Reprints Available directly from the Editor. Printed in New Zealand. CONSTRUCTION OF J–VARIATE DISTRIBUTION FUNCTIONS AND APPLICATIONS TO DISCRETE DECISION MODELS

The construction of J-variate distribution functions, introducing dependences among J random variables and keeping fixed the J marginal distribution functions, is important in the development of theoretical and empirical statistical analysis. Here, a method for generating such distribution functions is developed. Characteristics of the resulting distribution functions are discussed. An application to discrete regression models is presented. The latter is specialized to model choice of mode od transportation by travelers.


Introduction
In the development of probabilistic model and statistical analysis, the postulation of a joint cumulative distribution function (JCDF), keeping fixed some marginal cumulative distributions (MCDF), is often required (Yadlin, 1991;Long and Krysztofowicz, 1995). Some examples follow.
In survival models, it is usually assumed that the duration D of a component has an exponential distribution. If there are J components functioning in parallel, the formulation of a JCDF for the D j , J = 1, . . . , J, with exponential MCDF, that becomes the product of these MCDF under restrictions, may be convenient (Sarkar, 1987).
Here, we want to model the survival function P r ∩ J j=1 (D j ≥ d j ) . where the D j are not necessarily independent.
In dosage-response models, it is assumed that there exists a random tolerance function T r , r = 1, . . . , R, underlying each of the R possible response (Berkson, 1953). Examples are the response of an individual to some stimuli or the reaction of an insect to an insecticide. If there are J ≥ 2 stimuli, hence J tolerance functions, T dosages x (1) , . . . , x (j) is the rth response, that is, where the T (J) r may or may nor be independent. In discrete decision models (DDM), it is assumed that there exists a radom utility functions U m = i = 1, . . . , J, underlying each of the J possible decisions, for each sujects n = 1, . . . , N. Here U in = V in + in , where V in is non stochastic and in is a random perturbation. Examples are the decision by a high-school graduate about a career, the selection by a family of a vacation resort and the choice by a user of mode of transportation. Assuming that 1n , . . . , Jn are iid Extreme Value (EV) yields the prevailing methodology, which is the Multinomial Logit Model (MLM), which exhibits analytical and computational simplicity, but disregards interrelations among the J possible choices. A Nesting MLM, that overcomes the latter drawback, can be obtained by constructing a Nesting EV JCDF that keeps the J MCDF as EV. This Nesting MLM is also consistent with the principle of utility maximization, exhibits a closed functional form manageable for theoretical and empirical investigations, and allows for the introduction of relations among alternatives through the inclusion of a dependence parameter. Moreover, the Nesting MLM constitutes a parametric generalization of the MLM, in the sense that, under restrictions, the Nesting EV distribution becomes the product of J EV distributions, while the Nesting MLM reduces to the original MLM, so that classical testing procedures can be applied to decide between them. We will return to this example is Section 3.
Thus, the purpose is to build a JDCF for X 1 , . . . , X J such that: 1. The arbitrary MCDF of the X j are mantained. 2. Patterns of dependences are parametrized through a parameter α ∈ A ⊂ R L with 0 ∈ A. 3. The JCDF reduces to the product of the J marginals when α = 0. 4. Parametric generalizations of statistical models, with analytical and computational tractability, are accomplished.
Schematically, let F j be the MCDF of X j , F I be P J j=1 F j , M I be model under F I , F be the generalized JCDF of X 1 , . . . , X J , and M be the model under

Generating Method
Considering the JCDF for X 1 , . . . , X J , when the X j are grouped into subvectors where α m represents dependences among the X j , j ∈ B m . If the marginal densities f j exist, the joint density is given by Whatever the values of the α m , the MCDF and be marginal densities are F j and f j , respectively. Also, whatever the values of the α m , F has all the properties of a JCDF, except that it may assign negative probabilities to some rectangles in R J . Similarly, f integrates to 1, but it may take negative values. Thus, 2 M restrictions are required Hence, A is determined by (3).
The moment generating functions of (X 1 , . . . , X J ) is derived as Thus, where M j is the moment generating function of X J .
From (4), we can compute correlation (X i , , X l ) for any {i, l} ⊂ {1, . . . , J}. Setting α il be the dependence parameter associated to Hence, the correlation between X i and X l is zero whenever the dependence parameter α il is zero.

. An Application: Generalization of Discrete Decision Models
A DDM serves to explain the probabilities with which a subject will choose one and only one of J objects (Rust 1988;Yadlin 1991 and1993). Letting C = {1, . . . , J} be the choice set, a DDM specifies P n B (i), the probability that subject n will choose i from B, for all choice subsets B ⊂ C and all i ∈ B, on the basis of the principle of utility maximization. In this way, a DDM postulates that individuals' preferences can be expressed by means of a vector valued random utility function on C. It is assumed that subject n associates to alternative a random value U in , and them he chooses so as to maximize this value.
Thus, P n B (i) = P r(U in > U jn ; ∀j ∈ B − {i}). U in can be decomposed into a deterministic part V in and a random perturbation Thus, the functional form of a DDM is determined by F, the JCDF of 1n , . . . , jn . In particular, under independence It is usually assumed that where β is a vector of unknown parameters and Z in is a vector of observable exogeneous variables.
In what follows, we omit n and concentrate on P C (i), denoted by P (i), for shortness. No loss of generality is induced, because of the comment under (16) below.
It is clear that we can generalize any DDM based on independence of the j using (1). We obtain a model that embeds the independence model in a simple way, and whose decision probabilities are derived via (15) and are formulated in terms of the F j as Thus, If B ⊂ C is the union of some the B m , the probabilities P B (i), i ∈ B, belong to the same family (16).
The P (i) in (16)  The MLM is the prevailing model in applications of DDM. Denoting P (i) by Q(i), this model is given by Besides its great tractability, the MLM presents flexibility for interpreting the decision probabilities in terms of the relative mean utilities of the alternatives, and for evaluating the impact of manipulating the decision set C or the exogeneous variables Z in .
However, the likelihood odds for deciding alternative i over alternative j are constant whatever the set C is, so that these odds are independent of third alternatives available to the decision maker. In fact writing it can be seen that the evaluation of an alternative is based on binary comparisons only.
To derive the MLM from the principle of utility maximization, it is sufficient to assume that 1 , . . . , J are iid with EV distributions. Thus, replacing in (15) F j and f j , by (6) and (7), the following model is obtained The terms in the P (i) involve the MLM probabilities Q(i) and the dependence parameters α j , so that they consider interrelations among the elements inside each subser B m of the choice set C. The generalized probabilities in (19) constitute the Nesting EV Model. Unlike other generalizations of the MLM (Hausman and Wise, 1978;Small, 1994), the probabilities of the Nesting EV Model constitute simple functions of the probabilities of the MLM.
When the dependence parameters α m are zero for m = 1, . . . , M, the P (i) reduce to the Q(i). In this way, we have constructed a parametric generalization of the MLM, allowing us to test whether we can maintain the advantages of the MLM in any practical situation. In the event that the null hypothesis that all α m are zero, that is, the MLM is the true model, is rejected, we are still left with a manageable model, that takes into account relationship among the J possible decisions.
In the remainder of this section, an empirical illustration (Yadlin, 1991) is described. This illustration deals with the choice of mode of transportation by users who travel between two places. This problem has been usually approach with discrete regression models. Thus, the analysis of these models play an important role in the evaluation of transportation models, becoming fundamental to detect whether the simplest model provides an adequate fit to the available data.
Here we consider individuals who move from their homes to their working or studying place in Santiago.
The Others variables may have been considered. It was decided to have a very simple utility function because, the purpose is to evaluate the performance of probabilistic choice models under misspecification of the CDF F and not of the utility function, whose form is kept fixed. Thus the objetive is the comparison of the performance of models MLM and Nesting EV Model in the three-alternative, two-substitute situation, under misspecification of the probabilistic mechanism (CDF) underlyig the models, keeping the formulation of the utility function fixed, in the context of demand for transportation. The main steps of the methodology are the following: 1. Defining the choice set C described above and the target population, whose members are noncar owners transportation users, who travel from their resi-dence to their working or studying place by either of the modes in C at the morning peak hours.
2. The mean utility function is specified: When the population of decisions is derived from the Q n (i) (MLM) the null hypothesis, is accepted in the 25 samples, and when the population of decisions is derived from the P n (i) (Nesting EV Model), the null hypothesis is rejected in the 25 samples.
Thus, the Likelihhod Ratio Test rejects the false model always.

Conclusion
A straightforward method to build parametric generalizations of statistical models, that hold under independence of certain random variables have been presented.
In this way, a nesting statistical framework for non-separate models, compatible with likelihood methods is provided.
The importance of these developments becomes apparent in their applications in DDM.

Call for Papers
Intermodal transport refers to the movement of goods in a single loading unit which uses successive various modes of transport (road, rail, water) without handling the goods during mode transfers. Intermodal transport has become an important policy issue, mainly because it is considered to be one of the means to lower the congestion caused by single-mode road transport and to be more environmentally friendly than the single-mode road transport. Both considerations have been followed by an increase in attention toward intermodal freight transportation research.
Various intermodal freight transport decision problems are in demand of mathematical models of supporting them. As the intermodal transport system is more complex than a single-mode system, this fact offers interesting and challenging opportunities to modelers in applied mathematics. This special issue aims to fill in some gaps in the research agenda of decision-making in intermodal transport.
The mathematical models may be of the optimization type or of the evaluation type to gain an insight in intermodal operations. The mathematical models aim to support decisions on the strategic, tactical, and operational levels. The decision-makers belong to the various players in the intermodal transport world, namely, drayage operators, terminal operators, network operators, or intermodal operators.
Topics of relevance to this type of decision-making both in time horizon as in terms of operators are: • Intermodal terminal design • Infrastructure network configuration • Location of terminals • Cooperation between drayage companies • Allocation of shippers/receivers to a terminal • Pricing strategies • Capacity levels of equipment and labour • Operational routines and lay-out structure • Redistribution of load units, railcars, barges, and so forth • Scheduling of trips or jobs • Allocation of capacity to jobs • Loading orders • Selection of routing and service Before submission authors should carefully read over the journal's Author Guidelines, which are located at http://www .hindawi.com/journals/jamds/guidelines.html. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http://mts.hindawi.com/, according to the following timetable:

Manuscript Due
June 1, 2009 First Round of Reviews September 1, 2009