Linear Probability Functions Family

A Linear Probability Functions Family is a probability function family that is a linear function family.

• AKA: Linear Statistical Distribution.
• Context:
  • …
• Counter-Example(s):
  • Non-Linear Probability Model.
• See: Bernoulli Trial, Unit Interval, Linear Regression Algorithm. Binomial Regression, Simple Linear Regression, Least Squares, Weighted Least Squares.

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References

2015

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/linear_probability_model Retrieved:2015-2-16.
  • In statistics, a linear probability model is a special case of a binomial regression model. Here the observed variable for each observation takes values which are either 0 or 1. The probability of observing a 0 or 1 in any one case is treated as depending on one or more explanatory variables. For the "linear probability model", this relationship is a particularly simple one, and allows the model to be fitted by simple linear regression.

    The model assumes that, for a binary outcome (Bernoulli trial), [math]\displaystyle{ Y }[/math], and its associated vector of explanatory variables, [math]\displaystyle{ X }[/math],  :[math]\displaystyle{ \Pr(Y=1 | X=x) = x'\beta . }[/math]

    For this model, :[math]\displaystyle{ E[Y|X] = \Pr(Y=1|X) =x'\beta, }[/math]

    and hence the vector of parameters β can be estimated using least squares. This method of fitting would be inefficient. This method of fitting can be improved by adopting an iterative scheme based on weighted least squares, in which the model from the previous iteration is used to supply estimates of the conditional variances, [math]\displaystyle{ Var(Y|X=x) }[/math], which would vary between observations. This approach can be related to fitting the model by maximum likelihood.

    A drawback of this model for the parameter of the Bernoulli distribution is that, unless restrictions are placed on [math]\displaystyle{ \beta }[/math], the estimated coefficients can imply probabilities outside the unit interval [math]\displaystyle{ [0,1] }[/math]. For this reason, models such as the logit model or the probit model are more commonly used.

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