## Bernoulli

- Bernoulli
- Volume 24, Number 4A (2018), 2693-2720.

### Max-linear models on directed acyclic graphs

Nadine Gissibl and Claudia Klüppelberg

#### Abstract

We consider a new recursive structural equation model where all variables can be written as max-linear function of their parental node variables and independent noise variables. The model is max-linear in terms of the noise variables, and its causal structure is represented by a directed acyclic graph. We detail the relation between the weights of the recursive structural equation model and the coefficients in its max-linear representation. In particular, we characterize all max-linear models which are generated by a recursive structural equation model, and show that its max-linear coefficient matrix is the solution of a fixed point equation. We also find the minimum directed acyclic graph representing the recursive structural equations of the variables. The model structure introduces a natural order between the node variables and the max-linear coefficients. This yields representations of the vector components, which are based on the minimum number of node and noise variables.

#### Article information

**Source**

Bernoulli, Volume 24, Number 4A (2018), 2693-2720.

**Dates**

Received: January 2016

Revised: February 2017

First available in Project Euclid: 26 March 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.bj/1522051222

**Digital Object Identifier**

doi:10.3150/17-BEJ941

**Mathematical Reviews number (MathSciNet)**

MR3779699

**Zentralblatt MATH identifier**

06853262

**Keywords**

directed acyclic graph graphical model max-linear model minimal representation path analysis structural equation model

#### Citation

Gissibl, Nadine; Klüppelberg, Claudia. Max-linear models on directed acyclic graphs. Bernoulli 24 (2018), no. 4A, 2693--2720. doi:10.3150/17-BEJ941. https://projecteuclid.org/euclid.bj/1522051222