihood method is outlined for the estimation of the parameters of these models. Comparison studies of these models are considered in light of goodness of fit test as well as model selection criteria through real-life data occurring in agricultural and toxicological fields. Keywords : Beta-binomial, Biological data, Double binomial, Negative binomial model, Extra dispersion parameter.

  

  Here we propose a kernel based density estimator for the weighted univariate data under multiplicative censoring that may be useful in the context of inference for certain agricultural data. Asymptotic formulae for the MSE and MISE of the new estimator are derived and it is shown that the optimal rate of convergence of MISE of the new estimator is slower than that in the case of i.i.d. data as may be expected. Keywords : Length-biased data, Multiplicative censoring, Kernel density, Size-biased data.

  Genomic studies and data have revolutionized genetic studies. Two kinds of studies, namely, microarrays for gene expression and SNP?s for possible association with various diseases, have become very popular. We survey briefly both of these areas, highlighting some recent theoretical and methodological work in Statistics. Keywords : High dimensional inference, Microarrays, SNP?s, Benjamini-Hochberg test, mBIC, Lasso.

  For the problem of multiple testing, the Benjamini-Hochberg (B-H) procedure has become a very popular method in applications. We show how the B-H procedure can be interpreted as a test based on the spacings corresponding to the p-value distributions. Using this equivalence, we develop a class of generalized B-H procedures that maintain control of the false discovery rate in finite-samples. We also consider the effect of correlation on the procedure; simulation studies are used to illustrate the methodology. Keywords : Dependence, Familywise error rate, High-dimensional data, Multiple comparisons, Simultaneous inference.

  In choosing a fractional factorial plan, one often decides on an orthogonal plan. Fractional factorial plans represented by orthogonal arrays provide orthogonal plans which also have strong optimality properties. While using such fractional factorial plans, it is possible in some situations that certain treatment combinations are infeasible or, even if these are feasible, no observations can be made on these. In such situations, it is desirable to have an orthogonal fractional factorial plan that does not include the infeasible treatment combinations. In this paper, we provide a method of obtaining such plans represented by two-symbol orthogonal arrays of strength two. Keywords : Debarred combination, Projectivity, Hadamard property.

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  Molecular selection indices such as the molecular eigen selection index method (MESIM) and other molecular marker score selection indices maximize the selection response by combining information on molecular markers linked to quantitative trait loci and phenotypic values of the traits. The standard restrictive selection index and the restrictive eigen selection index method (RESIM) maximize the selection response of only some traits while leaving others unchanged. This research extends the MESIM, RESIM, genomewide molecular selection index, and standard restrictive selection index to the case of a multitrait multienvironment genomewide molecular marker selection index. We used simulated data and real data for estimating the performance of various genomewide and molecular marker selection indices. Results showed that, in general, when several traits were selected in various environments simultaneously and all the markers were included in the indices, the multitrait multienvironment genomewide molecular marker selection index increased the genotypic means over the mean of individuals selected by other selection indices. The sampling properties of MESIM and RESIM in the context of multitrait multienvironment genomewide molecular marker selection indices and their selection responses are known, and their estimators showed desirable statistical properties such as consistency and asymptotic unbiasedness. We propose a general procedure for finding the asymptotic statistical sampling properties of the multitrait multienvironment genomewide molecular marker selection index and of other selection indices by applying the theory underlying MESIM and RESIM. Keywords : Selection index, Restrictive selection indices, Eigenanalysis, Multitrait multienvironment genomewide molecular marker indices.

  The theory of wavelets has found wide applications in nonparametric estimation, especially for density and related functionals. It has been adapted to many other situations in addition to density estimation for iid data. Such procedures may potentially be useful for nonparametric density estimation in agricultural setting such as in modeling yield of crops and crop insurance claims distribution. This article presents some recent developments in this area in a comprehensive way dealing with different data types in addition to the iid setup. Keywords : Biased data, Censored data, Components of a mixture, Decon-volution model, Density estimation, Multiplicative censoring, Wavelets.

  Due to the ever-presence of genotype-by-environment interaction (GE), multi-environmental trials (MET) are essential for effective breeding line selection and cultivar recommendation. AMMI (Additive Main effect and Multiplicative Interaction) analysis and GGE (Genotypic main effect plus genotype-by-environment interaction) biplot analysis are two popular graphical analysis systems for MET data analysis. This paper introduces and compares the AMMI graphs and the GGE biplots for three major aspects of MET data analysis: mega-environment delineation, genotype evaluation, and test environment evaluation. The conclusions are: 1) when used properly, both systems are capable of mega-environments delineation and genotype evaluation; 2) the GGE biplot is also effective in test environment evaluation; 3) the GGE biplots are simpler to construct than the AMMI graphs; while different views of the same GGE biplot can be used to address all three aspects of MET data analysis, a different graph has to be constructed in AMMI analysis to address each aspect; 4) the GGE biplots are more informative than the AMMI graphs because of its inner-product property, whereby information on the performance of each genotype in each environment is preserved. Therefore, the GGE biplot graphs are highly preferable over AMMI graphs in MET data analysis. Keywords : Additive Main effect and Multiplicative Interaction, Genotype main effect, Genotype-by-environment interaction, Interaction principal component, Multi-environment trials, Principal component, Singular-value decomposition, Singular-value partitioning.,

  Relatively little attention has been paid to optimal sampling design when the support of the sample is a linear transect. The D-optimality criterion allows the quantitative comparison of spatial sampling designs for samples with either point or transect support. D-optimality is applied to transects that are equally spaced point samples taken along a straight or curved path. For short transects containing three points, the optimal angle between adjacent transect segments can be determined analytically by setting the derivative of the D-optimality criterion with respect to the spatial covariance to zero. Results show that straight transects are suboptimal when the random variable being sampled has a Gaussian or spherical covariance function. By combining D-optimality with simulated annealing or Powell?s algorithm, optimal spatial designs for longer transects can be determined. For a Gaussian variogram, a zigzag pattern is nearly optimal and is better than a straight transect. For a spherical variogram, a transect that bends twice to the right then twice to the left maintaining a constant interior angle is nearly optimal and is better than a straight transect. Finally, for an exponential variogram, straight transects are optimal. Implementation of these results for use in practical design of field surveys is discussed. Keywords : Statistical design, Transects, Geostatistics, Optimization.

  Along with the fast developments in molecular biology and biotechnology, a large amount of biological data is available from genetic studies of important breeding traits in plants, which in turn provides an opportunity for undertaking genotypic selection in the breeding process. However, gene information has not been effectively used in crop improvement due to the lack of appropriate tools. The simulation approach can utilize the vast and diverse genetic information, predict the cross performance and compare different selection methods. Hence, the best performing crosses and effective breeding strategies can be identified. QuLine and QuHybrid are computer tools capable of defining a range from simple to complex genetic models and simulating breeding processes for developing final advanced lines and hybirds. Based on the results from simulation experiments, breeders can optimize their breeding methodology and greatly improve the breeding efficiency. In this paper, we first introduce the underlying principles of simulation modeling in crop enhancement, and then summarize several applications of QuLine in comparing different selection strategies, precision parental selection using known gene information, and the design approach in breeding. Breeding simulation allows the definition of complicated genetic models consisting of multiple alleles, pleiotropy, epistasis and gene-by-environment interaction, and provides a useful tool to efficiently use the wide spectrum of genetic data and information available to the breeders. Keywords : Breeding strategy and method, Computer simulation, Crop breeding, Genetic model.

  For modelling and forecasting of cyclical time-series data, linear time-series models, like Autoregressive integrated moving average (ARIMA) model of order more than one, and Nonlinear time-series models, like Xxponential autoregressive (EXPAR) and Self-exciting threshold autoregressive (SETAR) models are generally employed. In practical situations, exact data generating process of time-series observations is not known. Therefore, fitted values from linear and nonlinear models may be used as explanatory variables to empirically describe the same. In this paper, the ARIMA, EXPAR and SETAR models, which are capable of capturing the cyclical behaviour are studied. In order to improve modelling and forecasting capabilities of the models, these are combined by using the Constant coefficient regression method (A.E.S.(C)) as well as the Time-varying coefficient regression method (A.E.S.(T.V.)) through Kalman filter (KF) technique. As an illustration, the models are then applied to describe annual Mackerel catch time-series data of Karnataka. Performance of fitted models is examined by computing various measures of goodness of fit, viz. Normalized Akaike information criterion (NAIC), Bayesian information criterion (BIC) and Mean square error (MSE). Finally, forecasting performance of fitted models is evaluated by Mean square prediction error (MSPE) criterion. It is found that the combined model fitted by using the A.E.S.(T.V.) has performed best for the data under consideration. For modelling and forecasting of cyclical time-series data, linear time-series models, like Autoregressive integrated moving average (ARIMA) model of order more than one, and Nonlinear time-series models, like Xxponential autoregressive (EXPAR) and Self-exciting threshold autoregressive (SETAR) models are generally employed. In practical situations, exact data generating process of time-series observations is not known. Therefore, fitted values from linear and nonlinear models may be used as explanatory variables to empirically describe the same. In this paper, the ARIMA, EXPAR and SETAR models, which are capable of capturing the cyclical behaviour are studied. In order to improve modelling and forecasting capabilities of the models, these are combined by using the Constant coefficient regression method (A.E.S.(C)) as well as the Time-varying coefficient regression method (A.E.S.(T.V.)) through Kalman filter (KF) technique. As an illustration, the models are then applied to describe annual Mackerel catch time-series data of Karnataka. Performance of fitted models is examined by computing various measures of goodness of fit, viz. Normalized Akaike information criterion (NAIC), Bayesian information criterion (BIC) and Mean square error (MSE). Finally, forecasting performance of fitted models is evaluated by Mean square prediction error (MSPE) criterion. It is found that the combined model fitted by using the A.E.S.(T.V.) has performed best for the data under consideration.

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  Two forms of genotype × environment interaction (GEI) are of concern to plant breeders. One consists of fixed GEI associated with predictable environmental, geographical, or management factors that can be used to delineate a target population of environments (TPE) for cultivar development and testing. The other consists of random and unexplained rank changes among trials within the TPE which are not associated with any known factor. These two types of GEI must be managed differently by plant breeding programs; fixed GEI is managed by developing or identifying cultivars with adaptation to the specific fixed factor causing the interaction, while random GEI is a noise stratum that is managed through wide-scale testing that adequately samples environmental variation in the TPE, and through the use of best linear unbiased prediction (BLUP). There is substantial evidence that fixed GEI is of limited importance within well-designed TPE. Management of GEI in cultivar development programs, and the estimation of means from multi-environment trials with appropriate measures of precision (METs) has been hampered by the widespread use of inappropriate models that designate trials or trial locations as fixed effects in the combined analysis of cultivar testing data, resulting in unnecessary division of TPEs, identification of putative patterns of adaptation that are not repeated in subsequent testing, and over-estimation of the precision of entry means in multi-environment trials. Mixed model approaches to testing the relative importance of fixed and random GEI in METs are presented. Keywords : Genotype × environment interaction, BLUP, Mixed models, Cultivar development genetic correlation, Adaptation.