Lecture

Extreme Value Models: Technical Derivation

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Description

This lecture covers the technical derivation of Multivariate Extreme Value models, focusing on the generating function and key technical conditions such as the limit property, u-homogeneity property, and strong alternating sign property. The lecture also explains how the model is derived from first principles, the motivation behind the technical conditions, and the properties of the model including marginal distributions, variance-covariance matrix, normalization, and the expected Maximum Utility. Additionally, it introduces the Inheritance theorem and highlights that Logit and nested Logit are part of the Multivariate Extreme Value family.

In MOOC

Selected Topics on Discrete Choice

The logit model is the workhorse of choice modelers. But it has some limitations. In particular, some assumptions used to derive it may not be consistent with the behavioral reality. It may lead to erroneous forecast. We illustrated using the so-called "red bud-blue bus" paradox, and Multivariate Extre Value models, addressing some of these issues, are introduced. The sampling procedure used to collect choice data has a critical impact on the model estimation procedure. We introduce classical sampling proc

Official source

Ontological neighbourhood

Statistics

Statistical inference: Mathematical statistics

Related concepts (32)

Logit

In statistics, the logit (ˈloʊdʒɪt ) function is the quantile function associated with the standard logistic distribution. It has many uses in data analysis and machine learning, especially in data transformations. Mathematically, the logit is the inverse of the standard logistic function , so the logit is defined as Because of this, the logit is also called the log-odds since it is equal to the logarithm of the odds where p is a probability. Thus, the logit is a type of function that maps probability values from to real numbers in , akin to the probit function.

Discrete choice

In economics, discrete choice models, or qualitative choice models, describe, explain, and predict choices between two or more discrete alternatives, such as entering or not entering the labor market, or choosing between modes of transport. Such choices contrast with standard consumption models in which the quantity of each good consumed is assumed to be a continuous variable. In the continuous case, calculus methods (e.g. first-order conditions) can be used to determine the optimum amount chosen, and demand can be modeled empirically using regression analysis.

Covariance matrix

In probability theory and statistics, a covariance matrix (also known as auto-covariance matrix, dispersion matrix, variance matrix, or variance–covariance matrix) is a square matrix giving the covariance between each pair of elements of a given random vector. Any covariance matrix is symmetric and positive semi-definite and its main diagonal contains variances (i.e., the covariance of each element with itself). Intuitively, the covariance matrix generalizes the notion of variance to multiple dimensions.

Multinomial logistic regression

In statistics, multinomial logistic regression is a classification method that generalizes logistic regression to multiclass problems, i.e. with more than two possible discrete outcomes. That is, it is a model that is used to predict the probabilities of the different possible outcomes of a categorically distributed dependent variable, given a set of independent variables (which may be real-valued, binary-valued, categorical-valued, etc.).

Covariance

In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the lesser values (that is, the variables tend to show similar behavior), the covariance is positive. In the opposite case, when the greater values of one variable mainly correspond to the lesser values of the other, (that is, the variables tend to show opposite behavior), the covariance is negative.

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