Preservation properties of the moment generating function ordering of residual lives

The purpose of this note is to study some preservation properties of the moment generating function ordering of residual lives (mg-rl) under the reliability operations of mixture and convolution. Some examples of interest in reliability theory are also presented.     

Preservation properties for the mean residual life ordering

Sereral preservation results for the mean residual life (mr) ordering are given. In particular, we show that the mr-ordering is preserved under convolutions, mixtures and weak convergence.     

A Laplace transform inversion method for probability distribution functions

This paper introduces a new Laplace transform inversion method designed specifically for when the target function is a probability distribution function. In particular, we use fixed point theory and Mann type iterative algorithms to provide a means by which to estimate and sample from the target probability distribution.     

Laplace transform approach to option pricing for time-changed Brownian models

In this article, we consider European option pricing for time-changed Brownian models using Laplace transform. We obtain a general formula for the option price as the integral of a real-valued function involving the Laplace transform of the random time change. Unlike the usual Fourier transform technique, our method does not suffer from difficulties specific to complex integration, such as the evaluation of multiple-valued functions, and allows for a model-independent analysis of the truncation error. In the numerical analysis part, we compare option prices in variance gamma (VG), normal inverse Gaussian (NIG), and generalized hyperbolic (GH) models obtained by Laplace transform with those obtained by the Fourier transform method introduced by Carr and Madan in 1999. The results show that our method converges faster than the Fourier approach when the Laplace transforms of the subordinators decay exponentially, for examples like NIG and GH models.     

Tests for properties of residual life

The lOOα -percentile (0 < α < 1) residual life function at time x is defined to be the lOOα -percentile of the remaining life given survival up to time x. Joe and Proschan (1982b) develop tests for testing the alternatives representing decreasing 100α-percentile residual life (DPRL-α ) and the property ‘new better than used with respect to the lOOα -percentile’ (NBUP-α ). In this paper, tests are developed for DPRL[α, l) and NBUP[α, l) alternatives, where DPRL[α, l) is the class of life distributions which are DPRL-β distributions for all a ≤ β < 1 if 0 ≤ α < 1 and for all 0 < β < 1 if α = 0, and NBUP[α, l) is similarly defined. When α = 0, the DPRL[α, l) class of life distributions corresponds to the increasing failure rate class and the NBUP[α, l) class of life distributions corresponds to the new better than used class, and the test statistics are respectively asymptotically equivalent to the Hollander and Proschan (1975) test statistics for decreasing mean residual life and new better than used in expectation alternatives.     

Rate efficient estimation of realized Laplace transform of volatility with microstructure noise

In this paper, we consider the problem of estimating the Laplace transform of volatility within a fixed time interval [0,T] using high‐frequency sampling, where we assume that the discretized observations of the latent process are contaminated by microstructure noise. We use the pre‐averaging approach to deal with the effect of microstructure noise. Under the high‐frequency scenario, we obtain a consistent estimator whose convergence rate is , which is known as the optimal convergence rate of the estimation of integrated volatility functionals under the presence of microstructure noise. The related central limit theorem is established. The simulation studies justify the finite‐sample performance of the proposed estimator.     

Preservation of generalized ageing classes on the residual life at random time

In this paper, we provide some new preservation properties of generalized ageing classes (s-IFR) on the residual life at random time, where s is a nonnegative integer. We also obtain bounds of the residual life at exponential random time. Results are expected to be useful in the reliability, queue theory and actuarial science.     

Some properties of the generalized TTT transform

Recently Li and Shaked [2007. A general family of univariate stochastic orders. J. Statist. Plann. Inference 137, 3601–3610] introduced the generalized total time on test (GTTT) transform with respect to a given function ?$?$. In this paper we study some properties of it which are related with stochastic orderings. A concept of Lehmann and Rojo [1992. Invariant directional orderings. Ann. Statist. 20, 2100–2110] is applied to a new setting and the GTTT transform is used to define invariance properties and distances of some stochastic orders. Iterations of the GTTT transforms are also studied and their relations with exponential mixtures of gamma distributions are established.     

Inference for the Generalized Normal Laplace Distribution

The generalized normal Laplace distribution has been used in financial modeling because of its skewness and excess kurtosis. To estimate its parameters, we use a method based on minimizing the quadratic distance between the real and imaginary parts of the empirical and theoretical characteristic functions. The quadratic distance estimator (QDE) obtained is shown to be robust, consistent, and with an asymptotically normal distribution. The goodness-of-fit test statistics presented follow an asymptotic chi-square distribution. The performance of the QDE is illustrated by simulation results and an application to financial data.     

Further properties of generalized residual designs

The concept of ‘residuation’ is extended so that all ‘generalized residual designs’ (in the sense of Shrinkhande and Singhi) are in fact ‘residual’ with respect to the extended type of residuation. A measure of departure from the usual type of residuation is given in general, and stronger estimates of this measure are given for affine designs.     

Preservation properties of stochastic orders by transformation to the transmuted-G model

Abstract

The transmuted-G model is a useful technique to construct some new distributions by adding a parameter. This paper considers stochastic comparisons in the transmuted-G family with different parameters and different baseline distributions in the sense of the usual stochastic, shifted stochastic, proportional stochastic and shifted proportional stochastic orders. Also, we present a necessary and sufficient condition for existence of the moments of the transmuted-G model and then we obtain some bounds for the survival and aging intensity functions of the transmuted-G model conditioned on its parameter and its baseline distribution.     

Preservation of some reliability properties by certain record statistics

Record values are closely connected with the occurrence times of a corresponding non-homogeneous Poisson process and in reliability theory, they are used in shock models. In this paper, first, some distribution properties of current records are presented. It is then shown that the hazard (reversed hazard) function of the upper (lower) current records can be expressed in terms of the hazard (reversed hazard) function of the parent distribution. Some reliability relationships between the original variable X and the corresponding current records and with record ranges are established. Also, preservation of some uncertainty measure based on current records is discussed.     

Aging and ordering properties of multivariate lifetimes with Archimedean dependence structures

This paper further studies monotone aging properties of the multivariate random lifetime. We revise the sufficient condition for the negative monotone aging property in terms of the multivariate usual stochastic order in Theorem 3.3 of Rezapour et al. (2013) Rezapour, M., Alamatsaz, M.H., Pellerey, F. (2013). Multivariate aging with Archimedean dependence structures in high dimensions. Commun. Stat. - Theory Methods 42:2056–2070.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar] and derive the condition sufficient to the multivariate monotone aging properties in terms of the upper orthant order. Also we study the upper orthant order of multivariate residual lifetimes and inactivity times from populations sharing a common Archimedean survival copula and Archimedean survival copula, respectively. Two simple applications in multivariate stress-strength and frailty models are presented as well.     

Explosion time for some Laplace transforms of the Wishart process

In this article, we focus upon a family of matrix valued stochastic processes and study the problem of determining the smallest time such that their Laplace transforms become infinite. In particular, we concentrate upon the class of Wishart processes, which have proved to be very useful in different applications by their ability in describing non-trivial dependence. Thanks to this remarkable property we are able to explain the behavior of the explosion times for the Laplace transforms of the Wishart process and its time integral in terms of the relative importance of the involved factors and their correlations.     

A gradient search maximization algorithm for the asymmetric Laplace likelihood

The asymmetric Laplace likelihood naturally arises in the estimation of conditional quantiles of a response variable given covariates. The estimation of its parameters entails unconstrained maximization of a concave and non-differentiable function over the real space. In this note, we describe a maximization algorithm based on the gradient of the log-likelihood that generates a finite sequence of parameter values along which the likelihood increases. The algorithm can be applied to the estimation of mixed-effects quantile regression, Laplace regression with censored data, and other models based on Laplace likelihood. In a simulation study and in a number of real-data applications, the proposed algorithm has shown notable computational speed.     

Some ordering properties of series and parallel systems with dependent component lifetimes

Abstract

In this paper, we consider series systems and parallel systems with the dependence between the component lifetimes modelled by an Archimedean copulas. We obtain sufficient and necessary conditions of relative ageing orders between series (parallel) systems with different component numbers, which partially generalize some main results of Misra and Francis. When the component lifetimes follow the scale model, we also characterize the ordering properties between the series systems and (n–1)-out-of-n systems (parallel systems and 2-out-of-n systems) by mixture distribution.     

Tests of Fit for Exponentiality based on the Empirical Laplace Transform

The Laplace transform \psi(t)=E[{\rm exp}(-tX)] of a random variable X with exponential density u exp( m u x ), x S 0, satisfies the equation (\lambda+t)\psi(t)-\lambda=0 , t S 0. We study the behavior of a class of consistent tests for exponentiality based on a suitably weighted integral of [({\hat\lambda}_n+t)\psi_n(t)-{\hat\lambda}_n]^2 , where {\hat\lambda}_n is the maximum-likelihood estimate of u , and é n is the empirical Laplace transform, each based on an i.i.d. sample X 1 , …, X n . As the decay of the weight function tends to infinity, the test statistic approaches the square of the first nonzero component of Neyman's smooth test for exponentiality. The new tests are compared with other omnibus tests for exponentiality.