Neyman-Pearson Lemma and Receiver Operating Characteristic Curve - Rhea

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Introduction

The purpose of this slecture is understanding Neyman-Pearson Lemma and Receiver Operating Characteristic (ROC) curve from theory to application. The fundamental theories stem from statistics and these can be used for signal detection and classification. In order to firmly understand these concepts, we will review statistical concept first including False alarm and Miss. After that, we will consider Bayes' decision rule using cost function. Next, we visit Neyman-Pearson test and minimax test. Lastly, we will discuss ROC curves. Note that we only consider two classes case in this slecture, but the concept can be extended to multiple classes.

The General Two Classes Case Problem

Before starting our discussion, we have the following setup:

• X a measure random variable, random vector, or random process
• $ x \in \mathcal{X} $ is a realization of X
• $\theta$ $\in$ $\Theta$ are unknown parameters
• f(x; $\theta$) is pdf of X (a known function) \\

Two distinct hypotheses on $\theta$ such that $H_0$: $\theta$ $\in$ $\Theta_0$ versus $H_1$: $\theta$ $\in$ $\Theta_1$ where $\Theta_0$, $\Theta_1$ is partition of $\Theta$ into two disjoint regions \begin{eqnarray*} \Theta_0 \cup \Theta_1 = \Theta, \qquad \Theta_0 \cap \Theta_1 = \phi \end{eqnarray*}

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