The analysis of some algorithms for generating random variates with a given hazard rate

Abstract

We analyze the expected time performance of two versions of the thinning algorithm of Lewis and Shedler for generating random variates with a given hazard rate on [O,m). For thinning with fixed dominating hazard rate g(.r) = c for example, it is shown that the expected number of iterations is cE(X) where X is the random variate that is produced. For DHR distributions, we can use dynamic thinning by adjusting the dominating hazard rate as we proceed. With the aid of some inequalities, we show that this improves the performance dramatically. For example, the expected number of iterations is bounded by a constant plus E(log+(h(O)X)) (the logarithmic moment of X ) .

Probabilistic Autoencoder Using Fisher Information

“…We can choose this prior without loss of generality since for every set of latent space vectors we can find a transformation into a coordinate system where they are distributed according to a standard Gaussian distribution. This transformation is also known as random variable generation with inverse transform sampling [ 30 ] and is the basis for the reparametrization trick [ 5 , 6 , 31 ]. It allows us to solely use the standard Gaussian distribution for our calculations, which significantly simplifies them.…”

Non-Uniform Random Variate Generation.

“…This may pose a problem for random variate generation. Various algorithms for simulation in the presence of hazard rates are surveyed by Devroye (1986c), the most useful among which is the thinning method of Lewis and Shedler (1979): assume that h ≤ g, where g is another hazard rate. If 0 < Y 1 < Y 2 < • • • is a nonhomogeneous Poisson point process with rate function g, and U 1 , U 2 , .…”

“…In that case, the expected number of iterations before halting is {h(0)X}. However, we can dynamically thin (Devroye, 1986c) by lowering g as values h(Y i ) trickle in. In that case, the expected time is finite when X has a finite logarithmic moment, and in any case, it is not more than 4 + 24 {h(0)X}.…”

Simulating nonhomogeneous poisson processes with proportional intensities

“…Nonhomogeneous Poisson processes with proportional intensities are gaining attention in the literature, in the areas of both medicine and, more recently, reliability engineering (e.g., Ascher and Feingold [4], Lawless [22], Williams [37]). There exists a fairly rich literature in the statistical analysis and simulation algorithms for purely time-varying nonhomogeneous Poisson processes, that is, without covariate information (e.g., Brown [9]; Cox and Lewis [13]; Crow [14]; Devroye [15]; Kaminsky and Rumpf [19]; Lewis [25]; Lewis and Shedler [26]; Ripley [31]). Only a few articles have contributed to the statistical analysis and simulation of nonhomogeneous Poisson processes in the presence of covariates (e.g., Guo and Love [17]; Lawless [22]; Leemis [23]; Leemis,Shih,and Reynertson [24]; Love and Guo [27,281;Whitaker and Samaniego [36]).…”

“…Furthermore, in most industrial situations, it general and may gain computational efficiency in certain circumstances (e.g. , Devroye [15], Ripley [31]). Section 4 extends these algorithms to the general Poisson process studied by Cinlar [ll].…”