LENGTH-BIASING, CHARACTERIZATIONS OF LAWS AND THE MOMENT PROBLEM

A positive random variable X with a finite mean has an induced length-biased law represented by Y, and Y is stochastically larger than X. An independent uniform random contraction of Y, UY, has the same law as X if and only if the latter is exponential. This property is extended to non-uniform contractions and a more general notion of length-biasing. The distributional equality of X and W leads to a functional equation for the moment function of X, which has either Infinitely many solutions or none. When U is constant, X can have a log-normal law, but it can also have laws with the same moment sequence as this log-nod law. The case where U has a certain beta, or generalized beta, law give t3 characterizations of generalized gamma laws, or to products of independent copies of them. This occurs even when these laws are not determined by their moment sequences.     

On Comparison Of Estimates Of Dispersion Using Generalized Pitman Nearness Criterion

In this paper, estimates QP dispersion matrix and its functions are compared based on generalized Pitman nearness criterion, Various Iosa functions are considered for the purpose. Locally superior estimates are defined and obtained. Comparison of these estimates are made with other standard ones. It is snown that within certain classes, defined in the paper, these are the best estimatcrs ia the generalized Fitman nearness sense     

Comparison of estimators of the location parameter using pitman's closeness criterion

Necessary and sufficient conditions for a linear estimator to dominate another linear estimator of a location parameter under the Pitman's criterion of comparison are discussed. Consequently it is demonstrated that a linear biased estimator can not dominate a linear unbiased estimator under Pitman's criterion and that the sample mean is the Closest Linear Unbiased Estimator (CLUE). It is also shown that the ridge regression estimator with a known biasing constant can not dominate the ordinary least squares estimator. If an estimator δdominates an estimator δin the average loss sense then sufficient conditions are obtained under which δis also preferred over δunder Pitman's criterion. Further we obtain sufficient conditions under which preference under the Pitman's criterion will lead to preference under the mean squared error sense.     

An alternative estimator for multiple characteristics in PPS sampling

The problem considered relates to large-scale sample surveys. A new estimator of population total for the characteristics that are poorly correlated with the selection probabilities has been developed for the PPSWR sampling scheme. The relative efficiency of the proposed estimator has been studied under a super-population model. A numerical investigation into the performance of the estimator has also been made.     

Estimation of signal to noise ratio using mahalanobis distance

In this paper, we propose some alternative estimatiors to that given by C. G. Khatri and C. R. Rao (1985), for estimating Signal to Noise ratio. Using Pitman Nearness, Condition for prefering one estimator over the other is estabilished. It is shown numerically that estimators corresponding to Entropy loss function are better more oftern than those corresponding to Squared Error loss.     

A NOTE ON OPTIMUM STRATIFICATION FOR EQUAL ALLOCATION WITH RATIO AND REGRESSION METHODS OF ESTIMATION

When the information on a highly positively correlated auxiliary variable x is used to construct stratified regression (or ratio) estimates of the population mean of the study variable y, the paper considers the problem of determining approximately optimum strata boundaries (AOSB) on x when the sample size in each stratum is equal. The form of the conditional variance function V(y/x) is assumed to be known. A numerical investigation into the relative efficiency of equal allocation with respect to the Neyman and proportional allocations has also been made. The relative efficiency of equal allocation with respect to Neyman allocation is found to be nearly equal to one.     

ESTIMATION OF GUARANTEE TIME AND MEAN LIFE AFTER WARRANTY FOR TWO-PARAMETER EXPONENTIAL FAILURE MODEL

Suppose manufactured items have failure times distributed with the two-parameter exponential density function à-^le-(x-μ)/à (x≥μ≥0). The paper considers estimation of the guarantee time μ and mean life after warranty à. Properties of various estimators are given for both separate and simultaneous estimation of μ and à. Criteria for comparing estimators are entropy loss, LINEX loss, and (generalized) Pitman nearness.     

The Parshvanath yantram yields a constant-sum partially balanced incomplete block design

Yantrams have been used to generate mixture designs in the interior of a simplex. In this note, we show a connection between Parshvanath yantram and a particular partially balanced incomplete block design. This block design is rather special and somewhat unexpected due to the feature that sum of the treatment symbols in any block is constant.     

Inequality restricted linear selection indices

The problem of construction of selection indices with arbitrary linear inequality constraints on covariances is considered. The selection indices suggested by Smith (1936), Kempthorne Nordskog (1959) Tall is (1962) are shown as special cases. A method of solution is presented and prediction errors of various indices are compared. The method is applied to the example given in Kempthorne Nordskog (1959).     

THE INFINITELY DEEP DAM WITH SEASONAL MARKOV INFLOWS—A SIMPLIGYING MODIFICATION

This is an extension of the work reported in Phatarfod (1981). Here we make the further assumption that the inputs to a bottomless dam are independent from year to year, although they are dependent within a year. It is shown that this assumption drastically simplifies, computationally and analytically, the limiting distribution of the depletions.