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Prof. C. R. Rao’s Birth Centenary is being celebrated by statistical communities all over the world from 10 September 2019. In this paper, we shall present a brief biographical sketch of the living legend C. R. Rao and discuss some of his contributions in the areas of theory and practice of sample surveys which have resulted in certain important aspects.

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.

Certain discrepancies in data sources for estimating the household consumption expenditure to derive Gross Domestic Product(GDP) of India are discussed in this paper. Simple implementable strategies are suggested to improve the estimation of GDP.

In this article we shall briefly describe the main functions of the National Statistical Commission, the apex statistical body of the country, which was established in 2006 following the recommendations of the Rangarajan Commission. We shall then discuss the important role of the Commission in the Official Statistical System of India.

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.

In this paper, we shall discuss the problem of optimum allocation of sample size to strata under the constraints of fixed total sample size and fixed total cost in a general setting. We shall also obtain near-optimum allocations for use in practice. This will result in minor loss of efficiency due to deviations from the optimum. We shall also provide illustrative examples.

In this paper we study some order relations between the selection and the inclusion probabilities for PPSWOR Sampling Scheme. We also establish some interesting bounds on the inclusion probabilities in terms of the selection probabilities.

We first consider Neyman's optimum allocation of sample size to strata in the light of available auxiliary information for which a suitable random permutation model is assumed. For a special case of this model the allocation of the sample size reduces to the same as when a certain superpopulation regression model is assumed. Motivated by this, more generally, we discuss some optimality results under random permutation models and compare them with the corresponding results when a superpopulation regression model is assumed.

In this note we observe that Brewer's (1979) result concerning the asymptotically design unbiased strategy which has minimum expected variance under a super population model can be established in a more general setting.

The problem of estimating the variance of the ratio estimator for theMidzuno-Sen sampling scheme is further studied in this paper. Sufficient conditions are derived for which the suggested variance estimator is always positive definite.

The problem of optimum allocation of sample size to strata is examined in the light ofa priori distributions. In this context, we discuss with illustrations the justification for the assumption that the unknown proportionate values of σ 2 ’s can be replaced by the proportionate values of the known σ 2 ’s, which are estimates of σ 2 ’s. The sample size is allocated so as to minimize the expected variance of the strategy consisting of πPS sampling scheme and the Horvitz-Thompson estimator under a general super-population model. It is further shown that, in the sense of expected variance, πPS sampling for unstratified sampling is inferior to πPS stratified sampling with this type of allocation, unless the superpopulation parameterg attains the value 2, in which case both schemes are equivalent.

In an earlier paper [Rao 1966] an exact expression for the variance of the ratio estimator under theMidzuno-Sen sampling scheme is obtained and here we study some of the interesting properties of the coefficients involved in this expression which depend on the auxiliary information. Use of these coefficients is made of in finding out an exact expression for the Bias and Mean Square Error of the ratio estimator under Simple Random Sampling With-Out Replacement (SRSWOR) scheme.

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.

For the sampling scheme ofMidzuno [3] andSen [4], which provides unbiased ratio estimators an expression for the variance of the estimator does not seem to be available in literature. An expression for the same is derived in this note.