Conditional Poisson Sampling Research Articles

Maximum Entropy Design by a Markov Chain Process

Abstract In this article, we study an implementation of maximum entropy (ME) design utilizing a Markov chain. This design, which is also called the conditional Poisson sampling design, is difficult to implement. We first present a new method for calculating the weights associated with conditional Poisson sampling. Then, we study a very simple method of random exchanges of units, which allows switching from one sample to another. This exchange system defines an irreducible and aperiodic Markov chain whose ME design is the stationary distribution. The design can be implemented without enumerating all possible samples. By repeating the exchange process a large number of times, it is possible to select a sample that respects the design. The process is simple to implement, and its convergence rate has been investigated theoretically and by simulation, which led to promising results.

Ranked set sampling in finite populations with bivariate responses: An application to an osteoporosis study.

The majority of the research on rank-based sampling designs in finite populations has been concerned with univariate situations. In this article, we study design-based estimation using a bivariate ranked set sampling (BIRSS) for finite populations when we have bivariate response variables. We derive the first and second-order inclusion probabilities associated with a BIRSS design. We show that the size of a BIRSS sample is random and propose using a conditional Poisson sampling (CPS) design to rectify this problem. We then use calculated inclusion probabilities to obtain design-based estimators of correlation coefficients between the bone mineral density (BMD) levels at the baseline and followup of a longitudinal BMD study in the province of Manitoba in Canada. We also study the problem of estimating the parameters of a regression model between the followup BMD and easy to obtain auxiliary information from the underlying population. Finally, we study the problem of classifying patients as those with or without osteoporosis using BIRSS and various CPS designs. We show that BIRSS designs are very flexible and can be used to obtain more efficient design-based estimators in sample surveys when dealing with response variables that are hard to measure or expensive to obtain.

Limiting properties of an equiprobable sampling scheme for 0–1 matrices

A sampling scheme that selects a random 0–1 matrix of size N×M uniformly in the set of 0–1 matrices with predetermined row and column totals is investigated. The limits, as M goes to ∞ and N is fixed, of the column relative frequencies is derived. The limiting values give a sampling design for a population of N units that generalizes the conditional Poisson sampling design introduced by Hajek. A method to calculate the joint selection probabilities for this new design using the known single inclusion probabilities is presented. Numerical examples show that the limiting theorem provides good approximations to the fixed M column probabilities of a random matrix.

Bernstein-type exponential inequalities in survey sampling: Conditional Poisson sampling schemes

This paper is devoted to establishing exponential bounds for the probabilities of deviation of a sample sum from its expectation, when the variables involved in the summation are obtained by sampling in a finite population according to a rejective scheme, generalizing simple random sampling without replacement, and by using an appropriate normalization. In contrast to Poisson sampling, classical deviation inequalities in the i.i.d. setting do not straightforwardly apply to sample sums related to rejective schemes, due to the inherent dependence structure of the sampled points. We show here how to overcome this difficulty, by combining the formulation of rejective sampling as Poisson sampling conditioned upon the sample size with the Esscher transformation. In particular, the Bennett/Bernstein type bounds thus established highlight the effect of the asymptotic variance of the (properly standardized) sample weighted sum and are shown to be much more accurate than those based on the negative association property shared by the terms involved in the summation. Beyond its interest in itself, such a result for rejective sampling is crucial, insofar as it permit to obtain tail bounds for many other sampling schemes, namely those that can be accurately approximated by rejective plans in the sense of the total variation distance.

Coordination of Conditional Poisson Samples

Abstract Sample coordination seeks to maximize or to minimize the overlap of two or more samples. The former is known as positive coordination, and the latter as negative coordination. Positive coordination is mainly used for estimation purposes and to reduce data collection costs. Negative coordination is mainly performed to diminish the response burden of the sampled units. Poisson sampling design with permanent random numbers provides an optimum coordination degree of two or more samples. The size of a Poisson sample is, however, random. Conditional Poisson (CP) sampling is a modification of the classical Poisson sampling that produces a fixed-size πps sample. We introduce two methods to coordinate Conditional Poisson samples over time or simultaneously. The first one uses permanent random numbers and the list-sequential implementation of CP sampling. The second method uses a CP sample in the first selection and provides an approximate one in the second selection because the prescribed inclusion probabilities are not respected exactly. The methods are evaluated using the size of the expected sample overlap, and are compared with their competitors using Monte Carlo simulation. The new methods provide a good coordination degree of two samples, close to the performance of Poisson sampling with permanent random numbers.

An efficient non-rejective implementation of the πps sampling designs

Poisson sampling is a method for unequal probabilities sampling with random sample size. There exist several implementations of the Poisson sampling design, with fixed sample size, which almost all are rejective methods, that is, the sample is not always accepted. Thus, the existing methods can be time-consuming or even infeasible in some situations. In this paper, a fast and non-rejective method, which is efficient even for large populations, is proposed and studied. The method is a new design for selecting a sample of fixed size with unequal inclusion probabilities. For the population of large size, the proposed design is very close to the strict πps sampling which is similar to the conditional Poisson (CP) sampling design, but the implementation of the design is much more efficient than the CP sampling. And the inclusion probabilities can be calculated recursively.

Maximum likelihood estimation in log-linear models

We study maximum likelihood estimation in log-linear models under conditional Poisson sampling schemes. We derive necessary and sufficient conditions for existence of the maximum likelihood estimator (MLE) of the model parameters and investigate estimability of the natural and mean-value parameters under a nonexistent MLE. Our conditions focus on the role of sampling zeros in the observed table. We situate our results within the framework of extended exponential families, and we exploit the geometric properties of log-linear models. We propose algorithms for extended maximum likelihood estimation that improve and correct the existing algorithms for log-linear model analysis.

Repeated Poisson sampling

The repeated Poisson sampling design is a new design for selecting a sample of fixed size with unequal inclusion probabilities. The design is very close to the conditional Poisson sampling design, but the implementation of the RP-design is much more efficient.

A comparison of conditional Poisson sampling versus unequal probability sampling with replacement

The variance of the Horvitz–Thompson estimator for a fixed size Conditional Poisson sampling scheme without replacement and with unequal inclusion probabilities is compared to the variance of the Hansen–Hurwitz estimator for a sampling scheme with replacement. We show, using a theorem by Gabler, that the sampling design without replacement is more efficient than the sampling design with replacement.

A nonsymmetric matrix with integer eigenvalues

A nonsymmetric N × N matrix with elements as certain simple functions of N distinct real or complex numbers r 1, r 2, …, rN is presented. The matrix is special due to its eigenvalues − the consecutive integers 0,1,2, …, N−1. Theorems are given establishing explicit expressions of the right and left eigenvectors and formulas for recursive calculation of the right eigenvectors. A special case of the matrix has appeared in sampling theory where its right eigenvectors, if properly normalized, give the inclusion probabilities of the conditional Poisson sampling design.

A Guide to the Fortran Programs to Calculate Inclusion Probabilities for Conditional Poisson Sampling and Pareto πps Sampling Designs

Conditional Poisson Sampling and Pareto źps Sampling designs are sampling methods with fixed sample size and with inclusion probabilities proportional to given size measures. Algorithms were introduced to calculate first and second exact inclusion probabilities for both schemes. Methods were also provided to adjust the parameters to get predetermined inclusion probabilities. In this paper, the Fortran procedures are introduced and documented. Moreover, guidelines are provided for their use as well as examples and the programs codes commented.

Comparisons between conditional Poisson sampling and Pareto πps sampling designs

Abstract Conditional Poisson sampling (CPS) and Pareto πps sampling (PPS) design belong to a class of sampling schemes which have inclusion probabilities proportional to given size measures (πps). Algorithms were introduced to calculate first- and second-order exact inclusion probabilities for both schemes. In this paper using those algorithms, both methods are compared by calculating Horvitz–Thompson estimators of populations totals and computing their biases and variances. They are also illustrated by examples, and the results in these are compared with known asymptotics.

Algorithms to Find Exact Inclusion Probabilities for Conditional Poisson Sampling and Pareto πps Sampling Designs

Conditional Poisson Sampling Design as developed by Haje´k may be defined as a Poisson sampling conditioned by the requirement that the sample has fixed size. In this paper, an algorithm is implemented to calculate the conditional inclusion probabilities given the inclusion probabilities under Poisson Sampling. A simple algorithm is also given for second order inclusion probabilities in Conditional Poisson Sampling. Furthermore a numerical method is introduced to compute the unconditional inclusion probabilities when the conditional inclusion probabilities are predetermined. Simultaneously, we study the Pareto πps sampling design. This method, introduced by Rose´n, belongs to a class of sampling schemes called Order Sampling with Fixed Distribution Shape. Methods are provided to compute the first and second order inclusion probabilities numerically also in this case, as well as two procedures to adjust the parameters to get predetermined inclusion probabilities.

Probability Inequalities for Sums of Independent Random Variables

Probability Inequalities for Sums of Independent Random Variables