title: Using finite mixtures to robustify statistical models

authors: Rohan, Maheswaran

subjects: Robust Statistics

abstract: This thesis is concerned with robust estimation of the parameters of statistical models. Although robust estimation is a very good idea, it has some shortcomings when seen from the statistical modelling point of view. For example, there are no easily applicable principles for creating robust estimates in new situations. In this thesis, we are trying to introduce a unified method to obtain the robustified statistics for various situations such as the linear model and the generalized linear model. We wish to modify the maximum likelihood estimation procedure, which is very sensitive to the outliers. In order to reduce the effect on these estimates by outliers, we add an additional component, which would be of no interest but would contain all outliers, to the regular component forming a finite mixture. In fact, we use the finite mixture model to obtain a ``robustified'' estimate for a model parameter $\theta$, where the finite mixture form is being used as a mathematical tool to have a tractable form of analysis rather than being used as a serious model for the data. We employ the EM algorithm to obtain the our proposed robustified estimates for the parameters. Our estimates are compared with some other estimates defined in the robust statistics literature. This thesis examines the robustness of the proposed estimates using the concept of influence function. The estimates are defined iteratively, so that the implicit differentiation method of Jorgensen is used to obtain the influence functions of the estimates. We give example plots of these influence functions which are bounded. In this thesis, we give mathematical results for all cases and we use well known real data sets to investigate our method. The statistical software R is used for all investigation. Finally, we hope that this method may give a unified approach for making parameter estimation in statistical models more robust.

text: Using finite mixtures to robustify statistical models This thesis is concerned with robust estimation of the parameters of statistical models. Although robust estimation is a very good idea, it has some shortcomings when seen from the statistical modelling point of view. For example, there are no easily applicable principles for creating robust estimates in new situations. In this thesis, we are trying to introduce a unified method to obtain the robustified statistics for various situations such as the linear model and the generalized linear model. We wish to modify the maximum likelihood estimation procedure, which is very sensitive to the outliers. In order to reduce the effect on these estimates by outliers, we add an additional component, which would be of no interest but would contain all outliers, to the regular component forming a finite mixture. In fact, we use the finite mixture model to obtain a ``robustified'' estimate for a model parameter $\theta$, where the finite mixture form is being used as a mathematical tool to have a tractable form of analysis rather than being used as a serious model for the data. We employ the EM algorithm to obtain the our proposed robustified estimates for the parameters. Our estimates are compared with some other estimates defined in the robust statistics literature. This thesis examines the robustness of the proposed estimates using the concept of influence function. The estimates are defined iteratively, so that the implicit differentiation method of Jorgensen is used to obtain the influence functions of the estimates. We give example plots of these influence functions which are bounded. In this thesis, we give mathematical results for all cases and we use well known real data sets to investigate our method. The statistical software R is used for all investigation. Finally, we hope that this method may give a unified approach for making parameter estimation in statistical models more robust.

year: 2011

title: A conjecture of De Koninck regarding particular square values of the sum of divisors function

authors: Broughan, Kevin A.

subjects: Science & Technology

abstract: We study integers n > 1 satisfying the relation σ(n) = γ(n)², where σ(n) and γ(n) are the sum of divisors and the product of distinct primes dividing n, respectively. If the prime dividing a solution n is congruent to 3 modulo 8 then it must be greater than 41, and every solution is divisible by at least the fourth power of an odd prime. Moreover at least 2/5 of the exponents a of the primes dividing any solution have the property that a + 1 is a prime power. Lastly we prove that the number of solutions up to x > 1 is at most x¹/⁶⁺є, for any є > 0 and all x > xє.

text: A conjecture of De Koninck regarding particular square values of the sum of divisors function We study integers n > 1 satisfying the relation σ(n) = γ(n)², where σ(n) and γ(n) are the sum of divisors and the product of distinct primes dividing n, respectively. If the prime dividing a solution n is congruent to 3 modulo 8 then it must be greater than 41, and every solution is divisible by at least the fourth power of an odd prime. Moreover at least 2/5 of the exponents a of the primes dividing any solution have the property that a + 1 is a prime power. Lastly we prove that the number of solutions up to x > 1 is at most x¹/⁶⁺є, for any є > 0 and all x > xє.

year: 2014

title: The Effects of Contaminated Rena Sediments on Juvenile Paua (Haliotis iris)

authors: McSweeney, Caleb S.

subjects: contamiant

abstract: The grounding of the MV Rena on Otaiti resulted in the release of heavy fuel oil and container debris contaminants into the surrounding environments including the rocky shores of the adjacent Mōtītī Island. This is the habitat where significant populations of benthic paua reside. Paua (Haliotis iris) are a staple and consistent food source for Mōtītī Island. Being an offshore island with no amenities, Mōtītī Island residents are reliant on the ocean as a pataka kai (food cupboard) and are therefore acutely aware of environmental influences to the harvest of kaimoana.
This thesis aimed to address concerns relating to the effects of contaminated boundary layer water emanating from contaminated ‘Rena’ sediment on juvenile paua. Research was focused in two areas: 1) the sublethal behavioural effects of contaminated Rena sediment to Paua and 2) the accumulation of trace metals in the edible tissue and viscera mass. The experiments were carried out with the use of a close circuit aquaria in a laboratory environment, followed by a field experiment.
In all experiments, paua in control treatments were healthy by comparison to paua exposed to treatments with Rena contaminated sediments and copper as judged by survivorship and behaviour. The most likely cause of of behavioural aberrations and mortality observed was deemed to be copper as demonstrated by Diffusive Gradient in thin film (DGT) and ambient water analyses in both experiments. Copper that is bioavailable can increase quickly in the edible tissue and viscera mass as was identified as the visceral mass of Rena and copper exposed paua had a higher mean concentration of this and other trace metals.
On Otaiti, the effects to paua from the Rena ship wreck and lost container contents, know to include a medly of metals and other contaminants, is not likely to be limited to copper alone. Results demosntrate the relevance of examining the effects of water borne contaminated plumes emanating from complex mixtures of contaminants. This is rarely done in ecotoxicological studies which tend to focus on individual contaminant compounds.

text: The Effects of Contaminated Rena Sediments on Juvenile Paua (Haliotis iris) The grounding of the MV Rena on Otaiti resulted in the release of heavy fuel oil and container debris contaminants into the surrounding environments including the rocky shores of the adjacent Mōtītī Island. This is the habitat where significant populations of benthic paua reside. Paua (Haliotis iris) are a staple and consistent food source for Mōtītī Island. Being an offshore island with no amenities, Mōtītī Island residents are reliant on the ocean as a pataka kai (food cupboard) and are therefore acutely aware of environmental influences to the harvest of kaimoana.
This thesis aimed to address concerns relating to the effects of contaminated boundary layer water emanating from contaminated ‘Rena’ sediment on juvenile paua. Research was focused in two areas: 1) the sublethal behavioural effects of contaminated Rena sediment to Paua and 2) the accumulation of trace metals in the edible tissue and viscera mass. The experiments were carried out with the use of a close circuit aquaria in a laboratory environment, followed by a field experiment.
In all experiments, paua in control treatments were healthy by comparison to paua exposed to treatments with Rena contaminated sediments and copper as judged by survivorship and behaviour. The most likely cause of of behavioural aberrations and mortality observed was deemed to be copper as demonstrated by Diffusive Gradient in thin film (DGT) and ambient water analyses in both experiments. Copper that is bioavailable can increase quickly in the edible tissue and viscera mass as was identified as the visceral mass of Rena and copper exposed paua had a higher mean concentration of this and other trace metals.
On Otaiti, the effects to paua from the Rena ship wreck and lost container contents, know to include a medly of metals and other contaminants, is not likely to be limited to copper alone. Results demosntrate the relevance of examining the effects of water borne contaminated plumes emanating from complex mixtures of contaminants. This is rarely done in ecotoxicological studies which tend to focus on individual contaminant compounds.

year: 2015