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| In [[statistics]], the '''inverse Mills ratio''', named after [[John P. Mills]], is the [[ratio]] of the [[probability density function]] to the [[cumulative distribution function]] of a distribution.
| | Wilber Berryhill is the name his mothers and fathers gave him and he completely digs that title. What me and my family members adore is doing ballet but I've been taking on new things lately. I am currently a travel agent. My wife and I reside in Mississippi and I adore each working day living here.<br><br>my blog post; [http://cspl.postech.ac.kr/zboard/Membersonly/144571 psychics online] |
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| ==Uses==
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| Use of the inverse Mills ratio is often motivated by the following property of the [[truncation (statistics)|truncated]] [[normal distribution]]. If ''X'' is a [[random variable]] having a [[normal distribution]] with mean ''μ'' and variance ''σ''<sup>2</sup>, then
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| : <math>\begin{align}
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| & \operatorname{E}[\,X\,|\ X > \alpha \,] =
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| \mu + \sigma \frac {\phi\big(\tfrac{\alpha-\mu}{\sigma}\big)}{1-\Phi\big(\tfrac{\alpha-\mu}{\sigma}\big)}, \\
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| & \operatorname{E}[\,X\,|\ X < \alpha \,] =
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| \mu + \sigma \frac {-\phi\big(\tfrac{\alpha-\mu}{\sigma}\big)}{\Phi\big(\tfrac{\alpha-\mu}{\sigma}\big)},
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| \end{align}</math>
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| where ''α'' is a constant, ''ϕ'' denotes the standard normal density function, and ''Φ'' is the standard normal cumulative distribution function. The two fractions are the inverse Mills ratios.<ref>Greene, W. (2003) ''Econometric Analysis'', Prentice-Hall, p.759</ref>
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| ===Use in regression===
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| A common application of the inverse Mills ratio (sometimes also called 'selection hazard') arises in [[regression analysis]] to take account of a possible [[selection bias]]. If a dependent variable is [[censoring (statistics)|censored]] (i.e., not for all observations a positive outcome is observed) it causes a concentration of observations at zero values. This problem was first acknowledged by Tobin (1958), who showed that if this is not taken into consideration in the estimation procedure, an ordinary [[least squares]] estimation ([[linear regression|OLS]]) will produce [[bias (statistics)|biased]] parameter estimates.<ref>Tobin, J. 1958. Estimation of relationships for limited dependent variables. ''Econometrica'', 26(1): 24–36.</ref> With censored dependent variables there is a violation of the [[Gauss–Markov theorem|Gauss–Markov]] assumption of zero [[correlation]] between independent variables and the [[errors and residuals in statistics|error term]].
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| Heckman (1976) proposed a two-stage estimation procedure using the inverse Mills ratio to take account of the selection bias. In a first step, a regression for observing a positive outcome of the dependent variable is modeled with a [[probit]] model.
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| The inverse Mills ratio must be generated from the estimation of a [[probit model]], a [[logit]] cannot be used. The [[probit model]] assumes that the error term follows a [[standard normal distribution]].<ref>Heckman, James. (1979) "Sample Selection as a Specification Error". ''Econometrica'', 47 (1), 153–161</ref> The estimated parameters are used to calculate the inverse Mills ratio, which is then included as an additional explanatory variable in the OLS estimation.<ref>Heckman, J. J. (1976) "The common structure of statistical models of truncation, sample selection and limited dependent variables and a simple estimator for such models". ''Annals of Economic and Social Measurement'', 5(4): 475–492.</ref>
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| See [[Heckman correction]] for more details.
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| ==See also==
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| *[[Mills ratio]]
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| ==References==
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| {{Reflist}}
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| {{DEFAULTSORT:Inverse Mills Ratio}}
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| [[Category:Data analysis]]
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| [[Category:Statistical ratios]]
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Wilber Berryhill is the name his mothers and fathers gave him and he completely digs that title. What me and my family members adore is doing ballet but I've been taking on new things lately. I am currently a travel agent. My wife and I reside in Mississippi and I adore each working day living here.
my blog post; psychics online