On interpenetrating samples of unequal sizes
TL;DR: In this paper, the optimum choice of individual sample sizes has also been discussed for a given (i) total sample size, (ii) cost and (iii) precision, with an assumed cost structure.
Abstract: Koop [1967] proved that interpenetrating samples of unequal sizes are more efficient than those with equal sizes for estimating a finite population total. After observing that there is a serious lacuna present in his proof, a correct proof has been suggested. The optimum choice of individual sample sizes has also been discussed for a given (i) total sample size, (ii) cost and (iii) precision, with an assumed cost structure. Finally, the resulting estimators have been compared with those based on a single sample.
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TL;DR: In this paper, it was shown that sampling with unequal sized plots generally leads to a more precise estimate of the tree parameter totals than that with plots of equal sizes, assuming that the trees in the forests are randomly distributed.
Abstract: Assuming that the trees in the forests are randomly distributed, it is demonstrated that plot sampling with unequal sized plots generally leads to a more precise estimate of the tree parameter totals than that with plots of equal sizes. The optimUlll choice of individual plot sizes has been discussed for a given (i) total plot size and (ii) cost (with an assumed cost structure). The problem of variance estimation when the plots are of unequal sizes has also been addressed .
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...Si mil?. r st udif's in a different context were made , as for example, by Srikantan (1963) , Koop (1967) and Sengupta (1980, 1982) ....
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TL;DR: In this article, the authors present a sampling procedure with unequal probabilities and without replacement for a given population of units with characteristics Yt (t = 1,2,..., N) whose total Y = Yi +Y2 +... +YN is estimated.
Abstract: GIVEN is a finite population of N units with characteristics Yt (t = 1,2, ..., N) whose total Y = Yi +Y2 + . .. +YN is to be estimated. If a sample of size n is to be drawn from such a population, it is often advantageous to select the units with unequal probability. For example, such a procedure may be useful when measures of sizes xt are known for all N units in the population which are positively correlated with the characteristics yt. In such cases, one may utilize the knowledge of the xt by selecting units with probabilities proportional to sizes xt, although this is, of course, not the only way of using the known xt. Of the literature on sampling with unequal probabilities and without replacement we mention papers by Horvitz and Thompson (1952), Narain (1951), Yates and Grundy (1953), Des Raj (1956) and Hartley and Rao (1962). There are some limitations, of varying importance, attached to all these methods. Briefly speaking, the method of Horvitz and Thompson (1952) is applicable only under severe restrictions on the prescribed probabilities, the unbiased procedures of Narain (1951), Yates and Grundy (1953) and Des Raj (1956) require a cumbersome evaluation of working probabilities, and Hartley and Rao (1962) give only asymptotic variance formulae for the estimates of Y for large and moderate size populations N. The present method is an attempt to avoid all these disadvantages at the expense of a slight loss in efficiency. It has the following properties: (i) It permits the computation of an estimator of the population total which has always a smaller variance than the standard estimator in sampling with unequal probabilities and with replacement. (ii) Unlike the unbiased procedures of Narain (1951), Yates and Grundy (1953) and Des Raj (1956), the present method does not entail heavy computations, even for sample size n > 2, for drawing the sample or computation of the estimator and its variance estimate.
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TL;DR: In this article, the authors proposed a sampling scheme for any value of n, i.e. for all values of the 'auxiliary character y' of interest, having the property of yielding an unbiased variance-estimator for that stratum.
Abstract: 1.1. IN order to estimate a .finite population total with the help of HorvitzThompson Estimator (HTE) many fixed sample size sampling schemes have been proposed in the literature employing the values of first order inclusion-probabilities '1r(,\\)'s, proportional to the values of some 'auxiliary character x' (called size-measure). The motivations under such sampling . schemes (called 1rps-sampling schemes henceforth) are wellknown (see e.g. Hanurav (1967)). Some of these schemes are applicable for sample size, n, equal to two only (e.g. Desraj (1956), Brewer (1963), Durbin (1967)) while others may be applied for any value of n (e.g. Madow (1949), Narain (1951), Midzuno (1952), Grundy (1954), Hanurav (1962), Felligi (1963), Carroll and Hartley (1964), Samford (1967), Vijayan (1968), Mukhopadhyay (1972)). Although applicability of a sampling scheme for n = 2 only is not a serious drawback in practice, because in actual situations it is generally convenient to break down large strata into several small strata which is more homogeneous than the original population and a sample of size two from each of these stratum may be quite representative of the population, having the property of yielding an unbiased variance-estimator for that stratum, yet the applicability of a sampling scheme for any value of n is a quite desirable property of the scheme at least for general flexibility. Also yielding non-negative values of Yates-Grundy (1952) estimator of variance of HTE ( vyc (HTE)) always, i.e. for all values of the 'original character y' of interest is another desirable property of a sampling scheme.
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TL;DR: The interpenetrating sub-sampling procedure with unequal sizes of the samples has been compared with an equicost procedure based on equal sized samples and it has been observed that unequal sized samples lead to more precise estimates of a finite population mean in almost all the cases dealt with in this paper.
Abstract: The interpenetrating sub-sampling procedure with unequal sizes of the samples has been compared with an equicost procedure based on equal sized samples and it has been observed that unequal sized samples lead to more precise estimates of a finite population mean in almost all the cases dealt with in this paper. Simple random sampling has b:en considered throughout and it has been assumed that the cost of the survey is proportional to the number of distinct units in the sample.
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