Interval Estimation of Simple Difference under Independent Negative Binomial Sampling

When the sample size is not large or when the underlying disease is rare, to assure collection of an appropriate number of cases and to control the relative error of estimation, one may employ inverse sampling, in which one continues sampling subjects until one obtains exactly the desired number of cases. This paper focuses discussion on interval estimation of the simple difference between two proportions under independent inverse sampling. This paper develops three asymptotic interval estimators on the basis of the maximum likelihood estimator (MLE), the uniformly minimum variance unbiased estimator (UMVUE), and the asymptotic likelihood ratio test (ALRT). To compare the performance of these three estimators, this paper calculates the coverage probability and the expected length of the resulting confidence intervals on the basis of the exact distribution. This paper finds that when the underlying proportions of cases in both two comparison populations are small or moderate (≤0.20), all three asymptotic interval estimators developed here perform reasonably well even for the pre-determined number of cases as small as 5. When the pre-determined number of cases is moderate or large (≥50), all three estimators are essentially equivalent in all the situations considered here. Because application of the two interval estimators derived from the MLE and the UMVUE does not involve any numerical iterative procedure needed in the ALRT, for simplicity we may use these two estimators without losing efficiency.     

Estimation of Difference of Population Means with Missing Observations

Sampling strategies for estimating difference of population means are proposed. The original sample is fragmented into three subsamples. The use of a purposive device permits to measure only one of the variables in two samples. Bootstrap procedures are employed to evaluate distribution of the errors of estimators.     

Estimation of Finite and Closed Population Size: An Inverse Sampling Approach with Unequal Recapture Probability

Based on capture-mark-recapture sampling methods the problem of estimating unknown population size was considered. The sampling started with the assumption that at the beginning of the experiment all the individuals were unmarked, and the unmarked individuals caught in each sample will be marked and returned to the original population before the next sample is drawn. It is also assumed that the population is closed by birth, death, emigration and immigration. Using a general inverse sampling approach, the unknown population size N is estimated by a maximum likelihood estimator (MLE), and a simple form for approximate MLE is obtained. The probability function for S (the minimum number of samples required to be drawn to have L (L ≥ 1) samples, each of which contains at least one marked individual) and the form for E[S] are also obtained. In addition, corrections and improvements of some previous works in this field are given.     

Estimation of Finite Population Variance Using Ratio and Product Methods of Estimation

For estimating finite population variance σ_y² of a character y under our study, estimators using auxiliary information on a character x in the form of ratio, product, ratio-type or product-type estimators have been suggested, and their comparative study with the conventional unbiased estimator s_y² of σ_y² has been made in simple random sampling with replacement. A generalized estimator representing a class of estimators for the finite populations variance, has also been studied.     

Estimation of Population Regression Coefficient in Successive Sampling

An attempt has been made to derive the theory of successive sampling for estimation of regression coefficient. The situations considered are estimation of regression coefficient for current occasion, estimation of change of regression coefficients over two occasions and estimation of average of regression coefficients over two occasions. The expressions of optimum estimators along with their variances have been worked out. On comparing their efficiencies empirically, it has been observed that similar to the estimation of mean, successive sampling can also be used with advantage for estimation of regression coefficient.     

On a restricted occupancy model and its applications

With the multivariate hypergeometric distribution as a background certain occupancy distributions useful in practical applications are derived. More specifically it is assumed that a sample of n individuals is drawn from a population consisting of m types with r individuals in each type, (i) without replacement and (ii) by returning the selected individual in the population and with it another individual of the same type. The distributions of the number Z of distinct types observed in the sample are obtained in both cases in terms of the numbers. Assuming, in addition to the m equiprobable types of individuals, the existence of a control type, say, with s individuals, the joint distribution of the number U of distinct types observed in the sample and the number V of individuals of the control type present in the sample is obtained in terms of the numbers C(n, k, r) and the marginal distribution of U in terms of the Gould-Hopper numbers. Using these distributions minimum variance unbiased estimators of the number m of types are derived. Moreover small sample tests based on the zero frequency are constructed.     

Product Cum Difference Method of Estimation Using two Auxiliary Variables

Product method of estimation (MURTHY, 1964) using supplementary information on an auxiliary variable having high negative correlation with the main variable under our study, is well known. In this paper, we propose product-cum-difference method of estimation for the population total and the product of population parameters when supplementary information is available on two auxiliary variables. A comparison of product-cum-difference method of estimation with the usual product method of estimation using single auxiliary character and the estimators by SINGH (1965) for the estimation of product of population parameters has been made along with an empirical study.     

Some Early Developments in Ratio Estimation

John Graunt (1662) was the first to estimate the ratio y/x where y represents the total population and x the known total number of registered births in the same areas during the preceding year. About 1765 Messance (Stephan, 1948) and Moheau (1778) published very carefully prepared estimates for France based on enumeration of population in certain districts and on the count of births, deaths and marriages as reported for the whole country. The districts from which the ratio of inhabitants to birth was determined only constituted a sample. Laplace (1786) prepared similar estimates in 1802 based on a two-stage sampling plan. Recently Hansen and Hurwitz (1943) showed that the ratio estimate (y_i/n_i)X of Y is unbiased where all x_i's are known and the n^th cluster is selected with p.p.s. More recently Hájek (1949), Lahiri (1951), Midzuno (1952) and Sen (1952) developed independently the sampling of n clusters with p.p.s to the totals of the sizes of the sample clusters S(x_i). Des Raj (1954) and Sen (1952, 1953) gave unbiased estimate of the variance of the estimator which was generally non-negative for samples with smaller probabilities. Rao and Vijayan (1977) gave an unbiased estimator which is non-negative for samples with larger probabilities. Hájek (1949) provided an almost unbiased estimator of the variance of the estimator. The paper discusses situations where Hájek's estimator of variance should be preferred to the Rao-Vijayan estimator and vice versa.     

Comparing Stochastically Restricted Estimators in a Linear Regression Model

The minimum dispersion linear unbiased estimators of the vector of parameters in a linear regression model are compared when the parameters of the model are subject to stochastic linear restrictions with different dispersion matrices of the disturbances involved in them.     

Efficient Estimators for the verianoe of a Finite Population

For the estimation of finite population variance of a character, DAS and TRIPATHI (1978) proposed some efficient estimators when population mean of variance of some other character (associated with main character) is known. We propose here some estimators for population variance using same information and compare with those given by DAS and TRIPATHI (1978). Some examples are given for illustration.     

Point Estimation on Relative Risk under Inverse Sampling

When the underlying disease is rare, to control the coefficient of variation for the sample proportion of cases, we may wish to apply inverse sampling. In this paper, we derive the uniformly minimum variance unbiased estimator (UMVUE) of relative risk and its variance in closed form under inverse sampling. On the basis of a Monte Carlo simulation, we demonstrate that using the UMVUE of relative risk can substantially reduce the mean-squared-error of using the maximum likelihood estimator, especially when the number of index cases in both comparison samples is small. For a given fixed total cost, we include a program that can be used to find the optimal allocation for the number of index cases to minimize the variance of the UMVUE as well.     

On a Simple Procedure of Inclusion Probability Proportional to Sizes (IPPS) Sampling

A simple method of inclusion probability proportional to sizes is proposed for samples of size three units. It is shown that the variance of the HORVITZ -THOMPSON estimator based on the proposed sampling scheme is uniformly smaller than that of the customary estimator used in the probability proportional to sizes with replacement sampling. Further, its performance over RAO -HARTLEY -COCHRAN and SAMPFORD sampling schemes has been studied empirically for some of the natural populations.     

Determination of Number and Size of Spherical Particles from Extraction Replicas

For spherical particles randomly dispersed in the space of a specimen the estimators of the parameters of the space structure from the measurements obtained from extraction replicas are given. First an arbitrary form of the probability density function f(x) of the diameter X and then the generalized RAYLEIGH and lognormal distributions of X are considered. Unbiased estimators of the space parameters and of parameters of these distributions are found. The variances of these estimators are given and unbiased estimators of these variances are determined.     

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).     

Estimation and Verification of Hypotheses in Some Zyskind-Martin Models with Missing Values

Several theorems on estimation and verification of linear hypotheses in some Zyskind-Martin (ZM) models are given. The assumptions are as follows. Let y = Xβ + e or (y, Xβ, σ²V) be a fixed model where y is a vector of n observations, X is a known matrix nXp with rank r(X) = r ≦ p < n, where p is a number of coordinates of the unknown parameter vector β, e is a random vector of errors with covariance matrix σ²V, where σ² is unknown scalar parameter, V is a known non-negative definite matrix such that R(X) ? R(V). Symbol R(A) denotes a vector space generated by columns of matrix A. The expected value of y is Xβ. In this paper four following Zyskind-Martin (ZM) models are considered: ZMd, ZMa, ZMc and ZMqd (definitions in sec. 1) when vector y y₁ y₂ involves a vector y₁ of m missing values and a vector y₂ with (n — m) observed values. A special transformation of ZM model gives again ZM model (cf. theorem 2.1). Ten properties of actual (ZMa) and complete (ZMc) Zyskind-Martin models with missing values (cf. theorem 2.2) test functions F are given in (2.11)) are presented. The third propriety constitutes a generalization of R. A. Fisher's rule from standard model (y, Xβ, σ²I) to ZM model. Estimation of vector y₁ (cf. 3.3) of vector β (cf. th. 3.2) and of scalar σ² (cf. th. 3.4) in actual ZMa model and in diagonal quasi-ZM model (ZMqd) are presented. Relation between y? ₁ and β is given in theorem 3.1. The results of section 2 are illustrated by numerical example in section 4.     

Associated LAH Numbers,Factorial Series Distributions and Unbiased Estimation of a Size Parameter

A finite population consists of kN individuals of N different categories with k individuals each. It is required to estimate the unknown parameter N, the number of different classes in the population. A sequential sampling scheme is considered in which individuals are sampled until a preassigned number of repetitions of already observed categories occur in the sample. Corresponding fixed sample size schemes were considered by Charalambides (1981). The sequential sampling scheme has the advantage of always allowing unbiased estimation of the size parameter N. It is shown that relative to Charalambides' fixed sample size scheme only minor adjustments are required to account for the sequential scheme. In particular, MVU estimators of parametric functions are expressible in terms of the C-numbers introduced by Charalambides.     

Predictive Estimation of Finite Population Mean Using a Product Estimator for Two-Stage Sampling

In an attempt to estimate a finite population mean under the predictive approach described in Basu (1971) through the product method of estimation, we created a new product-type estimator for a two-stage sampling procedure. We also report a simulation study that is made in order to understand better the performance of the new estimator compared to the classical product estimator.     

Interval Estimation of Simple Difference in Dichotomous Data with Repeated Measurements

When comparing two treatments, we often use the simple difference between the probabilities of response to measure the efficacy of one treatment over the other. When the measurement of outcome is unreliable or the cost of obtaining additional subjects is high relative to that of additional measurements from the obtained subjects, we may often consider taking more than one measurement per subject to increase the precision of an interval estimator. This paper focuses discussion on interval estimation of simple difference when we take repeated measurements per subject. This paper develops four asymptotic interval estimators of simple difference for any finite number of measurements per subject. This paper further applies Monte Carlo simulation to evaluate the finite‐sample performance of these estimators in a variety of situations. Finally, this paper includes a discussion on sample size determination on the basis of both the average length and the probability of controlling the length of the resulting interval estimate proposed elsewhere.     

A Note on Estimation of Several Missing Values in Two Way Classified Data with Many Observations Per Cell

In this paper an attempt has been made to reduce the computational complexities involved in estimation of several missing values. As a result it has been shown that one can estimate m missing values by developing only k (≤m) linear equations, where m and k are respectively the number of missing values and missing cells. The procedure is also illustrated with the help of a numerical example.     

On the Use of Jack‐Knife Techniques in Systematic Sampling

We have proposed two general classes of ratio and product type estimators to estimate an unknown population parameter of a response variable y under systematic sampling strategy. Jack‐Knife technique is employed to make the classes almost/exactly unbiased and sampling variance of the proposed estimators are derived to the first order of approximation. The merits of the proposed estimators over other estimators are discussed in this paper.