BIAS-CORRECTED MAXIMUM LIKELIHOOD ESTIMATION OF THE PARAMETERS OF THE WEIGHTED LINDLEY DISTRIBUTION

This report discusses the calculation of analytic second-order bias techniques for the maximum likelihood estimates (for short, MLEs) of the unknown parameters of the distribution in quality and reliability analysis. It is well-known that the MLEs are widely used to estimate the unknown parameters of the probability distributions due to their various desirable properties; for example, the MLEs are asymptotically unbiased, consistent, and asymptotically normal. However, many of these properties depend on an extremely large sample sizes. Those properties, such as unbiasedness, may not be valid for small or even moderate sample sizes, which are more practical in real data applications. Therefore, some bias-corrected techniques for the MLEs are desired in practice, especially when the sample size is small. Two commonly used popular techniques to reduce the bias of the MLEs, are ‘preventive’ and ‘corrective’ approaches. They both can reduce the bias of the MLEs to order O(n−2), whereas the ‘preventive’ approach does not have an explicit closed form expression. Consequently, we mainly focus on the ‘corrective’ approach in this report. To illustrate the importance of the bias-correction in practice, we apply the bias-corrected method to two popular lifetime distributions: the inverse Lindley distribution and the weighted Lindley distribution. Numerical studies based on the two distributions show that the considered bias-corrected technique is highly recommended over other commonly used estimators without bias-correction. Therefore, special attention should be paid when we estimate the unknown parameters of the probability distributions under the scenario in which the sample size is small or moderate.

Hilbert Space Valued Signal Plus Noise Models: Analysis of Structural Breaks under High Dimensionality and Temporal Dependence

This thesis is concerned with change point analysis for time series, i.e. with detection of structural breaks in time-ordered, random data. This long-standing research field regained popularity over the last few years and is still undergoing, as statistical analysis in general, a transformation to high-dimensional problems. We focus on the fundamental »change in the mean« problem and provide extensions of the classical non-parametric Darling-Erdős-type cumulative sum (CUSUM) testing and estimation theory within highdimensional Hilbert space settings. In the first part we contribute to (long run) principal component based testing methods for Hilbert space valued time series under a rather broad (abrupt, epidemic, gradual, multiple) change setting and under dependence. For the dependence structure we consider either traditional m-dependence assumptions or more recently developed m-approximability conditions which cover, e.g., MA, AR and ARCH models. We derive Gumbel and Brownian bridge type approximations of the distribution of the test statistic under the null hypothesis of no change and consistency conditions under the alternative. A new formulation of the test statistic using projections on subspaces allows us to simplify the standard proof techniques and to weaken common assumptions on the covariance structure. Furthermore, we propose to adjust the principal components by an implicit estimation of a (possible) change direction. This approach adds flexibility to projection based methods, weakens typical technical conditions and provides better consistency properties under the alternative. In the second part we contribute to estimation methods for common changes in the means of panels of Hilbert space valued time series. We analyze weighted CUSUM estimates within a recently proposed »high-dimensional low sample size (HDLSS)« framework, where the sample size is fixed but the number of panels increases. We derive sharp conditions on »pointwise asymptotic accuracy« or »uniform asymptotic accuracy« of those estimates in terms of the weighting function. Particularly, we prove that a covariance-based correction of Darling-Erdős-type CUSUM estimates is required to guarantee uniform asymptotic accuracy under moderate dependence conditions within panels and that these conditions are fulfilled, e.g., by any MA(1) time series. As a counterexample we show that for AR(1) time series, close to the non-stationary case, the dependence is too strong and uniform asymptotic accuracy cannot be ensured. Finally, we conduct simulations to demonstrate that our results are practically applicable and that our methodological suggestions are advantageous.

A new approach for computing a flood vulnerability index using cluster analysis

A Flood Vulnerability Index (FloodVI) was developed using Principal Component Analysis (PCA) and a new aggregation method based on Cluster Analysis (CA). PCA simplifies a large number of variables into a few uncorrelated factors representing the social, economic, physical and environmental dimensions of vulnerability. CA groups areas that have the same characteristics in terms of vulnerability into vulnerability classes. The grouping of the areas determines their classification contrary to other aggregation methods in which the areas' classification determines their grouping. While other aggregation methods distribute the areas into classes, in an artificial manner, by imposing a certain probability for an area to belong to a certain class, as determined by the assumption that the aggregation measure used is normally distributed, CA does not constrain the distribution of the areas by the classes. FloodVI was designed at the neighbourhood level and was applied to the Portuguese municipality of Vila Nova de Gaia where several flood events have taken place in the recent past. The FloodVI sensitivity was assessed using three different aggregation methods: the sum of component scores, the first component score and the weighted sum of component scores. The results highlight the sensitivity of the FloodVI to different aggregation methods. Both sum of component scores and weighted sum of component scores have shown similar results. The first component score aggregation method classifies almost all areas as having medium vulnerability and finally the results obtained using the CA show a distinct differentiation of the vulnerability where hot spots can be clearly identified. The information provided by records of previous flood events corroborate the results obtained with CA, because the inundated areas with greater damages are those that are identified as high and very high vulnerability areas by CA. This supports the fact that CA provides a reliable FloodVI.

Raman microscopy of formamide-intercalated kaolinites treated by controlled-rate thermal analysis technology

Raman spectroscopy of formamide-intercalated kaolinites treated using controlled-rate thermal analysis technology (CRTA), allowing the separation of adsorbed formamide from intercalated formamide in formamide-intercalated kaolinites, is reported. The Raman spectra of the CRTA-treated formamide-intercalated kaolinites are significantly different from those of the intercalated kaolinites, which display a combination of both intercalated and adsorbed formamide. An intense band is observed at 3629 cm-1, attributed to the inner surface hydroxyls hydrogen bonded to the formamide. Broad bands are observed at 3600 and 3639 cm-1, assigned to the inner surface hydroxyls, which are hydrogen bonded to the adsorbed water molecules. The hydroxyl-stretching band of the inner hydroxyl is observed at 3621 cm-1 in the Raman spectra of the CRTA-treated formamide-intercalated kaolinites. The results of thermal analysis show that the amount of intercalated formamide between the kaolinite layers is independent of the presence of water. Significant differences are observed in the CO stretching region between the adsorbed and intercalated formamide.

Detailed analysis of a conservative difference approximation for the time fractional diffusion equation

Diffusion equations that use time fractional derivatives are attractive because they describe a wealth of problems involving non-Markovian Random walks. The time fractional diffusion equation (TFDE) is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α ∈ (0, 1). Developing numerical methods for solving fractional partial differential equations is a new research field and the theoretical analysis of the numerical methods associated with them is not fully developed. In this paper an explicit conservative difference approximation (ECDA) for TFDE is proposed. We give a detailed analysis for this ECDA and generate discrete models of random walk suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation. The stability and convergence of the ECDA for TFDE in a bounded domain are discussed. Finally, some numerical examples are presented to show the application of the present technique.

Determination of preventive maintenance lead time using hybrid analysis

The time for conducting Preventive Maintenance (PM) on an asset is often determined using a predefined alarm limit based on trends of a hazard function. In this paper, the authors propose using both hazard and reliability functions to improve the accuracy of the prediction particularly when the failure characteristic of the asset whole life is modelled using different failure distributions for the different stages of the life of the asset. The proposed method is validated using simulations and case studies.

Hydrazine-hydrate intercalated halloysite under controlled-rate thermal analysis conditions

The thermal behaviour of halloysite fully expanded with hydrazine-hydrate has been investigated in nitrogen atmosphere under dynamic heating and at a constant, pre-set decomposition rate of 0.15 mg min-1. Under controlled-rate thermal analysis (CRTA) conditions it was possible to resolve the closely overlapping decomposition stages and to distinguish between adsorbed and bonded reagent. Three types of bonded reagent could be identified. The loosely bonded reagent amounting to 0.20 mol hydrazine-hydrate per mol inner surface hydroxyl is connected to the internal and external surfaces of the expanded mineral and is present as a space filler between the sheets of the delaminated mineral. The strongly bonded (intercalated) hydrazine-hydrate is connected to the kaolinite inner surface OH groups by the formation of hydrogen bonds. Based on the thermoanalytical results two different types of bonded reagent could be distinguished in the complex. Type 1 reagent (approx. 0.06 mol hydrazine-hydrate/mol inner surface OH) is liberated between 77 and 103°C. Type 2 reagent is lost between 103 and 227°C, corresponding to a quantity of 0.36 mol hydrazine/mol inner surface OH. When heating the complex to 77°C under CRTA conditions a new reflection appears in the XRD pattern with a d-value of 9.6 Å, in addition to the 10.2 Ĺ reflection. This new reflection disappears in contact with moist air and the complex re-expands to the original d-value of 10.2 Å in a few h. The appearance of the 9.6 Å reflection is interpreted as the expansion of kaolinite with hydrazine alone, while the 10.2 Å one is due to expansion with hydrazine-hydrate. FTIR (DRIFT) spectroscopic results showed that the treated mineral after intercalation/deintercalation and heat treatment to 300°C is slightly more ordered than the original (untreated) clay.