A new parametric (3-parameter) survival distribution, the lognormal-power function distribution, with flexible behaviour is introduced. Its hazard rate function can be either unimodal, monotonically decreasing or can exhibit a bathtub shape. Special cases include the lognormal distribution and the power function distribution, with finite support. Regions of parameter space where the various for...

For β = 1 the Weibull distribution coincides with the exponential distribution with mean α. In general, α represents the .632-quantile of the Weibull distribution regardless of the value of β since Fα,β(α) = 1 − exp(−1) ≈ .632 for all β > 0. Figure 1 shows a representative collection of Weibull densities. Note that the spread of the Weibull distributions around α gets smaller as β increases. Th...

Abstract We establish the optimal order of Malliavin-type remainders in asymptotic density approximation formula for Beurling generalized integers. Given $\alpha \in (0,1]$ and $c>0$ (with $c\leq 1$ if =1$ ), a number system is constructed with Riemann prime counting function $ \Pi (x)= \operatorname {\mathrm {Li}}(x)+ O(x\exp (-c \log ^{\alpha } x ) +\log _{2}x), whose integer satisfies ext...

This function was first studied by Ramanujan [6]. He wrote, for every prime p, z(p) = 2p I */2 COS Op and conjectured that 0v is real. This was proved by Deligne [2]. it is known that r(p~) =p11~/2 sin(a+ 1) 0p sin 0p If d(n) denotes the number of divisors of n, then it follows that It(n)[ _-< n 11/2 d(n), as r is a multiplicative function. Therefore, for some constant cl >0, { (CI 1ON F/.~ ~ "...

We unify two approaches that have been taken to explain the non-Gaussian probability distribution functions (PDFs) obtained in measurements of longitudinal velocity differences in turbulence, and we apply our approach to Couette-Taylor turbulence data. The first approach we consider was developed by Castaing and co-workers, who obtained the non-Gaussian velocity difference PDF from a superposit...

Supplementary material for estimation of a parametric function associated with the lognormal distribution Jiangtao Gou and Ajit C. Tamhane Department of Statistics, Northwestern University, Evanston, IL 60208, USA Department of Mathematics and Statistics, Hunter College, New York, NY 10065, USA Department of Industrial Engineering and Management Sciences, Northwestern University, Evanston, IL 6...

This paper examines the hypothesis that short run changes in tne U.S. size distribution of personal income have been systematically related to aggregate levels of output~ personal income~ and employment during the posh7ar period. Part one proposes a procedure for describing the size distribution of income. The observed distribution is approximated by a displaced lognormal distribution vmich cor...