The Pitman Inequality For Exchangeable Random Vectors

In this short article the following inequality called the “Pitman inequality” is proved for the exchangeable random vector (X1,X2,…,Xn) without the assumption of continuity and symmetry for each component Xi: P(|1n∑i=1nXi|≤|∑i=1nαiXi|)≥12 , where all αi≥0 are special weights with ∑i=1nαi=1

Sequential boundaries approach in clinical trials with unequal allocation ratios

BACKGROUND: In clinical trials, both unequal randomization design and sequential analyses have ethical and economic advantages. In the single-stage-design (SSD), however, if the sample size is not adjusted based on unequal randomization, the power of the trial will decrease, whereas with sequential analysis the power will always remain constant. Our aim was to compare sequential boundaries approach with the SSD when the allocation ratio (R) was not equal. METHODS: We evaluated the influence of R, the ratio of the patients in experimental group to the standard group, on the statistical properties of two-sided tests, including the two-sided single triangular test (TT), double triangular test (DTT) and SSD by multiple simulations. The average sample size numbers (ASNs) and power (1-β) were evaluated for all tests. RESULTS: Our simulation study showed that choosing R = 2 instead of R = 1 increases the sample size of SSD by 12% and the ASN of the TT and DTT by the same proportion. Moreover, when R = 2, compared to the adjusted SSD, using the TT or DTT allows to retrieve the well known reductions of ASN observed when R = 1, compared to SSD. In addition, when R = 2, compared to SSD, using the TT and DTT allows to obtain smaller reductions of ASN than when R = 1, but maintains the power of the test to its planned value. CONCLUSION: This study indicates that when the allocation ratio is not equal among the treatment groups, sequential analysis could indeed serve as a compromise between ethicists, economists and statisticians

A General Norm on Extension of a Hilbert’s Type Linear Operator

The main purpose of this paper is to study a general norm on extension of a Hilbert’s type linear operator in the continuous and discrete form. In addition to expressing the norm of a Hilbert’s type linear operator T : L 2 (0,∞) → L 2 (0,∞), a more general case with λ > 0, for the continuous form has been studied. By putting λ = 1 a norm of extension of Hilbert’s integral linear operator is obtained. Similar results have been expressed for series when 0 < λ 6

Some Properties of Entropy for the Exponentiated Pareto Distribution (EPD) Based on Order Statistics

In this paper, we derived the exact form of the entropy for Exponentiated Pareto Distribution (EPD). Some properties of the entropy and mutual information are presented for order statistics of EPD. Also, the bounds are computed for the entropies of the sample minimum and maximum for EP

Skorohod's theorem and convergence of types

This note presents a new proof for the convergence of types theorem by using Skorohod's theorem. This proof is simpler than the usual proofs and the method may be useful for some related results.Convergence in distribution Covariable of a random variable Skorohod's theorem Convergence of types

A Note on Sub-Independence and S Class of Bivariate Mixtures

First, we recall a concept which is called sub-independence. This concept is stronger than that of uncorrelatedness but a lot weaker than independence. The concept of sub-independence, unlike that of uncorrelatedness, does not depend on the existence of any moments. Then, we consider a particular bivariate mixture to construct a pair (X, Y), which is sub-independent but not independent