As an alternative series of second order response surface designs, Das and Dey (1967) introduced Group-Divisible Second Order Rotatable Designs (GDSORD) by modifying the restrictions on the levels of the factors in a second order rotatable design. In these designs the v-dimensional space corresponding to v-factors is divided into two mutually orthogonal spaces, one of p-dimensional and the other of (v-p) dimension. Given any point i.e., a treatment combination, in the v-dimensional space, we can visualize its projection. Let the distances of the projection of the points in each of the subspaces from a suitable origin be 2 2 1 2 and d d respectively.

Recent years have been an upsurge of interest in the role of statistical ideas and methods in improving the quality and productivity of industrial processes and products. Response surface methodology is, one such idea, useful for analysing problems where several independent variables influence a dependent variable or response. The earlier study of response surface designs mainly emphasized the estimation of absolute response. Estimation of differences in response at different points in the factor space will often be of great importance. If differences at points closed together are involved, estimation of local slope i.e., the rate of change of the response surface is of interest. This problem, estimation of slopes, occurs frequently in practical situations, particularly in fitting second order response surface. This enables the determination of the best operating conditions for the process i.e., the best combination of the levels of the controllable factors which give the "optimum" value of the second order response function. This would also enables us to determine the best way to control the process. In this paper, we made an attempt to study the role of second order slope rotatability to improve the quality of a product. Anjaneyulu et al. (1993) introduced embedding in Second-Order-Slope Rotatable Designs and constructed the same using embedding techniques similar to those of Draper (1960). Herzberg (1967), Park (1987) studied the necessary and sufficient conditions for second order slope rotatability overall directions, Anjaneyulu et al. (1997) showed that these designs have the variance sum property. In this paper an attempt is also made to explain the role of embedding in SOSRDOAD for Quality Improvement.

Hader and Park (1978) introduced second order slope rotatability in axial directions. Park (1987) introduced second order slope rotatability over all directions. It is shown that these designs have the additional property that the sum of the variances of estimates of slopes in all directions at any point is a function of the distance of the point from the design origin.

The problem of determining the asymptotically optimal group limits of a sample, for maximum likelihood estimation of the parameters in a 2-parameter Weibull distribution, is studied in this paper. The asymptotically optimal group limits (optimal interval length) in unequi-class (equi-class) grouped sample are (is) computed with various number of groups. The asymptotic relative efficiencies of the maximum likelihood estimates of the parameters are also computed for the optimally grouped sample.

The expressions for moments of order statistics from the generalized gamma distribution are derived. Coefficients to get the BLUEs of location and scale parameters in the generalized gamma distribution are computed. Some simple alternative linear unbiased estimates of location and scale parameters are also proposed and their relative efficiencies compared to the BLUEs are studied.

SUMMARY. In this paper we evaluate the moments of order statistics in the reflected gamma distribution and discuss the linear estimation of its location and scale parameters from complete as well as censored samples. The best linear unbiased estimators (BLUEs) are obtained and some competitive simplified linear unbiased estimators are proposed. The necessary numeri cal Tables of coefficients, variances and efficiencies are given.

In this paper a new method of construction of second order slope rotatable designs (SOSRDs) through balanced incomplete block designs is suggested. In this method the number of design points required is in some cases less than the number required in Hader and Park slope rotatable central composite design (SRCCD).

The first two moments and product moments of absolute values of order statistics are obtained for the double exponential and the double Weibull distributions. In both of the distributions an optimum linear unbiased estimator of the scale parameter, by absolute values of the order statistics, is obtained from complete and censored samples of size n=3(1)10. It is found that the new estimator is generally more efficient than the best linear unbiased estimator (BLUE) of the scale parameter by order statistcs in both of the distributions.

A new estimator of the scale parameter by the optimum linear combination of absolute values of order statistics in symmetric location-scale families with known location parameter (without loss of generality assumed to be zero) from complete and Type II censored samples is introduced and is termed as optimum unbiased absolute estimator of the scale parameter. The new estimator of the scale parameter is compared with the corresponding best linear unbiased estimator (BLUE) in the rectangular and normal distributions. Generally it is found that the new estimator is more efficient than the BLUE.

The differential cross-sections of atomic hydrogen for elastic scattering of electrons and positrons have been rederived with the help of a method using a single parameter-dependent unitary shift operator for the calculation of the direct contribution. When the parameter approaches zerc the new method leads to the well-known conventional Glauber results. The numerical calculations include polarization effects and the exchange corrections obtained according to alternative approximation methods. Results calculated with Franco’s exchange show a definite improvement over the earlier results for medium energy electrons at large angles of scattering. Total elastic cross-sections have been calculated for 50 and 100eV electrons and positrons.

The elastic differential cross section of electrons from molecular hydrogen has been calculated in the independent-atom model using the Glauber approximation for both direct and exchange scattering amplitudes. Polarization effects have been considered through the modified Glauber amplitude for the direct scattering. For the exchange amplitude the result obtained by Franco and Halpern has been used. Numerical results for various incident electron energies have been carried out. The polarization contribution has been found to have negative effects in bringing the calculated results close to the experimental values.

The exchange correction to the differential scattering cross section for the electron-hydrogen-molecule scattering is derived. In the independent scattering center and Glauber approximation our expressions do not agree with those used in the published literature. The overall agreement between the calculated and the measured cross sections improves at higher angles and lower incident electron energies, where the exchange contribution is important.

Rotatable designs were introduced by Box and Hunter (1954, 1957) for the exploration of response surfaces. They constructed these designs through geometrical configurations and obtained several second order designs. Afterwards, Gardiner and others (1959) obtained some third order designs through the same technique for two and three factors and a third order design for four factors. Bose and Draper (1959) obtained some second order designs by using a different method. Draper (1960 a) gave a method of construction of an infinite series of second order designs in three and more factors. Recently, Box and Behnken (1960 a) have obtained a class of second order rotatable designs from those of first order. Draper (1960 b) has obtained some third order rotatable designs in three dimensions and a third order rotatable design in four dimensions. Das (1961) has obtained such designs, both second and third orders up to 8 factors as fractional replicates of factorial designs. The method of construction of the designs presented in this paper is essentially based on that presented by Das (1961). After the manuscript of this paper was submitted for publication the authors' attention was drawn to the work of Box and Behnken (1960 b). They have obtained some second order designs by following a procedure which uses balanced incomplete block designs in the same manner as described below. They did not, however, extend the method to include other complementary sets of points which, as will be shown, allow one to obtain rotatable second and third order designs based on any balanced incomplete block design. In the present paper a method has been given by using the properties of balanced incomplete block designs through which second order rotatable designs with any number of factors, with a reasonably small number of points, can be obtained. By extending the method, third order rotatable designs, both sequential and nonsequential, up to 15 factors have been obtained with the help of doubly balanced incomplete block designs and complementary B.I.B. designs.