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Rao–Blackwellization, a term credited to C. R. Rao based on his 1945 ‘breakthrough’ paper published in the Bulletin of the Calcutta Mathematical Society, besides providing improved estimators in conventional, adaptive, link-tracing, size-biased sampling theories, found applications in post simulation improvement of Monte Carlo methods, cross validation and non parametric boot strapping, particle filtering, stereology, data compression, Rao-Blackwellized Field Goal percentage estimator (RB-FG %), Rao-Blackwellized Gaussian Smoothing, Rao–Blackwellized Parts-Constellation Tracker, Rao-Blackwellized Tempered Sampling (RTS) and a host of others. In this paper, we shall consider applications related to improving of estimators in finite population sampling theory. Taking cue from Basu (1958) wherein he showed that the ‘order statistic ‘(sample units in ascending order of their labels) is a sufficient statistic, Pathak (Sankhya A 23:409–414, 1961) in the context of sampling from finite populations,first noticed that ‘any estimator which is not a function of the order statistic’, can be uniformly improved by the use of Rao–Blackwellization technique. See also Sinha and Sen (Sankhya [B] 51: 65–83, 1989) who go beyond variance comparisons and generalize to convex loss functions. In this paper, we shall revisit some of the estimators in Probability Proportional to Size sampling With Out Replacement (PPSWOR) schemes and show how Rao–Blackwellization provides improved estimators.

We recall Godambe's theorem on the nonexistence of an umvue in finite population sampling for estimation of a population total and that the existence of a uniformly minimum mse estimator without any further conditions is not possible. Assuming model-unbiasedness with respect to a well known super population model, we give an alternative derivation of Royall's optimum estimator. Finally, the role of design unbiasedness criterion is briefly discussed.

Sugden and Smith [2002. Exact linear unbiased estimation in survey sampling. J. Stat. Plann. Inf. 102, 25–38] and Rao [2002. Discussion of “Exact linear unbiased estimation in survey sampling”. J. Stat. Plann. Inf. 102, 39–40] suggested some useful techniques of deriving a linear unbiased estimator of a finite population total by modifying a given linear estimator. In this paper we suggest various generalizations of their results. In particular, we search for estimators satisfying the calibration property with respect to a related auxiliary variable and obtain some new calibrated unbiased ratio-type estimators for arbitrary sampling designs. We also explore a few properties of one of the estimators suggested in Sugden and Smith [2002. Exact linear unbiased estimation in survey sampling. J. Stat. Plann. Inf. 102, 25–38].

Consider a finite population of units (U1,U2,…,UN). On each unit Ui, variates of interest y and x are defined taking values Yi and Xi, respectively, i=1,2,…,N. In certain surveys, it is of interest to estimate the population ratio R=Y/X (or equivalently, Y / X ), whereY=∑1NYi and X=∑1NXi, based on a sample of size n selected according to a sampling design p(s). Under simple random sampling scheme, the usual choices for the estimation of R are (i) a (single) ratio of sample means given by R 1 = y / x or (ii) the mean of (n) ratios, viz. R n = ∑ 1 n (y i /x i )/n . It is well known that both R 1 and R n are biased for R. Using the extent of biases, we shall first discuss the role of R 1 and R n in the construction of unbiased ratio estimators. When y is considered as the study variate and x is an auxiliary variate related to y, the problem of estimation of the population mean Ȳ or the population total Y is dealt by constructing Y = R X or Y = R X . For the estimation of the population total Y, we shall consider a class of Symmetrized Des Raj (SDR) strategies and look for a choice of a model-optimum estimator when design-unbiasedness is not demanded, among those utilising ‘mean of ratios’ and ‘ratio of means’.

In this paper we have examined certain techiques suggested in the literature for improving the ratio and regression methods of estimation and demonstrated that these techniques are not very profitable.

In the estimators t 3 , t 4 , t 5 of Mukerjee, Rao & Vijayan (1987), b yx and b yz are partial regression coefficients of y on x and z, respectively, based on the smaller sample. With the above interpretation of b yx and b yz in t 3 , t 4 , t 5 , all the calculations in Mukerjee at al.(1987) are correct. In this connection, we also wish to make it explicit that b xz in t 5 is an ordinary and not a partial regression coefficient. The 'corrected' MSEs of t 3 , t 4 , t 5 , as given in Ahmed (1998 Section 3) are computed assuming that our b yx and b yz are ordinary and not partial regression coefficients. Indeed, we had no intention of giving estimators using the corresponding ordinary regression coefficients which would lead to estimators inferior to those given by Kiregyera (1984). We accept responsibility for any notational confusion created by us and express regret to readers who have been confused by our notation. Finally, in consideration of the above, it may be noted that Tripathi & Ahmed's (1995) estimator t 0 , quoted also in Ahmed (1998), is no better than t 5 of Mukerjee at al.(1987).

Bansal and Singh (1985) have developed an alternative estimator for the probability proportional to size with replacement sampling scheme when certain characteristics under study are poorly correlated with the selection probabilities. We suggest, in this paper, a more intuitive and meaningful estimator and study its efficiency compared with other related estimators.

Summary In this paper we consider a practical situation where information on two auxiliary variables related to the study variable is available at different levels. Following Kiregyera (1980, 1984) who has obtained a chain ratio-to-regression estimator and regression to regression estimator, we shall study several estimators that arise naturally in this context and compare them under the mean square error criterion. We extend these results to the case when multiple auxiliary information is available.

The method of ratio estimation for estimating the population mean Ȳ 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 = Ȳ/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.

Summary In this paper it is shown that the generalized πPS sampling strategy consisting of the design with πi, the probability of inclusion of the ith unit in the sample, proportional to the modified size together with the corresponding Horvitz-Thompson estimator (Rao, 1971), is superior to the symmetrized Des Raj strategy under a general super-population set-up for all values of the super-population parameter g, when the samples are of size two.

Using the criterion of minimum expected variance and the Horvitz-Thompson estimator, we study various πPS (πi, the probability of inclusion of the ith unit, Proportional to Size) strategies and make a comparison between these strategies under a general super population model. This discussion further motivates the search for a class of designs which are best suited for the use of the Horvitz-Thompson estimator. A new class of such designs is obtained.

We consider the two ratio type estimators $$\frac{{\bar y}}{{\bar x}}\bar X$$ and $$\frac{{\bar X}}{m}\Sigma \frac{{y_i }}{{x_i }}$$ for the estimation of $$\bar Y$$ and show how the estimators due to Murthy and Nanjamma, Hartley and Ross, and Nieto de Pascual can be obtained as linear combinations of these two. Some interesting relations among these estimators have been mentioned.