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E-raamat: Using the Weibull Distribution: Reliability, Modeling, and Inference

(Penn State Great Valley, USA)
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Using the Weibull Distribution: Reliability, Modeling, and InferenceSuurem pilt
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Understand and utilize the latest developments in Weibull inferential methods

While the Weibull distribution is widely used in science and engineering, most engineers do not have the necessary statistical training to implement the methodology effectively. Using the Weibull Distribution: Reliability, Modeling, and Inference fills a gap in the current literature on the topic, introducing a self-contained presentation of the probabilistic basis for the methodology while providing powerful techniques for extracting information from data.

The author explains the use of the Weibull distribution and its statistical and probabilistic basis, providing a wealth of material that is not available in the current literature. The book begins by outlining the fundamental probability and statistical concepts that serve as a foundation for subsequent topics of coverage, including:

Optimum burn-in, age and block replacement, warranties

and renewal theory

Exact inference in Weibull regression

Goodness of fit testing and distinguishing the Weibull

from the lognormal

Inference for the Three Parameter Weibull 

Throughout the book, a wealth of real-world examples showcases the discussed topics and each chapter concludes with a set of exercises, allowing readers to test their understanding of the presented material. In addition, a related website features the author's own software for implementing the discussed analyses along with a set of modules written in Mathcad®, and additional graphical interface software for performing simulations.

With its numerous hands-on examples, exercises, and software applications, Using the Weibull Distribution is an excellent book for courses on quality control and reliability engineering at the upper-undergraduate and graduate levels. The book also serves as a valuable reference for engineers, scientists, and business analysts who gather and interpret data that follows the Weibull distribution
Preface xiii
1 Probability
1(22)
1.1 Sample Spaces and Events
2(1)
1.2 Mutually Exclusive Events
2(1)
1.3 Venn Diagrams
3(1)
1.4 Unions of Events and Joint Probability
4(2)
1.5 Conditional Probability
6(2)
1.6 Independence
8(1)
1.7 Partitions and the Law of Total Probability
9(3)
1.8 Reliability
12(1)
1.9 Series Systems
12(1)
1.10 Parallel Systems
13(2)
1.11 Complex Systems
15(1)
1.12 Crosslinked Systems
16(3)
1.13 Reliability Importance
19(4)
References
20(1)
Exercises
21(2)
2 Discrete and Continuous Random Variables
23(50)
2.1 Probability Distributions
24(2)
2.2 Functions of a Random Variable
26(2)
2.3 Jointly Distributed Discrete Random Variables
28(4)
2.4 Conditional Expectation
32(2)
2.5 The Binomial Distribution
34(5)
2.5.1 Confidence Limits for the Binomial Proportion p
38(1)
2.6 The Poisson Distribution
39(2)
2.7 The Geometric Distribution
41(1)
2.8 Continuous Random Variables
42(9)
2.8.1 The Hazard Function
49(2)
2.9 Jointly Distributed Continuous Random Variables
51(1)
2.10 Simulating Samples from Continuous Distributions
52(2)
2.11 The Normal Distribution
54(6)
2.12 Distribution of the Sample Mean
60(6)
2.12.1 P[ X < Y] for Normal Variables
65(1)
2.13 The Lognormal Distribution
66(1)
2.14 Simple Linear Regression
67(6)
References
69(1)
Exercises
69(4)
3 Properties of the Weibull Distribution
73(24)
3.1 The Weibull Cumulative Distribution Function (CDF), Percentiles, Moments, and Hazard Function
73(9)
3.1.1 Hazard Function
75(2)
3.1.2 The Mode
77(1)
3.1.3 Quantiles
77(1)
3.1.4 Moments
78(4)
3.2 The Minima of Weibull Samples
82(1)
3.3 Transformations
83(3)
3.3.1 The Power Transformation
83(1)
3.3.2 The Logarithmic Transformation
84(2)
3.4 The Conditional Weibull Distribution
86(3)
3.5 Quantiles for Order Statistics of a Weibull Sample
89(3)
3.5.1 The Weakest Link Phenomenon
92(1)
3.6 Simulating Weibull Samples
92(5)
References
94(1)
Exercises
95(2)
4 Weibull Probability Models
97(33)
4.1 System Reliability
97(6)
4.1.1 Series Systems
97(2)
4.1.2 Parallel Systems
99(3)
4.1.3 Standby Parallel
102(1)
4.2 Weibull Mixtures
103(2)
4.3 P(Y < X)
105(3)
4.4 Radial Error
108(2)
4.5 Pro Rata Warranty
110(2)
4.6 Optimum Age Replacement
112(7)
4.6.1 Age Replacement
115(2)
4.6.2 MTTF for a Maintained System
117(2)
4.7 Renewal Theory
119(4)
4.7.1 Block Replacement
121(1)
4.7.2 Free Replacement Warranty
122(1)
4.7.3 A Renewing Free Replacement Warranty
122(1)
4.8 Optimum Bidding
123(1)
4.9 Optimum Burn-In
124(2)
4.10 Spare Parts Provisioning
126(4)
References
127(1)
Exercises
128(2)
5 Estimation in Single Samples
130(50)
5.1 Point and Interval Estimation
130(1)
5.2 Censoring
130(2)
5.3 Estimation Methods
132(4)
5.3.1 Menon's Method
132(2)
5.3.2 An Order Statistic Estimate of xb0.10
134(2)
5.4 Graphical Estimation of Weibull Parameters
136(9)
5.4.1 Complete Samples
136(4)
5.4.2 Graphical Estimation in Censored Samples
140(5)
5.5 Maximum Likelihood Estimation
145(9)
5.5.1 The Exponential Distribution
147(1)
5.5.2 Confidence Intervals for the Exponential Distribution---Type II Censoring
147(3)
5.5.3 Estimation for the Exponential Distribution---Interval Censoring
150(1)
5.5.4 Estimation for the Exponential Distribution---Type I Censoring
151(2)
5.5.5 Estimation for the Exponential Distribution---The Zero Failures Case
153(1)
5.6 ML Estimation for the Weibull Distribution
154(26)
5.6.1 Shape Parameter Known
154(1)
5.6.2 Confidence Interval for the Weibull Scale Parameter--- Shape Parameter Known, Type II Censoring
155(2)
5.6.3 ML Estimation for the Weibull Distribution---Shape Parameter Unknown
157(5)
5.6.4 Confidence Intervals for Weibull Parameters---Complete and Type II Censored Samples
162(5)
5.6.5 Interval Censoring with the Weibull
167(1)
5.6.6 Confidence Limits for Weibull Parameters---Type I Censoring
167(10)
References
177(2)
Exercises
179(1)
6 Sample Size Selection, Hypothesis Testing, and Goodness of Fit
180(33)
6.1 Precision Measure for Maximum Likelihood (ML) Estimates
180(2)
6.2 Interval Estimates from Menon's Method of Estimation
182(2)
6.3 Hypothesis Testing---Single Samples
184(4)
6.4 Operating Characteristic (OC) Curves for One-Sided Tests of the Weibull Shape Parameter
188(3)
6.5 OC Curves for One-Sided Tests on a Weibull Percentile
191(4)
6.6 Goodness of Fit
195(9)
6.6.1 Completely Specified Distribution
195(3)
6.6.2 Distribution Parameters Not Specified
198(3)
6.6.3 Censored Samples
201(1)
6.6.4 The Program ADStat
201(3)
6.7 Lognormal versus Weibull
204(9)
References
210(2)
Exercises
212(1)
7 The Program Pivotal.exe
213(22)
7.1 Relationship among Quantiles
216(1)
7.2 Series Systems
217(1)
7.3 Confidence Limits on Reliability
218(3)
7.4 Using Pivotal.exe for OC Curve Calculations
221(3)
7.5 Prediction Intervals
224(2)
7.6 Sudden Death Tests
226(4)
7.7 Design of Optimal Sudden Death Tests
230(5)
References
233(1)
Exercises
234(1)
8 Inference from Multiple Samples
235(41)
8.1 Multiple Weibull Samples
235(1)
8.2 Testing the Homogeneity of Shape Parameters
236(2)
8.3 Estimating the Common Shape Parameter
238(6)
8.3.1 Interval Estimation of the Common Shape Parameter
239(5)
8.4 Interval Estimation of a Percentile
244(5)
8.5 Testing Whether the Scale Parameters Are Equal
249(8)
8.5.1 The SPR Test
250(2)
8.5.2 Likelihood Ratio Test
252(5)
8.6 Multiple Comparison Tests for Differences in Scale Parameters
257(2)
8.7 An Alternative Multiple Comparison Test for Percentiles
259(2)
8.8 The Program Multi-Weibull.exe
261(5)
8.9 Inference on P (Y < X)
266(10)
8.9.1 ML Estimation
267(2)
8.9.2 Normal Approximation
269(2)
8.9.3 An Exact Simulation Solution
271(2)
8.9.4 Confidence Intervals
273(1)
References
274(1)
Exercises
274(2)
9 Weibull Regression
276(22)
9.1 The Power Law Model
276(2)
9.2 ML Estimation
278(1)
9.3 Example
279(1)
9.4 Pivotal Functions
280(1)
9.5 Confidence Intervals
281(1)
9.6 Testing the Power Law Model
281(1)
9.7 Monte Carlo Results
282(3)
9.8 Example Concluded
285(2)
9.9 Approximating u* at Other Stress Levels
287(2)
9.10 Precision
289(1)
9.11 Stress Levels in Different Proportions Than Tabulated
289(2)
9.12 Discussion
291(1)
9.13 The Disk Operating System (DOS) Program REGEST
291(7)
References
296(1)
Exercises
296(2)
10 The Three-Parameter Weibull Distribution
298(15)
10.1 The Model
298(2)
10.2 Estimation and Inference for the Weibull Location Parameter
300(1)
10.3 Testing the Two- versus Three-Parameter Weibull Distribution
301(1)
10.4 Power of the Test
302(1)
10.5 Interval Estimation
302(5)
10.6 Input and Output Screens of LOCEST.exe
307(2)
10.7 The Program LocationPivotal.exe
309(2)
10.8 Simulated Example
311(2)
References
311(1)
Exercises
312(1)
11 Factorial Experiments with Weibull Response
313(20)
11.1 Introduction
313(1)
11.2 The Multiplicative Model
314(3)
11.3 Data
317(1)
11.4 Estimation
317(2)
11.5 Test for the Appropriate Model
319(1)
11.6 Monte Carlo Results
320(1)
11.7 The DOS Program TWOWAY
320(1)
11.8 Illustration of the Influence of Factor Effects on the Shape Parameter Estimates
320(7)
11.9 Numerical Examples
327(6)
References
331(1)
Exercises
332(1)
Index 333
JOHN I. McCOOL, PhD, is Professor of Systems Engineering at Penn State Great Valley School of Graduate Professional Studies. A Fellow of the American Society for Quality, Dr. McCool previously served as principal engineering scientist at SKF Industries Inc., where he conducted corporate as well as federally sponsored research projects with the Wright-Patterson Air Force Base, the Office of Naval Research, the Naval Air Propulsion Center, the Department of Energy, and the Air Force Office of Scientific Research.
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