A note on exponential dispersion models which are invariant under length-biased sampling

Length-biased sampling situations may occur in clinical trials, reliability, queueing models, survival analysis and population studies where a proper sampling frame is absent.In such situations items are sampled at rate proportional to their length so that larger values of the quantity being measured are sampled with higher probabilities.More specifically, if f(x) is a p.d.f. presenting a parent population composed of nonnegative valued items then the sample is practically drawn from a distribution with p.d.f. g(x) = xf(x)/E(X) describing the lengthbiased population.In this case the distribution associated with g is termed a length-biased distribution.In this note we present a unified approach for characterizing exponential dispersion models which are invariant, up to translations, under various types of length-biased sampling.The approach is rather simple as it reduces such invariance problems into differential equations in terms of the derivatives of the associated variance functions.sampling;variance;models;distribution;statistics

Group Testing Models with Processing Times and Incomplete Identification

We consider the group testing problem for a finite population of possibly defective items with the objective of sampling a prespecified demanded number of nondefective items at minimum cost.Group testing means that items can be pooled and tested together; if the group comes out clean, all items in it are nondefective, while a "contaminated" group is scrapped.Every test takes a random amount of time and a given deadline has to be met.If the prescribed number of nondefective items is not reached, the demand has to be satisfied at a higher (penalty) cost.We derive explicit formulas for the distributions underlying the cost functionals of this model.It is shown in numerical examples that these results can be used to determine the optimal group size.testing;sampling

A characterization related to the equilibrium distribution associated with a polynomial structure

Let f be a probability density function on (a, b) C (0, infinity) and consider the class Cf of all probability density functions of the form Pf where P is a polynomial. Assume that if X has its density in Cf then the equilibrium probability density x -> P(X > x)/E(X) also belongs to Cf : this happens for instance when f(x) = Ce-¿x or f(x) = C(b-x) ¿-1. The present paper shows that actually they are the only possible two cases. This surprising result is achieved with an unusual tool in renewal theory, by using ideals of polynomials

A characterization related to the equilibrium distribution associated with a polynomial structure

Let f be a probability density function on (a, b) C (0, infinity) and consider the class Cf of all probability density functions of the form Pf where P is a polynomial. Assume that if X has its density in Cf then the equilibrium probability density x -> P(X > x)/E(X) also belongs to Cf : this happens for instance when f(x) = Ce-¿x or f(x) = C(b-x) ¿-1. The present paper shows that actually they are the only possible two cases. This surprising result is achieved with an unusual tool in renewal theory, by using ideals of polynomials

Applications of Likelihood-Based Methods for the Reliability Parameter of the Location and Scale Exponential Distribution

Based on a type-2 censored sample we consider a likelihood-based inference for the reliability parameter R(t) of the location and scale exponential distribution.More specifically, we derive the profile and marginal likelihoods of R(t).A numerical example is presented demonstrating the flavor of results that can be obtained by likelihood-based methods.

Tandem queues with impatient customers for blood screening procedures

We study a blood testing procedure for detecting viruses like HIV, HBV and HCV. In this procedure, blood samples go through two screening steps. The first test is ELISA (antibody Enzyme Linked Immuno-Sorbent Assay). The portions of blood which are found not contaminated in this first phase are tested in groups through PCR (Polymerase Chain Reaction). The ELISA test is less sensitive than the PCR test and the PCR tests are considerably more expensive. We model the two test phases of blood samples as services in two queues in series; service in the second queue is in batches, as PCR tests are done in groups. The fact that blood can only be used for transfusions until a certain expiration date leads, in the tandem queue, to the feature of customer impatience. Since the first queue basically is an infinite server queue, we mainly focus on the second queue, which in its most general form is an S-server M=G[k;K]=S + G queue, with batches of sizes which are bounded by k and K. Our objective is to maximize the expected profit of the system, which is composed of the amount earned for items which pass the test (and before their patience runs out), minus costs. This is done by an appropriate choice of the decision variables, namely, the batch sizes and the number of servers at the second service station. As will be seen, even the simplest version of the batch queue, the M=M[k;K]=1 + M queue, already gives rise to serious analytical complications for any batch size larger than 1. These complications are discussed in detail. In view of the fact that we aim to solve realistic optimization problems for blood screening procedures, these analytical complications force us to take recourse to either a numerical approach or approximations. We present a numerical solution for the queue length distribution in theM=M[k;K]=S+M queue and then formulate and solve several optimization problems. The power-series algorithm, which is a numerical-analytic method, is also discussed