Bayesian accelerated life tests for the Weibull distribution under non-informative priors
- Authors: Mostert, Philip
- Date: 2020
- Subjects: Accelerated life testing -- Statistical methods , Accelerated life testing -- Mathematical models , Failure time data analysis , Bayesian statistical decision theory , Monte Carlo method , Weibull distribution
- Language: English
- Type: text , Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/172181 , vital:42173
- Description: In a competitive world where products are designed to last for long periods of time, obtaining time-to-failure data is both difficult and costly. Hence for products with high reliability, accelerated life testing is required to obtain relevant life-data quickly. This is done by placing the products under higher-than-use stress levels, thereby causing the products to fail prematurely. Part of the analysis of accelerated life-data requires a life distribution that describes the lifetime of a product at a given stress level and a life-stress relationship – which is some function that describes the way in which the life distribution changes across different stress levels. In this thesis it is assumed that the underlying life distribution is the wellknown Weibull distribution, with shape parameter constant over all stress levels and scale parameter as a log-linear function of stress. The primary objective of this thesis is to obtain estimates from Bayesian analysis, and this thesis considers five types of non-informative prior distributions: Jeffreys’ prior, reference priors, maximal data information prior, uniform prior and probability matching priors. Since the associated posterior distribution under all the derived non-informative priors are of an unknown form, the propriety of the posterior distributions is assessed to ensure admissible results. For comparison purposes, estimates obtained via the method of maximum likelihood are also considered. Finding these estimates requires solving non-linear equations, hence the Newton-Raphson algorithm is used to obtain estimates. A simulation study based on the time-to-failure of accelerated data is conducted to compare results between maximum likelihood and Bayesian estimates. As a result of the Bayesian posterior distributions being analytically intractable, two methods to obtain Bayesian estimates are considered: Markov chain Monte Carlo methods and Lindley’s approximation technique. In the simulation study the posterior means and the root mean squared error values of the estimates under the symmetric squared error loss function and the two asymmetric loss functions: the LINEX loss function and general entropy loss function, are considered. Furthermore the coverage rates for the Bayesian Markov chain Monte Carlo and maximum likelihood estimates are found, and are compared by their average interval lengths. A case study using a dataset based on accelerated time-to-failure of an insulating fluid is considered. The fit of these data for the Weibull distribution is studied and is compared to that of other popular life distributions. A full simulation study is conducted to illustrate convergence of the proper posterior distributions. Both maximum likelihood and Bayesian estimates are found for these data. The deviance information criterion is used to compare Bayesian estimates between the prior distributions. The case study is concluded by finding reliability estimates of the data at use-stress levels.
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- Authors: Mostert, Philip
- Date: 2020
- Subjects: Accelerated life testing -- Statistical methods , Accelerated life testing -- Mathematical models , Failure time data analysis , Bayesian statistical decision theory , Monte Carlo method , Weibull distribution
- Language: English
- Type: text , Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/172181 , vital:42173
- Description: In a competitive world where products are designed to last for long periods of time, obtaining time-to-failure data is both difficult and costly. Hence for products with high reliability, accelerated life testing is required to obtain relevant life-data quickly. This is done by placing the products under higher-than-use stress levels, thereby causing the products to fail prematurely. Part of the analysis of accelerated life-data requires a life distribution that describes the lifetime of a product at a given stress level and a life-stress relationship – which is some function that describes the way in which the life distribution changes across different stress levels. In this thesis it is assumed that the underlying life distribution is the wellknown Weibull distribution, with shape parameter constant over all stress levels and scale parameter as a log-linear function of stress. The primary objective of this thesis is to obtain estimates from Bayesian analysis, and this thesis considers five types of non-informative prior distributions: Jeffreys’ prior, reference priors, maximal data information prior, uniform prior and probability matching priors. Since the associated posterior distribution under all the derived non-informative priors are of an unknown form, the propriety of the posterior distributions is assessed to ensure admissible results. For comparison purposes, estimates obtained via the method of maximum likelihood are also considered. Finding these estimates requires solving non-linear equations, hence the Newton-Raphson algorithm is used to obtain estimates. A simulation study based on the time-to-failure of accelerated data is conducted to compare results between maximum likelihood and Bayesian estimates. As a result of the Bayesian posterior distributions being analytically intractable, two methods to obtain Bayesian estimates are considered: Markov chain Monte Carlo methods and Lindley’s approximation technique. In the simulation study the posterior means and the root mean squared error values of the estimates under the symmetric squared error loss function and the two asymmetric loss functions: the LINEX loss function and general entropy loss function, are considered. Furthermore the coverage rates for the Bayesian Markov chain Monte Carlo and maximum likelihood estimates are found, and are compared by their average interval lengths. A case study using a dataset based on accelerated time-to-failure of an insulating fluid is considered. The fit of these data for the Weibull distribution is studied and is compared to that of other popular life distributions. A full simulation study is conducted to illustrate convergence of the proper posterior distributions. Both maximum likelihood and Bayesian estimates are found for these data. The deviance information criterion is used to compare Bayesian estimates between the prior distributions. The case study is concluded by finding reliability estimates of the data at use-stress levels.
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An analysis of the Libor and Swap market models for pricing interest-rate derivatives
- Authors: Mutengwa, Tafadzwa Isaac
- Date: 2012
- Subjects: LIBOR market model , Monte Carlo method , Interest rates -- Mathematical models , Derivative securities
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5573 , http://hdl.handle.net/10962/d1005535
- Description: This thesis focuses on the non-arbitrage (fair) pricing of interest rate derivatives, in particular caplets and swaptions using the LIBOR market model (LMM) developed by Brace, Gatarek, and Musiela (1997) and Swap market model (SMM) developed Jamshidan (1997), respectively. Today, in most financial markets, interest rate derivatives are priced using the renowned Black-Scholes formula developed by Black and Scholes (1973). We present new pricing models for caplets and swaptions, which can be implemented in the financial market other than the Black-Scholes model. We theoretically construct these "new market models" and then test their practical aspects. We show that the dynamics of the LMM imply a pricing formula for caplets that has the same structure as the Black-Scholes pricing formula for a caplet that is used by market practitioners. For the SMM we also theoretically construct an arbitrage-free interest rate model that implies a pricing formula for swaptions that has the same structure as the Black-Scholes pricing formula for swaptions. We empirically compare the pricing performance of the LMM against the Black-Scholes for pricing caplets using Monte Carlo methods.
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- Authors: Mutengwa, Tafadzwa Isaac
- Date: 2012
- Subjects: LIBOR market model , Monte Carlo method , Interest rates -- Mathematical models , Derivative securities
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5573 , http://hdl.handle.net/10962/d1005535
- Description: This thesis focuses on the non-arbitrage (fair) pricing of interest rate derivatives, in particular caplets and swaptions using the LIBOR market model (LMM) developed by Brace, Gatarek, and Musiela (1997) and Swap market model (SMM) developed Jamshidan (1997), respectively. Today, in most financial markets, interest rate derivatives are priced using the renowned Black-Scholes formula developed by Black and Scholes (1973). We present new pricing models for caplets and swaptions, which can be implemented in the financial market other than the Black-Scholes model. We theoretically construct these "new market models" and then test their practical aspects. We show that the dynamics of the LMM imply a pricing formula for caplets that has the same structure as the Black-Scholes pricing formula for a caplet that is used by market practitioners. For the SMM we also theoretically construct an arbitrage-free interest rate model that implies a pricing formula for swaptions that has the same structure as the Black-Scholes pricing formula for swaptions. We empirically compare the pricing performance of the LMM against the Black-Scholes for pricing caplets using Monte Carlo methods.
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Pricing exotic options using C++
- Authors: Nhongo, Tawuya D R
- Date: 2007
- Subjects: C++ (Computer program language) , Monte Carlo method , Simulation methods , Options (Finance) -- Mathematical models , Pricing -- Mathematical models
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5577 , http://hdl.handle.net/10962/d1008373 , C++ (Computer program language) , Monte Carlo method , Simulation methods , Options (Finance) -- Mathematical models , Pricing -- Mathematical models
- Description: This document demonstrates the use of the C++ programming language as a simulation tool in the efficient pricing of exotic European options. Extensions to the basic problem of simulation pricing are undertaken including variance reduction by conditional expectation, control and antithetic variates. Ultimately we were able to produce a modularized, easily extend-able program which effectively makes use of Monte Carlo simulation techniques to price lookback, Asian and barrier exotic options. Theories of variance reduction were validated except in cases where we used control variates in combination with the other variance reduction techniques in which case we observed increased variance. Again, the main aim of this half thesis was to produce a C++ program which would produce stable pricings of exotic options.
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- Authors: Nhongo, Tawuya D R
- Date: 2007
- Subjects: C++ (Computer program language) , Monte Carlo method , Simulation methods , Options (Finance) -- Mathematical models , Pricing -- Mathematical models
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: vital:5577 , http://hdl.handle.net/10962/d1008373 , C++ (Computer program language) , Monte Carlo method , Simulation methods , Options (Finance) -- Mathematical models , Pricing -- Mathematical models
- Description: This document demonstrates the use of the C++ programming language as a simulation tool in the efficient pricing of exotic European options. Extensions to the basic problem of simulation pricing are undertaken including variance reduction by conditional expectation, control and antithetic variates. Ultimately we were able to produce a modularized, easily extend-able program which effectively makes use of Monte Carlo simulation techniques to price lookback, Asian and barrier exotic options. Theories of variance reduction were validated except in cases where we used control variates in combination with the other variance reduction techniques in which case we observed increased variance. Again, the main aim of this half thesis was to produce a C++ program which would produce stable pricings of exotic options.
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