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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 will construct an alternative Hartley-Ross unbiased ratio estimator for the population mean of a study variable when related auxiliary information is available. We will also present some efficiency comparisons and related results and give a numerical illustration.

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.

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’.

Summary This is a brief history of the evolution of official and academic Statistics in India which focuses mainly on the period 1930 to 1960 but traces its origins in antiquity and recent history. We also comment on how Statistics has continued to evolve since the 1960's. This is a history of both institutions and people, who built and shaped them, and of ideas. Resume Ceci est une coute histoire de l'evolution de la Statistique officielle et acadeemique en Inde. Nous retracons l'hisoire a patir des origines de l'antiquite jusqu'a l'histoire plus recente mais potons plus d'atention sur la peiode des annees 1930 a 1960. Ceci et lhistoire d'institutions, des gens qui les batirent et les formerent eet d'idees.

A large body of evidence demonstrates that antenatal glucocorticoids can accelerate fetal lung maturation. The purpose of this study was to delineate the optimal dose of dexamethasone and to determine whether a single- or multiple-injection regimen of the same total dexamethasone dosage was more effective in accelerating pulmonary development. Pregnant rhesus monkeys were injected with varying doses of dexamethasone or vehicle as either a single-bolus injection or as four separate injections, each spaced 12 hours apart. All maternal treatments were begun exactly 72 hours prior to delivery, and all fetuses were delivered by cesarean section at 135 +/- 1 days gestation. When a single injection of dexamethasone was used, fetal liver weight increased in a dose-related fashion. Fetal and maternal cortisol and fetal blood glucose concentrations were also influenced by increasing dexamethasone dosages. Fetal pulmonary phospholipids, however, were unchanged at all steroid doses examined. Multiple injections of dexamethasone generally produced more pronounced effects, even though the total dexamethasone dose remained the same. Thus, liver weight and fetal and maternal cortisol, glucose, and insulin levels were significantly influenced by the multiple administration of dexamethasone. In addition, total lung phosphatidylcholine, surfactant phosphatidylcholine, the surfactant-phosphatidylcholine-to-total-phosphatidylcholine ratio, and the surfactant-disaturated-phosphatidylcholine-to-total-lung-disaturated- phosphatidylcholine ratio were elevated after the multiple-injection regimen. A total dose of 0.5 mg dexamethasone/kg maternal body weight given in four separate injections appeared to produce the most beneficial results. These data suggest that low-dose, antenatal glucocorticoid treatment can effectively accelerate the biochemical maturation of the fetal lung.

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.

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.

An attempt was made to study the usefulness of activated carbon fabric (ACF) mask to prevent lead absorption. Indigenous ACF masks were provided to eight workers involved in the manufacture of batteries. Their blood lead levels were determined before and after using these masks. There was a substantial decrease in blood lead level after using the mask among those who were under treatment for high blood lead levels. Three workers who were not under treatment for lead also showed a decrease in blood lead level, after using this ACF mask, suggesting the usefulness of ACF in preventing further exposure. Such a study has not been reported anywhere in the available literature and similar such studies with more number of masks is necessary to evaluate its effectiveness.

The effect of a pumpless arteriovenous bypass with a microporous membrane oxygenator (MO) on the cardiopulmonary dynamics of dogs revealed an increase in cardiac output and cardiac work proportional to the increase in extracorporeal blood flow. Several current MOs offer so little resistance to blood flow that shunt flows exceeding a third of the normal cardiac output can easily be obtained. This should be enough for CO2 extraction covering metabolic production, and would add a non negligible amount of O2 to the blood.

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).

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 paper, we shall study an alternative method of deriving Neyman's Optimum allocation of sample size to strata for simple random sampling which is intuitive and simple. In order to obtain this allocation, the within stratum standard deviations of the study variate are required which are unknown and hence are substituted by known quantities. We shall discuss the effects of deviations from the optimum allocation and other related problems.

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.

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 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 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.

Cassel (1977) discussed the correspondence between the Horvitz-Thompson strategy and the stratefied random sampling strategy for a special case of the general superpopulation model. We consider a more general model and discuss the correspondence between these strategies in a more formal way.

In this paper we shall consider the problem of judging how good an actual compromise allocation is compared to Neyman's optimum allocation for the case of stratified random sampling by obtaining a bound to the ratio of the corresponding variances. A solution to a more general version of the problem considered by COCHRAN (1977) and RAO (1983) has been obtained and the results have been empirically demonstrated.

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.

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.

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.

Summary In this review paper, the problem of optimum allocation of sample size to strata is examined in the light of a priori distributions. We first consider Neyman's optimum allocation in the case of the Simple Random Sampling (SRS) scheme and review the results concerning the justification for the assumption that the unknown proportionate values of within stratum variances ca's can be replaced by the known proportionate values of aVc's which are the corresponding within stratum variances based on the auxiliary information. We next review the problem of optimum stratification with proportional and optimum allocations in the case of SRS in the light of a priori distributions and discuss the role of arranging the auxiliary variate in increasing order for stratification. We then turn our attention to varying probability sampling schemes and present a review of the results based on a study discussing whether a stratified ,rPS (PPS) strategy with various non-optimal allocations is likely to be worth while in practice.