Calcutta Statistical Association Bulletin

Weak convergence of randomly stopped log-likelihood ratio for general dependent processes has been obtained. We observe that under certain regularity conditions sequence of randomly stopped log-likelihood random fields converge weakly to a mixed Gaussian Shift process. Some applications are pointed out and two examples are also taken into account.

Let {x_n, n ⩾ 1} be a sequence of positive valued stable andon variables with exponent a ,   0   <   a   <   1.   Set S n = ∑ i = 1 n X i   and   T n   =   S n   +   a n   −   S n ,   n   ⩾   1 where a_n is non-decreasing sequence of positive integers. Write ξ n   =   { ( β n − 1   S n ) ,     ( γ n − 1   T n ) }   where   β n   =   θ a n 1 / α   ( log   log   n ) α − 1 α ,   γ n =   θ α a n 1 / α   ( log   n a n   +   log   log   n ) α − 1 α .   θ α   =   ( B ( a ) ) − ( α − 1 ) α a n d   B ( a )   =   ( 1   −   a ) a   α 1 − α   ( c o s ( π a 2 ) ) 1 α − 1   In this paper we obtain the almost sure limit set of the sequence (ξ _n ) under certain conditions on a_n.

The purpose of this article is to extend the concept of general efficiency balance from block designs to row-column designs. Some methods of construction of general efficiency balanced row-column designs are given. The canonical efficiencies of these designs are worked out.

For spherical experimental regions designs are considered which are D- and A-optimal among the rotatable designs for all the parameters and also for two subsets of the parameters in a second order model. Performance of these optimal designs under model variation is examined. The best central composite designs fot the above two criteria and for the three models are also presented.

Recently the analysis and the derivation of non-singularity conditions of fourth order rotatable designs in K-dimensions had been dealt by some authors. But this analysis is not easily comprehensible. In this paper the above is given in detail including the variance function of the fourth order rotatable design.

The paper studies an inventory model for items with exponential life, when demand for the item is made one at a time. The time between any two demands is assumed to be a continuous random variable with some general density function f(x). Assuming instantaneous replenishment on ordering, the model has been solved to obtain the optimal ordering quantity which minimises the total expected cost per unit time.

In this note, a lower bound of large deviation probabilities is derived; this extends the main results of Bahadur, Gupta and Zabell (1980). An application to the Sonov large deviation problem is considered.

For nonzero constants {a_n, n ⩾1} and i.i.d. random variables {Y, Y_n n⩾1} with EY² = ∞ and y → ∞ lim   y 2 P { |   Y   |   >   y } /   G ( y )   =   0   w h e r e   G ( y )   =   ∫ o y 2 t P   { |   Y   |   >   t }   d t ,   y ⩾ 0 Adler and Rosalsky (1989) proved a generalzed central limit theorem ( ∑ j = 1 m a j   Y j   −   A n )   /   B n → d   N   ( 0 ,   1 )   w h e r e   { A n , n ⩾ 1 } a n d { B n ,   n ⩾ 1 } are suitable numerical sequences. In addition, Adler and Rosalsky(l989) showed that the centering constants {A_n, n⩾1} can be taken to be A n =   ∑ j = 1 n a j   E y ,   n ⩾ 1 if the condition y → ∞ lim   y 2   P {   |   Y   |   >   y } /   G ( y )   =   0 is replaced by the stronger condition that p{ | Y | > y} is regularly varying with exponent - 2. In this note, it is shown that the generalized central limit theorem does indeed hold with A n   =   ∑ j = 1 n a j   E Y ,   n ⩾ 1 without any strengthening of the condition y → ∞ lim   y 2   P {   |   Y   |   >   y } /   G ( y )   = 0 .

Mukhopadhyay (1987) considered sequential procedure for the point estimation of the parameter θ in a U(0, θ) distribution under quadratic Joss. Given a random sample X₁, ..., X_n of size n, he used T_n = max (X₁ , ... ,X_n) to estimate θ In the present note, a sequential procedure based of UMVUE of θ is proposed. The asymptotic properties of the proposed procedure are established and its advantages over Mukhopadhyay's procedure are discussed.

A simple method of constsuction for Group divisible (GD) designs is given with the help of known bananced incomplete block (BIB) designs.

The idea of analsis of Balanced Incomplete Block (BIB) design of Puri (1984) nas been extended to row-coiumn designs eliminating two way heterogeneity and analysis of Youden Square with a missing column has been provided. It is shown that such designs are PEB row-column designs with three efficiency classes. The sets of basic contrasts and their corresponding losses have been provided.

This paper considers an inventory model where the policy is to place an order with a supplier whenever a demand is made or an item in stock deteriorates. Deterioration occurs randomly at a state dependent rate. Then the optimal stock to be maintained has been worked out by minimising the total expected cost involving holding cost, waiting cost of an order, shortage cost, and deterioration cost under state dependent markovian process for arrival of orders and arrival of supplies.