1. (Risk minimization and a simplified binary classifier) In lecture, we learnt about the Neyman-Pearson framework: we seek to find a "best" critical region that maximizes the power of the test while main- taining a given significance level. Another major paradigm in the literature to define a "best" critical region is known as risk minimization that we now introduce. Let X ~ N(μ, 1). We are interested in testing Ho:}=0, H₁ = 1. As we only have a single sample X, we intend to use the test statistic T and critical region C to be respectively T=X, C={X> c}, where c>0 is a constant. Let be the standard normal cdf. (a) Prove that the probability of type I error is a(c) = 1 − (c). The bracket of c on the left hand side is to indicate the dependence on c of a, that is, a is a function of c. (b) Prove that the probability of type II error is B(c) = (c-1). The bracket of c on the left hand side is to indicate the dependence on c of ẞ, that is, ẞ is a function of c. (c) Define the risk R to be the sum of the probability of type I and type II error, that is, R(c) = a(c) + ẞ(c) = 1 − Þ(c) + Þ(c − 1). In risk minimization, we seek to find an optimal critical region by minimizing R(c) with respect to c. Let c* be the resulting minimizer, that is, Prove that c* = arg min R(c). C 1 2 and hence R(c*) = 2(−1/2).
1. (Risk minimization and a simplified binary classifier) In lecture, we learnt about the Neyman-Pearson framework: we seek to find a "best" critical region that maximizes the power of the test while main- taining a given significance level. Another major paradigm in the literature to define a "best" critical region is known as risk minimization that we now introduce. Let X ~ N(μ, 1). We are interested in testing Ho:}=0, H₁ = 1. As we only have a single sample X, we intend to use the test statistic T and critical region C to be respectively T=X, C={X> c}, where c>0 is a constant. Let be the standard normal cdf. (a) Prove that the probability of type I error is a(c) = 1 − (c). The bracket of c on the left hand side is to indicate the dependence on c of a, that is, a is a function of c. (b) Prove that the probability of type II error is B(c) = (c-1). The bracket of c on the left hand side is to indicate the dependence on c of ẞ, that is, ẞ is a function of c. (c) Define the risk R to be the sum of the probability of type I and type II error, that is, R(c) = a(c) + ẞ(c) = 1 − Þ(c) + Þ(c − 1). In risk minimization, we seek to find an optimal critical region by minimizing R(c) with respect to c. Let c* be the resulting minimizer, that is, Prove that c* = arg min R(c). C 1 2 and hence R(c*) = 2(−1/2).
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 25EQ
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