1
$\begingroup$

I'm trying to run a logistic regression with a dependent variable $y \in \{0,1\}$ and one independent variable $x$. I'm trying to find the coefficient $\beta$ in the equation $Prob(y=1)=\frac{e^{\beta x}}{1+e^{\beta x}}.$

My data is given by $(y_1,x_1),...(y_n,x_n)$. If $y_i = 1$, then I get to see $x_i$. However, if $y_i=0$, then $x_i$ is unobserved. Only the fact that $y_i = 0$ is known about observation $i$.

Is there a way of recovering an unbiased and consistent estimator $\hat{\beta}$ of $\beta$?

$\endgroup$
  • 4
    $\begingroup$ Let's flip this around: you have two groups of data, one consisting of the $x_i$ for which $y_i=1$ and the other consisting of the $x_i$ for which $y_i=0$. This second group gives you no data at all: you haven't observed any of those $x_i$. Your question therefore amounts to "I would like to compare two groups of values but I know absolutely nothing about the second group except its size. What can I say?" I hope that makes the answer obvious. $\endgroup$ – whuber Sep 18 '17 at 20:22
1
$\begingroup$

Answered in comments: Let's flip this around: you have two groups of data, one consisting of the $x_i$ for which $y_i=1$ and the other consisting of the $x_i$ for which $y_i=0$. This second group gives you no data at all: you haven't observed any of those x$_i$. Your question therefore amounts to "I would like to compare two groups of values but I know absolutely nothing about the second group except its size. What can I say?" I hope that makes the answer obvious. – whuber

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.