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I want to do a regression to figure out the relationship between $y$ and $x$ given some sample data. My problem is that within the sample data, while $x_i$ is always observed, I do not directly observe $y_i$. Instead, I observe if $y_i>z_i$, where $z_i$ is some observed number.

For example, I will have to train the model $y=\hat{f}(x)$ on the below sample data: $$(x_1=2, y_1>1)$$ $$(x_2=3, y_2<4)$$ $$(x_3=10, y_3>2)$$ ...

Then I can use the trained $\hat{f}(x)$ to make predictions of $y$ on any new $x$, $y^*=\hat{f}(x^*)$

How do you think I can approach this problem?

Tom Bennett
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    Is this essentially the same question as http://stats.stackexchange.com/questions/202348/statistical-methods-for-data-where-only-a-minimum-maximum-value-is-known/202369 (with much the same answer), or is there another twist to your question that I did not realise? – Björn Apr 21 '16 at 08:22
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    Google censored regression. There is plenty of material around. The Tobit model is a common solution. – luchonacho Apr 21 '16 at 09:18

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