Showing papers in "Calcutta Statistical Association Bulletin in 1976"

On Construction of Neighbor Designs

Abstract: Recently a type of block designs called ‘Neighbor designs’ has been developed for serological experiments. It is described as an arrangement of v objects (antigens) on b circular plates (blocks) such that (i) every plate has k objects, not necessarily all distinct, (ii) each object appears r times (not necessarily on r different plates) and (iii) each object is a neighbor of every other exactly λ times. Complete block (i.e, k = v) neighbour design for any odd number of antigens (v) and incomplete block (i.e., k 1 are available in the literature. In this paper, we have suggested a systematic method of constructing complete block neighbor design for any even v. Moreover, we have presented four new methods of construction of incomplete block neighbor designs and have been able to obtain some series of these designs with minimum λ-value through one of the proposed methods.

Asymptotically Optimal Rank Order Tests For Progressive Censoring

Abstract: For progressive censoring schemes, Chatterjee and Sen (1973) have considered a general class of time-sequential tests based on suitable rank order statistics and have studied their asymptotic relative efficiency results in the light of the Bahadur-efficiency ,criterion. The object of the present note is to study the power properties of these tests for local (contiguous) alternatives, and, within this framework, to throw some light on their asymptotic optimality. Unlike the case of the Bahadur-efficiency, for our purpose, we confine ourselves to local altsrntatives and also need an additional regularity condition which localizes (to some extent) the scope of the comparisons. Along with the preliminary notions, the rank order tests are briefly presented in Section 2. In Section 3, a general class of contiguous location and scale alternatives is introduced and the asymptotic power properties of the rank order tests for such alternatives are studied. Asymptotic optimality results are then considered in Section 4.

On Balanced Sampling Schemes

Abstract: Balanced sampling schemes have been introduced by Roy and Chakravarti (1960) in the search for best regular estimators of the finite population means. Restricting to the clas-; of linear unbiased estimators of a finite population mean, they have introduced the regular estimato1·s as those having variances proportional to the population variance (of the study variable). The best regular estimatm· is the one whose variance function attains the lowe1· bonnd to the constant of propm·tionality. In this context, the idea of introducing balanced sampling schemes has been to produce automatically best regular estimators underlying such schemes.

Records and Oscillating Random Walk

Abstract: The distribution of the place of occurrence of the m-th record and of the maximum have been obtained in the case of an oRcillating random walk. Similar distributions have already been obtained for a simple random walk (both symmetric and unsymmetric) by F eller (1968) .

On Construction of a Series of Two-Way Designs:

Abstract: Designs for two-way elimination of heterogeneity are of fundamental importance. Any tlu·ee factor additive design is defined to be a design for two-way elimination of heterogeneity. Methods of construction of these designs have been considered by Shrikhande [1951], Youden [1937], Potthoff [1962], Shrivastava and Anderson [1971], Agrawal [1966(a),(b),(c),(d),(e)], Agrawal and Sharma [1975]. In this paper methods of construction of two-way designs based upon the series of BIB design v = 4t+ 1, b = 2v, 1' = 4t, k = 2t, A = 2t-1 are given, analyses of the designs are included as well. The notation followed are that of AgTawal [1966(a)]. In section 2 some theorems based on the BIB design v = 4t+1 (prime power), b = 2v, 1' = -4t, k = 2t, A= 2t-1 are proved. In section 3 various methods of construction of two-way designs along with their analyses based on these theorems are given. The basic advantage of these methods is that the designs so obtv,ined are either balanced or partially balanced and require lesser number of observations.

Balanced Optimal Plans for Factorial Experiments

Abstract: Raktoe and Federer ( 1973) obtained balanced optimal saturated main effect plans of 2m factorial system and characterised the plans in terms of the matrix of the selected treatment combinations. These balanced optimal plans were also called as Hadamard Plans by them. So far balanced optimal saturated main effect plans are not available in the existing literature for higher class factorial system of the type sm. The present paper is an attempt in this direction to obtain balanced optimal saturated main effect plans in an sm factorial system, where sis of the form 2m, m and n being positive integers. Raktoe and Federer plans referred to above appear as a particular case of the work in this paper by ta,king the value of n as 1. Balanced optimal Resolution V plans can be constructed in symmetrical factorial system of the above type i.e. sm, where s = 2n More generally, existence of balanced optimal Resolution L (odd) plans can be easily :Shown.

Admissibility and Minimaxity of Mvue When the Parameter Space is Restricted to Integers

Abstract: Ghosh (1975) in a recent paper studied the admissibility and minimaxity of maximum likelihood estimators (m.l.e's) in certain examples where the pa-rameter space is restricted to integers. Tn particular, in connection with his Example 1 when Xv X 2, • •• , X 11 are independent and indentically distributed random variables (i.i .d.r.v.s') each with truncated exponential probability density function (p.d.f.) p (x) = exp(O-x), x ;;;., 0 .. . (1.1) = 0, otherwise and 8 is restricted to belong to the set of integers 8 = {0, ±1 , ±2, .. . }, he observed that the m.l.e. of 8, b = [T,], the integer pa.rt. ofT,= min (X1 , ... ,X,), is ina.dmissible as an estimate of() wrt squared error loss, being dominated by the minimum variance estimator D-exp( -n)/{1-exp( -n)} = 9* (say) of e. However, since with probability 1, b* is not an integer, he conjectured {}* to be inadmissible and rema.rked that he had failed to prove so. In this note we show that this conjecture is false i.e., {}* is in fact admissible. The admissibility proof is given in section 2 along with a remark which proves the minimaxity as well.

On Selection Procedures Based on The Exceedance Probability

Abstract: A number of normal populations are to be compared in terms of exceedance probability. When comparing two populations, a population is to be designated as better than the other if it has a greater exceedance probability. A Bayes procedure is given for selecting the subset of populations which contains the best population. A Bayes sequential procedure for selection of the best population is also described.

An Inventory Model with Supply Breaks1

Abstract: Purchase-inventory models with price breaks are mathematically discussed by Churchman, Ackoff and Arnoff (I957). The same problem is also considered by Hadley and Whitin (I963). Johnson (I97I) described an easy graphical approach to price break analysis. The concept of Johnson is further developed by Wilkinson and Green (I972). However, all of them agreed in one basic assumption that the quantity received to meet the future demand is precisely the same as the quantity ordered, with purchased amount subject to quantity discounts. In practice, situations may sometimes arise where the amount received deviates from the amount ordered. However, in these circumstances, the whole-saler may ensure the supply of a fixed proportion of the ordered amount, the vf'Jue of the proportion being again dependent on the amount ordered-larger amount ordered will get higher proportion. Deficit amount will also be supplied depending on chance, i.e. amount received besides the fixed proportion of the original order is a random one. Thus, here supply is pgptly deterministic and partly probabilistic. For analytical simplicity the delivery lag is treated here to be virtually zero. While it is possible to develop this type of the inventory problem for each of the models considered in Chapter 8 of Churchman (I957); we will exhibit this problem only with respect to the so-called static, or single-period, model in which a single inventory decision is made in anticipation of future demand, which is viewed as a random variable.