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(д, 1). We are interested in testing Họ:g=0, Hi:p=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)+3(c) = 1 − (c) + (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). and hence = R(c) 24(-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(д, 1). We are interested in testing Họ:g=0, Hi:p=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)+3(c) = 1 − (c) + (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). and hence = R(c) 24(-1/2).
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
Related questions
Question
Transcribed Image Text: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(д, 1). We are interested in
testing
Họ:g=0, Hi:p=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)+3(c) = 1 − (c) + (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).
and hence
=
R(c) 24(-1/2).
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps
Recommended textbooks for you
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
MATLAB: An Introduction with Applications
Statistics
ISBN:
9781119256830
Author:
Amos Gilat
Publisher:
John Wiley & Sons Inc
Probability and Statistics for Engineering and th…
Statistics
ISBN:
9781305251809
Author:
Jay L. Devore
Publisher:
Cengage Learning
Statistics for The Behavioral Sciences (MindTap C…
Statistics
ISBN:
9781305504912
Author:
Frederick J Gravetter, Larry B. Wallnau
Publisher:
Cengage Learning
Elementary Statistics: Picturing the World (7th E…
Statistics
ISBN:
9780134683416
Author:
Ron Larson, Betsy Farber
Publisher:
PEARSON
The Basic Practice of Statistics
Statistics
ISBN:
9781319042578
Author:
David S. Moore, William I. Notz, Michael A. Fligner
Publisher:
W. H. Freeman
Introduction to the Practice of Statistics
Statistics
ISBN:
9781319013387
Author:
David S. Moore, George P. McCabe, Bruce A. Craig
Publisher:
W. H. Freeman