Problem statement: The D-optimal design is often used in clinical research. In multi-factor clinical experiments it is natural to restrict the experiment's design space so as not to give a patient the combination of several high dose treatments simultaneously. Under such design space restrictions it is unknown what designs are D-optimal. The goal of the thesis has been to find D-optimal designs for these design spaces. Approach: Two new algorithms for finding D-optimal designs with one, two or three factors with linear models has been developed and implemented in MATLAB. Two restricted design spaces were explored. In cases when the program could not find the D-optimal design an analytic approach was used. Results: Special attention was given to the two factor model with interaction. All of the D-optimal designs for this model, N less or equal to 30, and their permutations have been listed as well as their continous designs. Conclusion: In one of the restricted design regions a simple design pattern appeared for N greater than or equal to 7. In the other restricted design region no obvious pattern was found but its continuous design could be calculated through analysis. It turned out that the number of trials at the lowest dose combination did not change when moving from the full space design to the restricted design regions. / Frågeställning: D-optimala designer är vanliga i kliniska studier. När flera faktorer (läkemedel) prövas samtidigt kan det vara nödvändigt att begränsa försöksrummet så att patienterna undviker att få en hög dos av flera faktorer samtidigt. I sådana begränsade försöksrum är det okänt vilka designer som är D-optimala. Uppsatsens mål har varit att hitta D-optimala designer i begränsade försöksrum. Metod: Två nya algoritmer för att hitta D-optimala designer med en, två eller tre dimensioner och linjära modeller har utvecklats och implementerats i MATLAB. Två begränsade försöksrum har utforskats. I de fall då MATLAB-programmet inte kunde hitta de D-optimala designerna användes analytiska metoder. Resultat: Analys av en tvåfaktormodell med interaktion utforskades särskilt noggrant. Alla D-optimala designer och permutationer av dessa i de båda begränsade försöksrummen har listats för alla N mindre än eller lika med 30, samt även deras kontinuerliga designer. Slutsats: För det ena försöksrummet upptäcktes ett mönster i designen då N är större än eller lika med 7. I det andra försöksrummet upptäcktes inget mönster och det krävdes således analytiska metoder för att finna dess kontinuerliga design. Det visade sig att antalet försök i den lägsta doskombinationen förblev oförändrat då man bytte från det fulla designrummet till de båda begränsade designrummen.

In this thesis, we investigate several inferential issues regarding the lifetime data from exponential distribution under different censoring schemes. For reasons of time constraint and cost reduction, censored sampling is commonly employed in practice, especially in reliability engineering. Among various censoring schemes, progressive Type-I censoring provides not only the practical advantage of known termination time but also greater flexibility to the experimenter in the design stage by allowing for the removal of test units at non-terminal time points. Hence, we first consider the inference for a progressively Type-I censored life-testing experiment with k uniformly spaced intervals. For small to moderate sample sizes, a practical modification is proposed to the censoring scheme in order to guarantee a feasible life-test under progressive Type-I censoring. Under this setup, we obtain the maximum likelihood estimator (MLE) of the unknown mean parameter and derive the exact sampling distribution of the MLE through the use of conditional moment generating function under the condition that the existence of the MLE is ensured. Using the exact distribution of the MLE as well as its asymptotic distribution and the parametric bootstrap method, we discuss the construction of confidence intervals for the mean parameter and their performance is then assessed through Monte Carlo simulations. Next, we consider a special class of accelerated life tests, known as step-stress tests in reliability testing. In a step-stress test, the stress levels increase discretely at pre-fixed time points and this allows the experimenter to obtain information on the parameters of the lifetime distributions more quickly than under normal operating conditions. Here, we consider a k-step-stress accelerated life testing experiment with an equal step duration τ. In particular, the case of progressively Type-I censored data with a single stress variable is investigated. For small to moderate sample sizes, we introduce another practical modification to the model for a feasible k-step-stress test under progressive censoring, and the optimal τ is searched using the modified model. Next, we seek the optimal τ under the condition that the step-stress test proceeds to the k-th stress level, and the efficiency of this conditional inference is compared to the preceding models. In all cases, censoring is allowed at each change stress point iτ, i = 1, 2, ... , k, and the problem of selecting the optimal Tis discussed using C-optimality, D-optimality, and A-optimality criteria. Moreover, when a test unit fails, there are often more than one fatal cause for the failure, such as mechanical or electrical. Thus, we also consider the simple stepstress models under Type-I and Type-II censoring situations when the lifetime distributions corresponding to the different risk factors are independently exponentially distributed. Under this setup, we derive the MLEs of the unknown mean parameters of the different causes under the assumption of a cumulative exposure model. The exact distributions of the MLEs of the parameters are then derived through the use of conditional moment generating functions. Using these exact distributions as well as the asymptotic distributions and the parametric bootstrap method, we discuss the construction of confidence intervals for the parameters and then assess their performance through Monte Carlo simulations. / Thesis / Doctor of Philosophy (PhD)