Some improved ratio type estimators of population mean and ratio in finite population sample surveys

In this paper we present a class of ratio type estimators of the population mean and ratio in a finite population sample surveys with without replacement simple random sampling design, where information on an auxiliary variate x positively correlated with the main variate y is available. Large sample approximations to mean square errors (MSE) of these estimatorsare evaluated and their MSE's are compared with the MSE of the usual ratio estimator [ybar]_R of [ybar] the population mean of y. It is shown that under certain conditions these estimators are more efficient than [ybar]_R. When a prior knowledge of the value of thecoefficient of variation, c_y, of y is at hand, ratio type estimator, say [ybar]₁ of [ybar] is proposed. It is shown, under certain conditions, that [ybar]₁ is more efficient than [ybar]_R. When values of c_y, c_x and the population correlation coefficient ρ is at hand, then we have proposed another estimator, say [ybar]₂ of [ybar], which is always better than [ybar]_R as far as the efficiency is concerned. In fact, is [ybar] ₂ is shown to be even better than [ybar]₁. Finally estimators better than the usual ratio estimator [ybar]/[xbar] of [Ybar] are given.     

A note on unbiasedness in ratio estimation

The method of ratio estimation for estimating the population mean ? of a characteristic y when we have auxillary information on a characteristic x highly correlated with y, consists in getting an estimator of the population ratio R = ?/X? and then multiplying this estimator by the known population mean X?. Though efficient, ratio estimators are in general biased and in this article we review some of the unbiased ratio estimators and discuss a method of constructing them. Next we present the Jackknife technique for reducing bias and show how the generalized Jackknife could be interpreted by the same method.     

Sequential experimental design and response optimisation

We consider the situation where one wants to maximise a functionf(θ,x) with respect tox, with θ unknown and estimated from observationsy _k . This may correspond to the case of a regression model, where one observesy _k =f(θ,x _k )+ε _k , with ε _k some random error, or to the Bernoulli case wherey _k ∈{0, 1}, with Pr[y _k =1|θ,x _k |=f(θ,x _k ). Special attention is given to sequences given by , with an estimated value of θ obtained from (x₁, y₁),...,(x _k ,y _k ) andd _k (x) a penalty for poor estimation. Approximately optimal rules are suggested in the linear regression case with a finite horizon, where one wants to maximize ∑ _i=1 ^N w _i f(θ, x _i ) with {w _i } a weighting sequence. Various examples are presented, with a comparison with a Polya urn design and an up-and-down method for a binary response problem.     

Approximate optimal allocation in repeated sampling from a finite population

Samples of size n are drawn from a finite population on each of two occasions. On the first occasion a variate x is measured, and on the second a variate y. In estimating the population mean of y, the variance of the best linear unbiased combination of means for matched and unmatched samples is itself minimized, with respect to the sampling design on the second occasion, by a certain degree of matching. This optimal allocation depends on the population correlation coefficient, which previous authors have assumed known. We estimate the correlation from an initial matched sample, then an approximately optimal allocation is completed and an estimator formed which, under a bivariate normal superpopulation model, has model expected mean square error equal, apart from an error of order n^-2, to the minimum enjoyed by any linear, unbiased estimator.     

A study of ratio and product estimators under a super population model

In this paper ratio and product estimators are studied under a super population model considered by Durbin (1959. Biometrika) where a regression model of y (the characteristic variablel on x(the auxiliary variable) is assumed. The comparison of the ratio and the product estimators have been made in the literature (see Chaubey, Dwivedi and Singh (1984), Commun. Statist. - Theor. Meth.) When the auxiliary variable has a gamma distribution. In this paper similar analysis has been carried out when the auxiliary variable has an inverse Gaussian distribution.     

A characterization of invariant power-series abundance distributions

Consider a population the individuals in which can be classified into groups. Let y, the number of individuals in a group, be distributed according to a probability function f(y;ø_o) where the functional form f is known. The random variable y cannot be observed directly, and hence a random sample of groups cannot be obtained. Consider a random sample of N individuals from the population. Suppose the N individuals are distributed into S groups with x₁, x₂, …, x_S representatives respectively. The random variable x, the number of individuals in a group in the sample, will be a fraction of its population counterpart y, and the distributions of x and y need not have the same functional form. If the two random variables x and y have the same functional form for their distributions, then the particular common distribution is called an invariant abundance distribution. The paper provides a characterization of invariant abundance distributions in the class of power-series distributions.     

Partly linear models on Riemannian manifolds

In partly linear models, the dependence of the response y on (x ^T, t) is modeled through the relationship y=x ^T β+g(t)+?, where ? is independent of (x ^T, t). We are interested in developing an estimation procedure that allows us to combine the flexibility of the partly linear models, studied by several authors, but including some variables that belong to a non-Euclidean space. The motivating application of this paper deals with the explanation of the atmospheric SO₂ pollution incidents using these models when some of the predictive variables belong in a cylinder. In this paper, the estimators of β and g are constructed when the explanatory variables t take values on a Riemannian manifold and the asymptotic properties of the proposed estimators are obtained under suitable conditions. We illustrate the use of this estimation approach using an environmental data set and we explore the performance of the estimators through a simulation study.     

Integral Identities for Random Variables

Using a general method for deriving identities for random variables, we find a number of new results involving characteristic functions and generating functions. The method is simply to promote a parameter in an integral relation to the status of a random variable and then take expected values of both sides of the equation. Results include formulas for calculating the characteristic functions for x ², √x, 1/x, x ² + x, R ² = x ² + y ², and so forth in terms of integral transforms of the characteristic functions for x and (x, y), and so forth. Generalizations to higher dimensions can be obtained using the same method. Expressions for inverse/fractional moments, E{n!}, and so forth are also presented, demonstrating the method.     

The Characterization of the Symmetric Lorenz Curves by Doubly Truncated Mean Function

We characterize symmetric Lorenz curves by the relation m(x, μ²/x) = μ (where μ =E(X) and m(x, y) = E(X | x ≤ X ≤ y) is the doubly truncated mean function). We establish that the points of the r.v. which generate the symmetric points on the Lorenz curve are x and μ²/x, and that all the distribution functions defined on the same support which are generators of the symmetric Lorenz curves have the same mean. We obtain the conditions under which doubly truncated distributions generate symmetrical Lorenz curves.     

A Bivariate Distribution Family with Specified Correlation Coefficient and Given Marginals

For given continuous distribution functions F(x) and G(y) and a Pearson correlation coefficient ρ, an algorithm is provided to construct a sequence of continuous bivariate distributions with marginals equal to F(x) and G(y) and the corresponding correlation coefficient converges to ρ. The algorithm can be easily implemented using S-Plus or R. Applications are given to generate bivariate random variables with marginals including Gamma, Beta, Weibull, and uniform distributions.     

The Uniform Asymptotics of the Overshoot of a Random Walk with Light-Tailed Increments

Let {S _n : n ≥ 0} be a random walk with light-tailed increments and negative drift, and let τ(x) be the first time when the random walk crosses a given level x ≥ 0. Tang (2007 Tang , Q. ( 2007 ). The overshoot of a random walk with negative drift . Statist. Probab. Lett. 77 : 158 – 165 .[Crossref], [Web of Science ®] , [Google Scholar]) obtained the asymptotics of P(S _τ(x) ? x > y, τ(x) < ∞) as x → ∞, which is uniform for y ≥ f(x) for any positive function f(x) → ∞ as x → ∞. In this article, the uniform asymptotics of P(S _τ(x) ? x > y, τ(x) < ∞) as x → ∞, for 0 ≤ y ≤ N for any positive number N will be given. Using the above two results, the uniform asymptotics of P(S _τ(x) ? x > y, τ(x) < ∞) as x → ∞, for y ≥ 0, is presented.     

Modeling with Mixtures of Linear Regressions

Consider data (x ₁,y ₁),...,(x _n,y _n), where each x _i may be vector valued, and the distribution of y _i given x _i is a mixture of linear regressions. This provides a generalization of mixture models which do not include covariates in the mixture formulation. This mixture of linear regressions formulation has appeared in the computer science literature under the name Hierarchical Mixtures of Experts model.This model has been considered from both frequentist and Bayesian viewpoints. We focus on the Bayesian formulation. Previously, estimation of the mixture of linear regression model has been done through straightforward Gibbs sampling with latent variables. This paper contributes to this field in three major areas. First, we provide a theoretical underpinning to the Bayesian implementation by demonstrating consistency of the posterior distribution. This demonstration is done by extending results in Barron, Schervish and Wasserman (Annals of Statistics 27: 536–561, 1999) on bracketing entropy to the regression setting. Second, we demonstrate through examples that straightforward Gibbs sampling may fail to effectively explore the posterior distribution and provide alternative algorithms that are more accurate. Third, we demonstrate the usefulness of the mixture of linear regressions framework in Bayesian robust regression. The methods described in the paper are applied to two examples.     

Testing equality of variances after a preliminary test of significance for correlation coefficient

On the basis of the outcome of a preliminary test of significance for the population correlation coefficient it is decided as to whether the variance ratio or the sample correlation coefficient between u=(x+y)/2 and v=x?y)/2 is to be used as a test statistic for testing the equality of variances. A method for determining the critical points for the preliminary test and the main test has been suggested. The power of the test procedure is compared with those of standard tests.     

Bounds on Bivariate Distribution Functions with Given Margins and Known Values at Several Points

Let H(x, y) be a continuous bivariate distribution function with known marginal distribution functions F(x) and G(y). Suppose the values of H are given at several points, H(x _i , y _i ) = θ _i , i = 1, 2,…, n. We first discuss conditions for the existence of a distribution satisfying these conditions, and present a procedure for checking if such a distribution exists. We then consider finding lower and upper bounds for such distributions. These bounds may be used to establish bounds on the values of Spearman's ρ and Kendall's τ. For n = 2, we present necessary and sufficient conditions for existence of such a distribution function and derive best-possible upper and lower bounds for H(x, y). As shown by a counter-example, these bounds need not be proper distribution functions, and we find conditions for these bounds to be (proper) distribution functions. We also present some results for the general case, where the values of H(x, y) are known at more than two points. In view of the simplification in notation, our results are presented in terms of copulas, but they may easily be expressed in terms of distribution functions.     

VARIANCE ESTIMATION IN TWO-PHASE SAMPLING

Two‐phase sampling is often used for estimating a population total or mean when the cost per unit of collecting auxiliary variables, x, is much smaller than the cost per unit of measuring a characteristic of interest, y. In the first phase, a large sample s₁ is drawn according to a specific sampling design p(s₁) , and auxiliary data x are observed for the units i∈s₁ . Given the first‐phase sample s₁ , a second‐phase sample s₂ is selected from s₁ according to a specified sampling design {p(s₂∣s₁) } , and (y, x) is observed for the units i∈s₂ . In some cases, the population totals of some components of x may also be known. Two‐phase sampling is used for stratification at the second phase or both phases and for regression estimation. Horvitz–Thompson‐type variance estimators are used for variance estimation. However, the Horvitz–Thompson ( Horvitz & Thompson, J. Amer. Statist. Assoc. 1952 ) variance estimator in uni‐phase sampling is known to be highly unstable and may take negative values when the units are selected with unequal probabilities. On the other hand, the Sen–Yates–Grundy variance estimator is relatively stable and non‐negative for several unequal probability sampling designs with fixed sample sizes. In this paper, we extend the Sen–Yates–Grundy ( Sen , J. Ind. Soc. Agric. Statist. 1953; Yates & Grundy , J. Roy. Statist. Soc. Ser. B 1953) variance estimator to two‐phase sampling, assuming fixed first‐phase sample size and fixed second‐phase sample size given the first‐phase sample. We apply the new variance estimators to two‐phase sampling designs with stratification at the second phase or both phases. We also develop Sen–Yates–Grundy‐type variance estimators of the two‐phase regression estimators that make use of the first‐phase auxiliary data and known population totals of some of the auxiliary variables.     

Cross-correlation between simultaneously generated sequences of pseudo-random uniform deviates

Systematic patterns are revealed when sequences of pseudo-random uniform deviates are generated from multiplicative congruential generators. If the initial seeds, x ₀ and y ₀, for two such sequences are related by y ₀ = (n ₁/n ₂)x ₀, where n ₁ and n ₂ are relatively prime, positive integers, then an approximate argument suggests that the asymptotic correlation coefficient between corresponding members of the two sequences is (n ₁ n ₂)^–1. This unsettling phenomenon is discussed in the context of related, existing literature.     

Empirical Likelihood in a Semi-Parametric Model for Missing Response Data

Let Y be a response and, given covariate X,Y has a conditional density f(y | x, θ), where θ is a unknown p-dimensional vector of parameters and the marginal distribution of X is unknown. When responses are missing at random, with auxiliary information and imputation, we define an adjusted empirical log-likelihood ratio for the mean of Y and obtain its asymptotic distribution. A simulation study is conducted to compare the adjusted empirical log-likelihood and the normal approximation method in terms of coverage accuracies.     

A Note on Fitting a Regression without an Intercept Term

By entering the data (y _i ,x _i ) followed by (–y _i ,–x _i ), one can obtain an intercept-free regression Y = Xβ + ε from a program package that normally uses an intercept term. There is no bias in the resultant regression coefficients, but a minor postanalysis adjustment is needed to the residual variance and standard errors.     

HITTING LINES and CIRCLES WITH DIFFUSION PROCESSES

For (x(t),y(t)), a diffusion process starting from (x(0),y(0)) = (x,y), the problem of computing the moment generating function of the first passage time T(x, y) to a given subset D of IR²is considered. A particular case of the method of similarity solutions is used. The problems that can be solved explicitly are those for which D is either a straight line or a circle.     

Improved estimator of population total in PPS sampling

In this paper, we have considered an estimation of the population total Y of the study variable y, making use of information on an auxiliary variable x. A class of estimators for the population total Y using transformation on both the variables study as well as auxiliary has been suggested based on the probability proportional to size with replacement (PPSWR). In addition to many the usual PPS estimator, Reddy and Rao's (1977) estimator and Srivenkataramana and Tracy's (1979, 1984, 1986) estimators are shown to be members of the proposed class of estimators. The variance of the proposed class of estimators has been obtained. In particular, the properties of 75 estimators based on different known population parameters of the study as well as auxiliary variables have been derived from the proposed class of estimators. In support of the present study, numerical illustrations are given.