Linear Probability Functions Family

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



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]Y[/math], and its associated vector of explanatory variables, [math]X[/math],[1] :[math] \Pr(Y=1 | X=x) = x'\beta . [/math]

      For this model, :[math] 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.[1] This method of fitting can be improved by adopting an iterative scheme based on weighted least squares,[1] in which the model from the previous iteration is used to supply estimates of the conditional variances, [math]Var(Y|X=x)[/math], which would vary between observations. This approach can be related to fitting the model by maximum likelihood.[1]

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

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