### Abstract

We consider a problem in which exactly one of n+1 distinct signals {S_{0}, ..., S_{n}}, say S_{0}, may be present in a noisy environment and the presence or absence of S_{0} needs to be determined. The performance is given by the false alarm probabilities P_{f(i)}, i = 1, ..., n, where P_{f(i)} is the probability that S_{i} is declared as S_{0}, and the detection probability P_{d} which is the probability that S_{0} is correctly recognized. It is required that P_{d} be maximized while P_{f(i)}≤c_{i}, i = 1, ..., n, where c_{1}, ..., c_{n} are prescribed non-negative constants. The solution is an extended Neyman-Pearson test in which p_{0}(x) is tested against (w_{1}p_{1}(x)+...+w_{n}p_{n}(x)) where p_{i}(x) is the probability density function of observation X when S_{i} occurs. We propose two methods to obtain w_{1}, ..., w_{n} for any given c_{1}, ..., c_{n}. For certain simple cases, analytical or graphical solution is used. For the general case, we propose a search algorithm on w_{1}, ..., w_{n} which provides a sequence of extended Neyman-Pearson tests that converge to the optimal solution. Numerical examples are given to illustrate these methods.

Original language | English (US) |
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Title of host publication | Proceedings of SPIE - The International Society for Optical Engineering |

Publisher | Society of Photo-Optical Instrumentation Engineers |

Pages | 378-389 |

Number of pages | 12 |

Volume | 3720 |

State | Published - 1999 |

Event | Proceedings of the 1999 Signal Processing, Sensor Fusion, and Target Recognition VIII - Orlando, FL, USA Duration: Apr 5 1999 → Apr 7 1999 |

### Other

Other | Proceedings of the 1999 Signal Processing, Sensor Fusion, and Target Recognition VIII |
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City | Orlando, FL, USA |

Period | 4/5/99 → 4/7/99 |

### Fingerprint

### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Condensed Matter Physics

### Cite this

*Proceedings of SPIE - The International Society for Optical Engineering*(Vol. 3720, pp. 378-389). Society of Photo-Optical Instrumentation Engineers.

**On the design of extended Neyman-Pearson hypothesis tests.** / Zhang, Qian; Varshney, Pramod Kumar; Zhu, Yunmin.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Proceedings of SPIE - The International Society for Optical Engineering.*vol. 3720, Society of Photo-Optical Instrumentation Engineers, pp. 378-389, Proceedings of the 1999 Signal Processing, Sensor Fusion, and Target Recognition VIII, Orlando, FL, USA, 4/5/99.

}

TY - CHAP

T1 - On the design of extended Neyman-Pearson hypothesis tests

AU - Zhang, Qian

AU - Varshney, Pramod Kumar

AU - Zhu, Yunmin

PY - 1999

Y1 - 1999

N2 - We consider a problem in which exactly one of n+1 distinct signals {S0, ..., Sn}, say S0, may be present in a noisy environment and the presence or absence of S0 needs to be determined. The performance is given by the false alarm probabilities Pf(i), i = 1, ..., n, where Pf(i) is the probability that Si is declared as S0, and the detection probability Pd which is the probability that S0 is correctly recognized. It is required that Pd be maximized while Pf(i)≤ci, i = 1, ..., n, where c1, ..., cn are prescribed non-negative constants. The solution is an extended Neyman-Pearson test in which p0(x) is tested against (w1p1(x)+...+wnpn(x)) where pi(x) is the probability density function of observation X when Si occurs. We propose two methods to obtain w1, ..., wn for any given c1, ..., cn. For certain simple cases, analytical or graphical solution is used. For the general case, we propose a search algorithm on w1, ..., wn which provides a sequence of extended Neyman-Pearson tests that converge to the optimal solution. Numerical examples are given to illustrate these methods.

AB - We consider a problem in which exactly one of n+1 distinct signals {S0, ..., Sn}, say S0, may be present in a noisy environment and the presence or absence of S0 needs to be determined. The performance is given by the false alarm probabilities Pf(i), i = 1, ..., n, where Pf(i) is the probability that Si is declared as S0, and the detection probability Pd which is the probability that S0 is correctly recognized. It is required that Pd be maximized while Pf(i)≤ci, i = 1, ..., n, where c1, ..., cn are prescribed non-negative constants. The solution is an extended Neyman-Pearson test in which p0(x) is tested against (w1p1(x)+...+wnpn(x)) where pi(x) is the probability density function of observation X when Si occurs. We propose two methods to obtain w1, ..., wn for any given c1, ..., cn. For certain simple cases, analytical or graphical solution is used. For the general case, we propose a search algorithm on w1, ..., wn which provides a sequence of extended Neyman-Pearson tests that converge to the optimal solution. Numerical examples are given to illustrate these methods.

UR - http://www.scopus.com/inward/record.url?scp=0032685641&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032685641&partnerID=8YFLogxK

M3 - Chapter

VL - 3720

SP - 378

EP - 389

BT - Proceedings of SPIE - The International Society for Optical Engineering

PB - Society of Photo-Optical Instrumentation Engineers

ER -