Estimate constant under absolute loss
Company: Citadel
Role: Data Scientist
Category: Statistics & Math
Difficulty: easy
Interview Round: Technical Screen
Suppose you have observed target values \(y_1, y_2, \dots, y_n\), and you want to fit the simplest possible model that predicts the same constant value for every observation:
\[
\hat y_i = c \quad \text{for all } i.
\]
Define the objective function using absolute error:
\[
E(c) = \sum_{i=1}^{n} |y_i - c|.
\]
What value of \(c\) minimizes \(E(c)\)? Derive the result and explain whether the minimizer is always unique.
Overview: This question evaluates understanding of L1 loss and robust point estimation, testing concepts from statistics such as absolute deviation, order statistics, and properties of estimators under non-quadratic loss, and it belongs to the Statistics & Math domain for data scientist roles.
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