A NOTE ON OPTIMUM STRATIFICATION IN SAMPLING WITH VARYING PROBABILITIES1

Singh and Sukhatme [4] have considered the problem of optimum stratification on an auxiliary variable x when the units from the different strata are selected with probability proportional to the value of the auxiliary variable and the sample sizes for the different strata are determined by using Neyman allocation method. The present paper considers the same problem for the proportional and equal allocation methods. The rules for finding approximately optimum strata boundaries for these two allocation methods have been given. An investigation into the relative efficiency of these allocation methods with respect to the Neyman allocation has also been made. The performance of equal allocation is found to be better than that of proportional allocation and practically equivalent to the Neyman allocation.     

SAMPLING ON SEVERAL SUCCESSIVE OCCASIONS WITH EQUAL AND UNEQUAL PROBABILITIES AND WITHOUT REPLACEMENT1

The applicability of the technique of controlled selection with equal probabilities proposed by the authors, Avadhani and Sukhatme (1965, 1968), and that of unequal probability sampling suggested by Rao et al. (1962) to sampling on several successive occasions is investigated in this paper. Appropriate recurrence formulae for the optimum replacement fraction and the corresponding weight are also derived.     

A REGRESSION- TYPE ESTIMATOR IN SAMPLING WITH PPS AND WITH REPLACEMENT

A regression-type estimator is proposed for the population total of a character y , when sample-units are selected with probability proportional to some measure of size z (pps) and with replacement; and the information on yet another variate x is used. The proposed estimator is shown to be superior to the well-known estimator (pps with replacement), and the ratio estimator (pps with replacement). It is also shown to be superior to the simple random estimator under some general conditions.     

A MODEL OF SYSTEMATIC SAMPLING WITH UNEQUAL PROBABILITIES

A new method is described of drawing, without replacement, two sample units per stratum from any population. The method is developed from a consideration of the asymptotic properties of systematic sampling with unequal probabilities, as the sizes of the population units tend to zero. The essential properties of this method are very easily analysed. They also converge, over a large number of strata, to those of systematic sampling from the same strata with their population units arranged in random order. In proving this, the assumption is made that the underlying population is of the type to which it is appropriate to apply ratio estimation. The sampling method described is, however, simple enough to commend itself as an alternative to systematic sampling when the underlying population is not of this type. Consideration is given to the case where the sizes of some of the population units exceed the skip interval.     

VARIANCE ESTIMATION WITH HIGHLY STRATIFIED SAMPLING DESIGNS WITH UNEQUAL PROBABILITIES

It is common practice to design a survey with a large number of strata. However, in this case the usual techniques for variance estimation can be inaccurate. This paper proposes a variance estimator for estimators of totals. The method proposed can be implemented with standard statistical packages without any specific programming, as it involves simple techniques of estimation, such as regression fitting.     

ON ESTIMATING THE MEAN OF A FINITE POPULATION SAMPLED ON TWO OCCASIONS WITH VARYING PROBABILITIES

The problem considered is one of estimating the current value of a finite population mean from two samples selected with varying probabilities on the current and a previous occasion. Following the generalized least squares approach of Gurney and Daly (1965) we extend the works of Sin& (1968) and others to obtain two sampling strategies which are respectively more efficient than the ones suggested earlier by Raj (1965) and Chotai (1974).     

BAYESIAN ESTIMATION OF RUIN PROBABILITIES WITH A HETEROGENEOUS AND HEAVY-TAILED INSURANCE CLAIM-SIZE DISTRIBUTION

This paper describes a Bayesian approach to make inference for risk reserve processes with an unknown claim‐size distribution. A flexible model based on mixtures of Erlang distributions is proposed to approximate the special features frequently observed in insurance claim sizes, such as long tails and heterogeneity. A Bayesian density estimation approach for the claim sizes is implemented using reversible jump Markov chain Monte Carlo methods. An advantage of the considered mixture model is that it belongs to the class of phase‐type distributions, and thus explicit evaluations of the ruin probabilities are possible. Furthermore, from a statistical point of view, the parametric structure of the mixtures of the Erlang distribution offers some advantages compared with the whole over‐parametrized family of phase‐type distributions. Given the observed claim arrivals and claim sizes, we show how to estimate the ruin probabilities, as a function of the initial capital, and predictive intervals that give a measure of the uncertainty in the estimations.     

TESTING THE CONCORDANCE OF EVENTS WITH THEIR PROBABILITIES

A test due to D. R. Cox (1958a, 1958b) for the agreement between an observed set of independent events and independently prescribed probabilities is considered. Although the exact distribution can be found for small numbers of events this is often not possible. A method of modifying the scoring system slightly is shown to result in much simpler calculation with very little loss in power. The exact distribution is desirable because of slow convergence to normality. The logic of using a simulation test is considered.     

SAMPLING AND ESTIMATION IN THE PRESENCE OF CUT-OFF SAMPLING

Cut-off sampling consists of deliberately excluding a set of units from possible selection in a sample, for example if the contribution of the excluded units to the total is small or if the inclusion of these units in the sample involves high costs. If the characteristics of interest of the excluded units differ from those of the rest of the population, the use of naïve estimators may result in highly biased estimates. In this paper, we discuss the use of auxiliary information to reduce the bias by means of calibration and balanced sampling. We show that the use of the available auxiliary information related to both the variable of interest and the probability of being excluded enables us to reduce the potential bias. A short numerical study supports our findings.     

EXACT AND APPROXIMATED RELATIONS BETWEEN NEGATIVE HYPERGEOMETRIC AND NEGATIVE BINOMIAL PROBABILITIES

We derive orthogonal expansions in terms of the Meixner polynomials of the first kind for hypergeometric probabilities. We show how these expansions can be used to obtain negative binomial approximations to negative hypergeometric probabilities. Some limit properties of these approximations are studied and also the extension of these results to cumulative probabilities.     

A GALTON-WATSON BRANCHING PROCESS IN VARYING ENVIRONMENTS WITH ESSENTIALLY CONSTANT OFFSPRING MEANS AND TWO RATES OF GROWTH1

A Galton-Watson process in varying environments (Z_n), with essentially constant offspring means, i.e. E(Z_n)/mⁿ→α∈(0, ∞), and exactly two rates of growth is constructed. The underlying sample space Ω can be decomposed into parts A and B such that (Z_n)_n grows like 2ⁿon A and like mⁿon B (m > 4).     

ESTIMATING BIOMASS IN A VEGETATION MOSAIC USING DOUBLE SAMPLING WITH REGRESSION

In some vegetation types, total fuel loading (phytomass) can be predicted by easily-measured variables such as vegetation type and height. A double sampling scheme is proposed in which fuel loading is estimated on a particular site by using quadrat sampling within patches of similar vegetation to develop a general prediction equation, and then line intercept sampling is used to estimate the mean of the easily-measured variables on the site. This method is applied to estimate the total fine fuel loading on a heathland site.     

A NOTE ON THE APPLICATION OF THE KALMAN FILTER TO REGRESSION MODELS WITH SOME PARAMETERS VARYING OVER TIME AND OTHERS UNVARYING

The Kalman filter has been applied to estimation of the time-varying vector of regression parameters. I investigate the case where a portion of elements of the vector is invariant over time while others are varying as generated by the nonstationary, random walk model. Combined with the regression model it yields a state-space model in which observability holds but controllability does not. Under Grenan-der's condition on the exogenous variables I shall show that the estimate of the time-invariant portion is consistent, despite the seemingly unfavorable circumstances mentioned above, with the order equal to the reciprocal of sample size.