Relative Efficiency of Estimators of the Mean of a Normal Distribution when Coefficient of Variation is Known

A biased but simple and consistent estimator of the parameter ? has been obtained for the normal distribution N(?, a?₂), ?>0 where a is a known constant. It is shown that the estimator is more efficient than the sample mean or any suitably chosen constant multiple of the sample standard deviation. It is also proved to be more efficient than the mimumum variance unbiased estimator among a typical class of unbiased estimators derived by RASUL KHAN (1968).     

On Combination of Ratio and PPS Estimators

AGARWAL and KUMAR (1980) proposed an estimator, combining ratio and pps estimators of population mean and proved that the proposed estimator would always be better (in minimum mean square error sense) than the pps estimator or the ratio estimator under pps sampling scheme for optimum value of constant k (parameter). The optimum value of k is rarely known in practice, hence the alternative is to replace k from the sample-values. In this paper, an estimator depending on estimated optimum value of k based on sample-values, under pps sampling scheme is proposed and studied.     

A Monte Carlo Study of Robustness of Pretest and Shrinkage Estimators in Pooling Coefficients of Variation

Pooling data, when justified, is advantageous for estimating the true parameter. In this paper the problem of estimating the coefficient of variation is considered when it is a priori suspected that two coefficients of variation are the same. Various estimators based on pretest and shrinkage rules are considered. A comparison through the Simulated Mean Squared Error (SMSE) criterion is carried out among various proposed estimators of the target coefficient of variation. The relative simulated efficiencies of the restricted, shrinkage restricted and shrinkage pretest estimators are studied. It is found that the proposed estimators are quite robust when the sample sizes are not too large. The result of Monte Carlo study indicates that the proposed shrinkage pretest estimator is efficient than the usual estimator in a wider range.     

Estimation of introgression in cutthroat trout populations using microsatellites

Introgressive hybridization, mediated by anthropogenic activity, poses a threat to numerous and diverse taxa. The management of introgressed individuals or populations within species of conservation concern is currently the subject of scientific and political debate. We investigate the utility of 10 non-diagnostic microsatellite loci for investigating admixture from introduced Yellowstone cutthroat trout (Oncorhynchus clarkii bouvieri) and rainbow trout (O. mykiss) within 25 putative Rio Grande cutthroat trout (O. c. virginalis) populations. We apply five different approaches (correspondence analysis, maximum-likelihood assignment tests, an admixture estimator based on allele frequencies, an admixture estimator based on coalescent theory and an admixture estimator implementing a Bayesian method) and use two alternative O. c. virginalis reference samples. All approaches were capable of identifying one population that consisted entirely of introduced O. c. bouvieri, and three out of five approaches enabled us to discriminate those populations with relatively high levels of non-native introgression from those populations with little or none. Actual estimates of admixture coefficients within a test population, varied, however, with the approach and reference sample used. These results have important implications for policies dividing populations into different management categories according to the estimated proportion of non-native genetic material that they contain.     

A Class of Ratio-Cum-Product-Type Estimator

The problem of estimating the population mean using an auxiliary information has been dealt with in literature quite extensively. Ratio, product, linear regression and ratio-type estimators are well known. A class of ratio-cum-product-type estimator is proposed in this paper. Its bias and variance to the first order of approximation are obtained. For an appropriate weight ‘a’ and good range of α-values, it is found that the proposed estimator is superior than a set of estimators (i.e., sample mean, usual ratio and product estimators, SRIVASTAVA's (1967) estimator, CHAKRABARTY's (1979) estimator and a product-type estimator) which are, in fact, the particular cases of it. At optimum value of α, the proposed estimator is as efficient as linear regression estimator.     

Estimation and adjustment of microsatellite null alleles in nonequilibrium populations

Nonamplified (null) alleles are a common feature of microsatellite genotyping and can bias estimates of allele and genotype frequencies, thereby hindering population genetic analyses. The frequency of microsatellite null alleles in diploid populations can be estimated for populations that are in Hardy–Weinberg equilibrium. However, many microsatellite data sets are from nonequilibrium populations, often with known inbreeding coefficients (F) or fixation indices (F_IS or F_ST). Here, we propose a novel null allele estimator that can be used to estimate the null allele frequency and adjust visible allele frequencies in populations for which independent estimates of F, F_IS or F_ST are available. The algorithm is currently available as an Excel macro that can be downloaded at no cost from http://www.microchecker.hull.ac.uk/ and will be incorporated into the software micro ‐checker .     

A Nonparametric Estimator of Heterogeneity Variance with Applications to SMR‐ and Proportion‐Data

In this paper the situation of extra population heterogeneity is discussed from a analysis of variance point of view. We first provide a non‐iterative way of estimating the variance of the heterogeneity distribution without estimating the heterogeneity distribution itself for Poisson and binomial counts. The consequences of the presence of heterogeneity in the estimation of the mean are discussed. We show that if the homogeneity assumption holds, the pooled mean is optimal while in the presence of strong heterogeneity, the simple (arithmetic) mean is an optimal estimator of the mean SMR or mean proportion. These results lead to the problem of finding an optimal estimator for situations not represented by these two extreme cases. We propose an iterative solution to this problem. Illustrations for the application of these findings are provided with examples from various areas.     

Use of Transformed Auxiliary Variable in Estimating the Finite Population Mean

For estimating the finite population mean Y- of the study character y, an estimator using a transformed auxiliary variable has been defined. The bias and mean-squared error (MSE) of the proposed estimator have been obtained. The regions of preference have been obtained under which it is better than usual unbiased estimator y-, the ratio estimator y-_R = y-X-/x-, Sisodia and Dwivedi (1981) estimator y-_s = y-(X- + C_x)/(x- + C_x) and Singh and Kakran (1993) estimator y-_k = y[X- + β₂(x)]/[x- + β₂(x)]. An empirical study has been carried out to demonstrate the superiority of the suggested estimator over the others.     

Simultaneous Selection of Extreme Populations: A Subset Selection Approach

The problem of selecting a “best” (largest mean, or smallest mean) population from a collection of k independent populations was formulated and solved by Bechhofer (1954). Gupta (1965) solved another important problem, that of selecting a subset of populations containing the “best” population from the original collection of populations. Since then many variations of the problem have been considered. Tong (1969) and Lewis (1980) have investigated the problem of selecting extreme populations (populations with a largest, and populations with a smallest, mean) with respect to one and two standard populations, respectively. In this paper we study the selection of extreme populations in absence of any standard population. We formulate subset-selection procedures when variances are known and equal, and also in the most general case when they are unknown and unequal. Nonexistence of a single-stage procedure is noted for this latter case (even if variances are equal). A two-stage procedure and some of its associated properties are discussed. Tables needed for application are provided, as is a worked example.     

MAXIMUM-LIKELIHOOD ESTIMATION OF POPULATION DIVERGENCE TIMES AND POPULATION PHYLOGENY IN MODELS WITHOUT MUTATION

In this paper we present a method for estimating population divergence times by maximum likelihood in models without mutation. The maximum-likelihood estimator is compared to a commonly applied estimator based on Wright's F_ST statistic. Simulations suggest that the maximum-likelihood estimator is less biased and has a lower variance than the F_ST-based estimator. The maximum-likelihood estimator provides a statistical framework for the analysis of population history given genetic data. We demonstrate how maximum-likelihood estimates of the branching pattern of divergence of multiple populations may be obtained. We also describe how the method may be applied to test hypotheses such as whether populations have maintained equal population sizes. We illustrate the method by applying it to two previously published sets of human restriction fragment length polymorphism (RFLP) data.     

A Note on the Estimation of the Population Mean in Stratified Random Sampling Using Two Auxiliary Variables

In this paper, a two‐phase sampling estimator for a stratified population mean using two auxiliary variables x and z is considered when the stratum mean of x is unknown but that of z is known. The suggested estimator under its optimal condition is found to be more efficient than the one using only x.     

Accounting for missing data in the estimation of contemporary genetic effective population size (Ne)

Theoretical models are often applied to population genetic data sets without fully considering the effect of missing data. Researchers can deal with missing data by removing individuals that have failed to yield genotypes and/or by removing loci that have failed to yield allelic determinations, but despite their best efforts, most data sets still contain some missing data. As a consequence, realized sample size differs among loci, and this poses a problem for unbiased methods that must explicitly account for random sampling error. One commonly used solution for the calculation of contemporary effective population size (N_e) is to calculate the effective sample size as an unweighted mean or harmonic mean across loci. This is not ideal because it fails to account for the fact that loci with different numbers of alleles have different information content. Here we consider this problem for genetic estimators of contemporary effective population size (N_e). To evaluate bias and precision of several statistical approaches for dealing with missing data, we simulated populations with known N_e and various degrees of missing data. Across all scenarios, one method of correcting for missing data (fixed‐inverse variance‐weighted harmonic mean) consistently performed the best for both single‐sample and two‐sample (temporal) methods of estimating N_e and outperformed some methods currently in widespread use. The approach adopted here may be a starting point to adjust other population genetics methods that include per‐locus sample size components.     

The conformational properties of α,β‐dehydroamino acids with a C‐terminal ester group

α,β‐Dehydroamino acid esters occur in nature. To investigate their conformational properties, a systematic theoretical analysis was performed on the model molecules Ac‐ΔXaa‐OMe [ΔXaa = ΔAla, (E)‐ΔAbu, (Z)‐ΔAbu, ΔVal] at the B3LYP/6‐311+ + G(d,p) level in the gas phase as well as in chloroform and water solutions with the self‐consistent reaction field‐polarisable continuum model method. The Fourier transform IR spectra in CCl₄ and CHCl₃ have been analysed as well as the analogous solid state conformations drawn from The Cambridge Structural Database. The ΔAla residue has a considerable tendency to adopt planar conformations C5 (?, ψ ≈ ? 180°, 180°) and β2 (?, ψ ≈ ? 180°, 0°), regardless of the environment. The ΔVal residue prefers the conformation β2 (?, ψ ≈ ? 120°, 0°) in a low polar environment, but the conformations α (?, ψ ≈ ? 55°, 35°) and β (?, ψ ≈ ? 55°, 145°) when the polarity increases. The ΔAbu residues reveal intermediate properties, but their conformational dispositions depend on configuration of the side chain of residue: (E)‐ΔAbu is similar to ΔAla, whereas (Z)‐ΔAbu to ΔVal. Results indicate that the low‐energy conformation β2 is the characteristic feature of dehydroamino acid esters. The studied molecules constitute conformational patterns for dehydroamino acid esters with various side chain substituents in either or both Z and E positions. Copyright © 2011 European Peptide Society and John Wiley & Sons, Ltd.     

The significance of genetic erosion in the process of extinction

Summary The amount of genetic variation within a population is, among other things, related to population size. In small populations loss of genetic variation due to high levels of genetic drift and inbreeding may result in decline of individual fitness and increase the chance of population extinction. This chain of processes is known as genetic erosion. In this study we tested the genetic erosion hypothesis by investigating the relation between morphological variation and population size in two perennial, outbreeding plant species, Salvia pratensis and Scabiosa columbaria. To relate phenotypic variation to genetic variation the experiments were performed under common environmental conditions. For both species a positive correlation was observed between the amount of phenotypic variation and population size (Salvia r=0.915; Scabiosa r=0.703). Part of this variation is likely to have a genetic base, although maternal effects were present in the seedling and juvenile life stages. Differences between populations could in both species be attributed to parameters related to fitness, i.e. growth rate in Salvia and reproductive effort in Scabiosa. Discriminant functions reflecting these parameters did not however discriminate between large and small populations.Results are discussed in relation to the common environment approach and to electrophoretic results obtained earlier (Van Treuren et al. 1991).     

Genetic variation in sympatric populations of diploid and polyploid brine shrimp (Artemia parthenogenetica)

We examined genetic variation in sympatric diploid and polyploid brine shrimp Artemia parthenogenetica from each of three populations (China, Italy and Spain). Italian and Spanish tetraploids are closely related (I=0.964). Diploids and tetraploids within each of the two European populations are also closely related (mean I=0.905). Most alleles found in diploids also exist in sympatric polyploids. In contrast, the asexual Artemia (2N, 4N and 5N) in our study share few alleles with their close sexual relative, A. tunisiana (mean I=0.002). These results, as well as the work of other authors, strongly suggest that at least the tetraploid Artemia in our study have an autopolyploid origin.Clonal diversity of polyploid Artemia can be very high at least in some population. Both diploids and polyploids had low clonal diversities in the populations dominated by polyploids and high clonal diversities in the population dominated by diploids.The most common genotypes of sympatric diploid and polyploid Artemia frequently differed. Some alleles occurred only in diploids, while others were restricted to polyploids. These results suggest that polyploidy in Artemia has led to genetic divergence from diploid progenitors, and that ploidy-level variation must also be considered in developing an understanding of spatial and temporal allozyme polymorphism in asexual populations.     

Estimation of the Power of the Kruskal-Wallis Test

Power calculations of a statistical test require that the underlying population distribution(s) be completely specified. Statisticians, in practice, may not have complete knowledge of the entire nature of the underlying distribution(s) and are at a loss for calculating the exact power of the test. In such cases, an estimate of the power would provide a suitable substitute. In this paper, we are interested in estimating the power of the Kruskal-Wallis one-way analysis of variance by ranks test for a location shift. We investigated an extension of a data-based power estimation method presented by Collings and Hamilton (1988), which requires no prior knowledge of the underlying population distributions other than necessary to perform the Kruskal-Wallis test for a location shift. This method utilizes bootstrapping techniques to produce a power estimate based on the empirical cumulative distribution functions of the sample data. We performed a simulation study of the extended power estimator under the conditions of k = 3 and k = 5 samples of equal sizes m = 10 and m = 20, with four underlying continuous distributions that possessed various location configurations. Our simulation study demonstates that the Extended Average × & Y power estimation method is a reliable estimator of the power of the Kruskal-Wallis test for k = 3 samples, and a more conservative to a mild overestimator of the true power for k = 5 samples.     

Percentage Points of Multivariate t Distribution with Zero Correlations and their Application

The k-variate t distribution plays an important role in the application of Multiple Comparison Procedures, Ranking and Selection Procedures, and Estimation of Ranked Parameters. This distribution was considered by several authors in the past and is now widely used. In this paper, tables of some percentage points of the multivariate t distribution are tabulated for given probability level γ = 0.80, 0.90, 0.95, and 0.99; population size k = 2(1)20; and degrees of freedom v=2(1) 30(5)60 with common correlation coefficient ? = 0. The accuracy of the percentage points is specifically determined (and is up to the fourth decimal place). An application of these tables is considered for interval estimation of the largest mean of k(≧2) normal populations.     

Contrasting levels of genetic diversity between the historically rare orchid Cypripedium japonicum and the historically common orchid Cypripedium macranthos in South Korea

Many terrestrial orchids are historically rare and occur in small, spatially isolated populations. Theory predicts that such species will harbour low levels of genetic variation within populations and will exhibit a high degree of population genetic divergence, primarily as a result of genetic drift. If the origin of the present‐day populations is relatively recent from the same genetically depauperate source population, a complete lack of genetic differentiation between conspecific populations is expected. If a terrestrial orchid was historically common with moderate or high levels of genetic diversity, but has experienced more recent anthropogenic disturbance as a result of over‐collection, it would still exhibit initial levels of genetic variation within populations and a low degree of genetic divergence between populations. To test these predictions, we examined the genetic diversity in six populations (N = 131) of the historically and currently rare Cypripedium japonicum and in four populations (N = 94) of the historically common but now rare C. macranthos from South Korea. Fourteen putative allozyme loci resolved from eight enzyme systems revealed no variation either within or among populations of C. japonicum, which supports the first prediction. In contrast, populations of C. macranthos harboured high levels of genetic variation (mean percentage of polymorphic loci %P = 46.7; mean expected heterozygosity H_e = 0.185) and exhibited a low degree of population genetic divergence (G_ST = 0.059), supporting the second prediction. The lack of genetic variation both within and among conspecific populations of C. japonicum may suggest that populations originated from the same genetically depauperate ancestral population. The high levels of genetic diversity maintained in populations of C. macranthos suggest that the collection‐mediated decrease in the number of individuals is still too recent for long‐term effects on genetic variation. Based on current demographic and genetic data, in situ and ex situ conservation strategies should be provided to preserve genetic variation and to ensure the long‐term survival of the two species in the Korean Peninsula. © 2009 The Linnean Society of London, Botanical Journal of the Linnean Society, 2009, 160 , 119–129.     

An Improved Class of Estimators for Finite Population Mean in Sample Surveys

This paper proposes a class of estimators for estimating the finite population mean -Y of a study variate y using information on two auxiliary variates, one of which is positively and the other negatively correlated with the study variate y. An “asymptotically optimum estimator” (AOE) in the class is identified with its bias and mean square error formulae. It is observed that the proposed AOE is more efficient than Srivastava (1965), Srivastava (1974), Prasad (1989) and Gandge , Varghese , and Prabhu-Ajgaonkar (1993) estimators.