Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

Zeros of orthogonal polynomials

Special Functions and Their Applications. By N. N. Lebedev, translated by R. A. Silverman. Pp. xii, 308. 96s. 1965. (Prentice-Hall)

(1966). Mathematical Theory of Reliability. Journal of the Operational Research Society: Vol. 17, No. 2, pp. 213-215.

https://link.springer.com/content/pdf/10.1057%2Fjors.1966.43.pdf

Mathematical Theory of Reliability

Independent and stationary sequences of random variables

Let X λ1 , X λ2 , …, X λ n be independent Weibull random variables with X λ i ∼ W(α, λ i ), where λ i > 0 for i = 1, …, n. Let X n:n λ denote the lifetime of the parallel system formed from X λ1 , X λ2 , …, X λ n . We investigate the effect of the changes in the scale parameters (λ1, …, λ n ) on the magnitude of X n:n λ according to reverse hazard rate and likelihood ratio orderings.

/pdf/stochastic-order-relations-among-parallel-systems-from-3zl8s2g5j5.pdf

Stochastic Order Relations Among Parallel Systems from Weibull Distributions

In this work, we discuss stochastic comparisons between the largest order statistics from multiple-outlier models when the numbers of independent and identically distributed random variables are different. That is, if Xn:n(p,q) is the largest order statistic among X1,…,Xp,Xp+1,…,Xn and Xn∗:n∗(p∗,q∗) is the largest order statistic among X1,…,Xp∗,Xp∗+1,…,Xn∗, where q=n−p and q∗=n∗−p∗, we discuss whether Xn:n(p,q) and Xn∗:n∗(p∗,q∗) are ordered in some stochastic sense.

Comparisons between largest order statistics from multiple-outlier models

In this article, we focus on stochastic orders to compare the magnitudes of two parallel systems from Weibull distributions when one set of scale parameters majorizes the other. The new results obtained here extend some of those proved by Dykstra et al. (1997) and Joo and Mi (2010) from exponential to Weibull distributions. Also, we present some results for parallel systems from multiple-outlier Weibull models.

/pdf/stochastic-order-relations-among-parallel-systems-from-4k1nbf5ld9.pdf

Stochastic order relations among parallel systems from Weibull distributions

Let be independent non negative random variables with , i = 1, …, n, where λi > 0, i = 1, …, n and F is an absolutely continuous distribution. It is shown that, under some conditions, one largest order statistic Xλn: n is smaller than another one Xθn: n according to likelihood ratio ordering. Furthermore, we apply these results when F is a generalized gamma distribution which includes Weibull, gamma and exponential random variables as special cases.

/pdf/on-stochastic-comparisons-of-largest-order-statistics-in-the-21x2hhrbog.pdf

On Stochastic Comparisons of Largest Order Statistics in the Scale Model

Weibull distribution is a very flexible family of distributions which has been applied in a vast number of disciplines. In this work, we investigate stochastic properties of the smallest order statistics from two independent heterogeneous Weibull random variables with different scale and shape parameters. Furthermore, we study the hazard rate order of the smallest order statistics from lower-truncated Weibull distributions due to, in general, Weibull random variables are not ordered according to this ordering in the shape parameter.

Nuria Torrado

Papers

Stochastic Order Relations Among Parallel Systems from Weibull Distributions

Comparisons between largest order statistics from multiple-outlier models

Stochastic order relations among parallel systems from Weibull distributions

On Stochastic Comparisons of Largest Order Statistics in the Scale Model

Comparisons of smallest order statistics from Weibull distributions with different scale and shape parameters