Contents

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Reliability Engineering

• Estimation

• A Brief Introduction to Reliability

• Reliability Engineering and Business Plans

• Key Reasons for Reliability Engineering

• Disciplines Covered by Reliability Engineering

• A Few Common Sense Applications

• Advantages of a Reliability Engineering Program

• Summary: Key Reasons for Implementing a Reliability Engineering Program

• Reliability and Quality Control

Statistical Background

• Basic Statistical Definitions

• The Probability Density and Cumulative Density Functions

• The Reliability Function

• The Failure Rate Function

• The Mean Life Function

• The Median Life Function

• The Mode Function

• Distributions

• Most Commonly Used Distributions

• Other Distributions

• Parameter Estimation

• Probability Plotting

• Least Squares Parameter Estimation (Regression Analysis)

• MLE (Maximum Likelihood) Parameter Estimation for Complete Data)

• Bayesian Statistics

Data & Data Types

• Data Classification

• Complete Data

• Censored Data

• Data Types and Weibull++

• Analysis & Parameter Estimation Methods for Censored Data

• Analysis of Right Censored (Suspended) Data

• The Complete Likelihood Function

Confidence Bounds

• What are Confidence Bounds?

• Fisher Matrix Confidence Bounds

• Approximate Estimates of the Mean and Variance of a Function

• Approximate Confidence Intervals on the Parameters

• Fisher Matrix Confidence Bounds on Time (Type 1)

• Fisher Matrix Confidence Bounds on Reliability (Type 2)

• Beta Binomial Confidence Bounds

• Likelihood Ratio Confidence Bounds

• Likelihood Ratio Confidence Bounds on the Parameters

• Likelihood Ratio Confidence Bounds on Time (Type 1)

• Likelihood Ratio Confidence Bounds on Reliability (Type 2)

• Bayesian Confidence Bounds

• Bayesian Confidence Bounds on Time (Type 1)

• Bayesian Confidence Bounds on Reliability (Type 2)

• Simulation Based Bounds

The Weibull Distribution

• Weibull Probability Density Function

• Weibull Statistical Properties

• Characteristics of the Weibull Distribution

• Characteristic Effects of the Shape Parameter, β, for the Weibull Distribution

• Characteristic Effects of the Scale Parameter, η, for the Weibull Distribution

• Characteristic Effects of the Location Parameter, γ, for the Weibull Distribution

• Estimation of the Weibull Parameters

• Using Probability Plotting to Calculate the Parameters of the Weibull Distribution

• Using Rank Regression on Y to Calculate the Parameters of the Weibull Distribution

• Using Rank Regression on X to Calculate the Parameters of the Weibull Distribution

• Using Three-Parameter Weibull Regression to Calculate the Parameters of the Weibull Distribution

• Using Maximum Likelihood Estimation to Calculate the Parameters of the Weibull Distribution

• Confidence Bounds for the Weibull Distribution

• Fisher Matrix Confidence Bounds for the Weibull Distribution

• Fisher Matrix Confidence Bounds on the Parameters for the Weibull Distribution

• Fisher Matrix Confidence Bounds on Reliability for the Weibull Distribution

• Fisher Matrix Confidence Bounds on Time for the Weibull Distribution

• Likelihood Ratio Confidence Bounds for the Weibull Distribution

• Bayesian Confidence Bounds for the Weibull Distribution

• Weibull-Bayesian Analysis

• Posterior Distributions for Functions of Parameters in Weibull-Bayesian Analysis

• Confidence Bounds on R(T) for Weibull-Bayesian Analysis

• Confidence Bounds on Time T for Weibull-Bayesian Analysis

• General Examples Using the Weibull Distribution

• Published Examples Using the Weibull Distribution

The Exponential Distribution

• Exponential Probability Density Function

• Exponential Statistical Properties

• Characteristics of the Exponential Distribution

• Estimation of the Exponential Parameters

• Using Probability Plotting to Calculate the Parameters of the Exponential Distribution

• Using Rank Regression on Y to Calculate the Parameters of the Exponential Distribution

• Using Rank Regression on X to Calculate the Parameters of the Exponential Distribution

• Using Maximum Likelihood Estimation to Calculate the Parameters of the Exponential Distribution

• Confidence Bounds for the Exponential Distribution

• Fisher Matrix Confidence Bounds for the Exponential Distribution

• Likelihood Ratio Confidence Bounds for the Exponential Distribution

• Bayesian Confidence Bounds for the Exponential Distribution

• General Examples Using the Exponential Distribution

The Normal Distribution

• Normal Probability Density Function

• Normal Statistical Properties

• Characteristics of the Normal Distribution

• Estimation of the Normal Parameters

• Using Probability Plotting to Calculate the Parameters of the Normal Distribution

• Using Rank Regression on Y to Calculate the Parameters of the Normal Distribution

• Using Rank Regression on X to Calculate the Parameters of the Normal Distribution

• Using Maximum Likelihood Estimation to Calculate the Parameters of the Normal Distribution

• Confidence Bounds for the Normal Distribution

• Fisher Matrix Confidence Bounds for the Normal Distribution

• Likelihood Ratio Confidence Bounds for the Normal Distribution

• Bayesian Confidence Bounds for the Normal Distribution

• General Examples Using the Normal Distribution

The Lognormal Distribution

• Lognormal Probability Density Function

• Lognormal Statistical Properties

• Characteristics of the Lognormal Distribution

• Estimation of the Lognormal Distribution

• Using Probability Plotting to Calculate the Parameters of the Lognormal Distribution

• Using Rank Regression on Y to Calculate the Parameters of the Lognormal Distribution

• Using Rank Regression on X to Calculate the Parameters of the Lognormal Distribution

• Using Maximum Likelihood Estimation to Calculate the Parameters of the Lognormal Distribution

• Confidence Bounds for the Lognormal Distribution

• Fisher Matrix Confidence Bounds for the Lognormal Distribution

• Likelihood Ratio Confidence Bounds for the Lognormal Distribution

• Bayesian Confidence Bounds for the Lognormal Distribution

• General Examples Using the Lognormal Distribution

Other Distributions

• Mixed Weibull Distribution

• Background on the Mixed Weibull Distribution

• The Mixed Weibull Equations

• Mixed Weibull Parameter Estimation

• Mixed Weibull Confidence Bounds

• Using the Mixed Weibull Distributions in Weibull++

• About the Mixed Weibull Calculated Parameters

• Other Uses for the Mixed Weibull Distribution

• The Generalized Gamma Distribution

• Generalized Gamma Probability Density Function

• Generalized Gamma Statistical Properties

• Characteristics of the Generalized Gamma Distribution

• Confidence Bounds for the Generalized Gamma Distribution

• The Gamma Distribution

• Gamma Probability Density Function

• Gamma Statistical Properties

• Characteristics of the Gamma Distribution

• Confidence Bounds for the Gamma Distribution

• The Logistic Distribution

• Logistic Probability Density Function

• Logistic Statistical Properties

• Characteristics of the Logistic Distribution

• Logistic Probability Paper

• Confidence Bounds for the Logistic Distribution

• The Loglogistic Distribution

• Loglogistic Probability Density Function

• Loglogistic Statistical Properties

• Characteristics of the Loglogistic Distribution

• Confidence Bounds for the Loglogistic Distribution

• A Loglogistic Distribution Example

• The Gumbel/SEV Distribution

• Gumbel Probability Density Function

• Gumbel Statistical Properties

• Characteristics of the Gumbel Distribution

• Gumbel Probability Paper

• Confidence Bounds for the Gumbel Distribution

• A Gumbel Distribution Example

Recurrent Event Data Analysis

• Non-Parametric Recurrent Event Data Analysis

• The Mean Cumulative Function (MCF) for Recurrent Event Data

• Confidence Limits for the MCF

• Parametric Recurrent Event Data Analysis

• The GRP Model for Parametric Recurrent Event Data Analysis

• Confidence Bounds for Parametric Recurrent Event Data Analysis

Additional Reliability Analysis Tools

• Degradation Analysis

• Warranty Analysis

• Warranty Data Entry Types

• Warranty Analysis (Nevada Format)

• Warranty Prediction

• Analysis of Non-Homogeneous Warranty Data

• Monitoring Warranty Returns Using Statistical Process Control (SPC)

• Test Design

• Demonstration Test Design

• Constant Failure Rate/Chi-Squared Test Design

• SimuMatic

• Parameter Estimation and Confidence Bounds Techniques

• Using Simulation to Design Reliability Tests

• Nonparametric Life Data Analysis

• Kaplan-Meier Estimator

• Simple Actuarial Method

• Standard Actuarial Method

• Nonparametric Confidence Bounds

• Competing Failure Modes

• Plotting Competing Failure Modes

• Confidence Bounds for Competing Failure Modes

• Confidence Bounds on Reliability

• Confidence Bounds on Time

• Complex Competing Failure Modes

• Tests of Comparison

• Simple Plotting

• Using Contour Plots

• Estimating P[t2 >= t1] Using the Comparison Wizard

• Stress-Strength Analysis

• Risk Analysis and Probabilistic Design with Monte Carlo Simulation

Appendix A: Parameter Estimation

• Least Squares/Rank Regression Equations

• Maximum Likelihood Estimation

• Illustrating the MLE Method Using the Exponential Distribution

• Illustrating the MLE Method Using the Normal Distribution

Appendix B: Special Analysis Methods

• Grouped Data Analysis

• ReliaSoft's Alternate Ranking Method (RRM) Step-by-Step Example

Appendix C: Distribution Log-Likelihood Equations

• Weibull Log-Likelihood Functions and their Partials

• Exponential Log-Likelihood Functions and their Partials

• Normal Log-Likelihood Functions and their Partials

• Lognormal Log-Likelihood Functions and their Partials

• Mixed Weibull Log-Likelihood Functions and their Partials

• Logistic Log-Likelihood Functions and their Partials

• The Loglogistic Log-Likelihood Functions and their Partials

• The Gumbel Log-Likelihood Functions and their Partials

• The Gamma Log-Likelihood Functions and their Partials

Appendix D: References

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