Showing papers in "Calcutta Statistical Association Bulletin in 1969"

Some Results in Markov Renewal Processes

Abstract: The following results are obtained in this paper: (1) The probability generating function of the simultaneous distribution of all the Nj(t)'s, where Nj(t) represents the number of times the j­th state (j = 1, 2, ... , m) is visited in time t, in a Markov Renewal Process ; (2) the covariance between Nj(t) and Nk(t); (3) the probability generating function and moments of Nj(t)'s in a General Markov Renewal Process i.e., a Markov Renewal Process with a random origin; (4) Cumulative processes associated with a Markov Renewal Process along with its first passage time and (5) Equilibrium Markov Renewal Processes.

On a Robustness Property of a Class of Nonparametric Tests Based on U­Statistics*

Abstract: A CLASS of nonparametric tests based on Hoeffding's (1948) U-statisticsis shown to be robust for heterogeneity of the distributions of the sample observations ; the theory is illustrated by means of some known tests. Let X10 • •• , Xn be independent (real or vector valued) random variables having cumulative distribution functions (cdf's) F1(x), ... Fn(x) respectively. Let n be the set of all cdf's closed under the following (countable) set operation: if FidJ. for i = 1, 2, ... , then (1/n)Lt~tFtt:!2 for all n;>l. It is assumed that Fn = (F1, ••. , Fn) is an element of the product set Qn, and the null hypothesis states that

Some Further Results on Hodges­Lehmann Conditional Rank Tests

Abstract: IN 1962, Hodges-Lehmann proposed certain conditional rank tests for comparing two treatments in a two-way layout. These results were extended to the case of k > 2 treatments and for complete experimental designs by Mehra and Sarangi (1967). The asymptotic efficiency of these tests relative to competing test procedures using translation alternatives was also studied in this paper, for experimental designs with identical complete blocks. The present paper extends the results of Mehra and Sarangi (1967) to cover equi-replicated designs of a constant block size including the balanced and partially balanced incomplete block designs. In Section 3, large sample distribution of the test statistic has been studied under the null hypothesis, to extend similar results in Mehra and Sarangi to almost all incomplete designs. In Section 4, the asymptotic unconditional distribution of the test statistic has been found under a sequence of translation alternatives approaching the hypothesis, using equi-replicated designs of a constant block size. Section 5 contains explicit expressions for the asymptotic efficiency of the present test relative to the classical :J-test and the corresponding separate ranking test procedures, based on independent rankings within each block. The last section gives us a useful lower bound for the asymptotic efficiency of the present conditional test relative to the classical :J-test, showing that these procedures are fairly robust.