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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 $\beta$?

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    $\begingroup$ 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 ♦
    Commented Sep 18, 2017 at 20:22

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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

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