Kernel conditional quantile estimation via reduction revisited

Quantile regression refers to the process of estimating the quantiles of a conditional distribution and has many important applications within econometrics and data mining, among other domains. In this paper, we show how to estimate these conditional quantile functions within a Bayes risk minimization framework using a Gaussian process prior. The resulting non-parametric probabilistic model is easy to implement and allows non-crossing quantile functions to be enforced. Moreover, it can directly be used in combination with tools and extensions of standard Gaussian Processes such as principled hyperparameter estimation, sparsification, and quantile regression with input-dependent noise rates. No existing approach enjoys all of these desirable properties. Experiments on benchmark datasets show that our method is competitive with state-of-the-art approaches. © 2009 IEEE.

Nonparametric Estimation of Data Dimensionality Prior to Data Compression: the case of the Human Development Index

In many applications in applied statistics researchers reduce the complexity of a data set by combining a group of variables into a single measure using factor analysis or an index number. We argue that such compression loses information if the data actually has high dimensionality. We advocate the use of a non-parametric estimator, commonly used in physics (the Takens estimator), to estimate the correlation dimension of the data prior to compression. The advantage of this approach over traditional linear data compression approaches is that the data does not have to be linearized. Applying our ideas to the United Nations Human Development Index we find that the four variables that are used in its construction have dimension three and the index loses information.

The reasons behind the progression in PISA scores: An education production function approach using semi-parametric techniques

Research Masters

Estimation utilisant les polynômes de Bernstein

Ce mémoire porte sur la présentation des estimateurs de Bernstein qui sont des alternatives récentes aux différents estimateurs classiques de fonctions de répartition et de densité. Plus précisément, nous étudions leurs différentes propriétés et les comparons à celles de la fonction de répartition empirique et à celles de l'estimateur par la méthode du noyau. Nous déterminons une expression asymptotique des deux premiers moments de l'estimateur de Bernstein pour la fonction de répartition. Comme pour les estimateurs classiques, nous montrons que cet estimateur vérifie la propriété de Chung-Smirnov sous certaines conditions. Nous montrons ensuite que l'estimateur de Bernstein est meilleur que la fonction de répartition empirique en terme d'erreur quadratique moyenne. En s'intéressant au comportement asymptotique des estimateurs de Bernstein, pour un choix convenable du degré du polynôme, nous montrons que ces estimateurs sont asymptotiquement normaux. Des études numériques sur quelques distributions classiques nous permettent de confirmer que les estimateurs de Bernstein peuvent être préférables aux estimateurs classiques.

Identification and estimation of interventions using changes in inequality measures

This paper presents semiparametric estimators of changes in inequality measures of a dependent variable distribution taking into account the possible changes on the distributions of covariates. When we do not impose parametric assumptions on the conditional distribution of the dependent variable given covariates, this problem becomes equivalent to estimation of distributional impacts of interventions (treatment) when selection to the program is based on observable characteristics. The distributional impacts of a treatment will be calculated as differences in inequality measures of the potential outcomes of receiving and not receiving the treatment. These differences are called here Inequality Treatment Effects (ITE). The estimation procedure involves a first non-parametric step in which the probability of receiving treatment given covariates, the propensity-score, is estimated. Using the inverse probability weighting method to estimate parameters of the marginal distribution of potential outcomes, in the second step weighted sample versions of inequality measures are computed. Root-N consistency, asymptotic normality and semiparametric efficiency are shown for the semiparametric estimators proposed. A Monte Carlo exercise is performed to investigate the behavior in finite samples of the estimator derived in the paper. We also apply our method to the evaluation of a job training program.

Wavelet estimation in diffusions with periodicity

Wir betrachten einen zeitlich inhomogenen Diffusionsprozess, der durch eine stochastische Differentialgleichung gegeben wird, deren Driftterm ein deterministisches T-periodisches Signal beinhaltet, dessen Periodizität bekannt ist. Dieses Signal sei in einem Besovraum enthalten. Wir schätzen es mit Hilfe eines nichtparametrischen Waveletschätzers. Unser Schätzer ist von einem Wavelet-Dichteschätzer mit Thresholding inspiriert, der 1996 in einem klassischen iid-Modell von Donoho, Johnstone, Kerkyacharian und Picard konstruiert wurde. Unter gewissen Ergodizitätsvoraussetzungen an den Prozess können wir nichtparametrische Konvergenzraten angegeben, die bis auf einen logarithmischen Term den Raten im klassischen iid-Fall entsprechen. Diese Raten werden mit Hilfe von Orakel-Ungleichungen gezeigt, die auf Ergebnissen über Markovketten in diskreter Zeit von Clémencon, 2001, beruhen. Außerdem betrachten wir einen technisch einfacheren Spezialfall und zeigen einige Computersimulationen dieses Schätzers.

The accuracy of parameter estimation from noisy data, with application to resonance peak estimation in distributed Brillouin sensing

Distributed Brillouin sensing of strain and temperature works by making spatially resolved measurements of the position of the measurand-dependent extremum of the resonance curve associated with the scattering process in the weakly nonlinear regime. Typically, measurements of backscattered Stokes intensity (the dependent variable) are made at a number of predetermined fixed frequencies covering the design measurand range of the apparatus and combined to yield an estimate of the position of the extremum. The measurand can then be found because its relationship to the position of the extremum is assumed known. We present analytical expressions relating the relative error in the extremum position to experimental errors in the dependent variable. This is done for two cases: (i) a simple non-parametric estimate of the mean based on moments and (ii) the case in which a least squares technique is used to fit a Lorentzian to the data. The question of statistical bias in the estimates is discussed and in the second case we go further and present for the first time a general method by which the probability density function (PDF) of errors in the fitted parameters can be obtained in closed form in terms of the PDFs of the errors in the noisy data.

The accuracy of parameter estimation from noisy data, with application to resonance peak estimation in distributed Brillouin sensing

Distributed Brillouin sensing of strain and temperature works by making spatially resolved measurements of the position of the measurand-dependent extremum of the resonance curve associated with the scattering process in the weakly nonlinear regime. Typically, measurements of backscattered Stokes intensity (the dependent variable) are made at a number of predetermined fixed frequencies covering the design measurand range of the apparatus and combined to yield an estimate of the position of the extremum. The measurand can then be found because its relationship to the position of the extremum is assumed known. We present analytical expressions relating the relative error in the extremum position to experimental errors in the dependent variable. This is done for two cases: (i) a simple non-parametric estimate of the mean based on moments and (ii) the case in which a least squares technique is used to fit a Lorentzian to the data. The question of statistical bias in the estimates is discussed and in the second case we go further and present for the first time a general method by which the probability density function (PDF) of errors in the fitted parameters can be obtained in closed form in terms of the PDFs of the errors in the noisy data.

Estimation du paramètre d'une copule archimedienne en présence de censure dépendante

Les méthodes classiques d’analyse de survie notamment la méthode non paramétrique de Kaplan et Meier (1958) supposent l’indépendance entre les variables d’intérêt et de censure. Mais, cette hypothèse d’indépendance n’étant pas toujours soutenable, plusieurs auteurs ont élaboré des méthodes pour prendre en compte la dépendance. La plupart de ces méthodes émettent des hypothèses sur cette dépendance. Dans ce mémoire, nous avons proposé une méthode d’estimation de la dépendance en présence de censure dépendante qui utilise le copula-graphic estimator pour les copules archimédiennes (Rivest etWells, 2001) et suppose la connaissance de la distribution de la variable de censure. Nous avons ensuite étudié la consistance de cet estimateur à travers des simulations avant de l’appliquer sur un jeu de données réelles.

Forecasting time-varying value-at-risk

In this thesis we are interested in financial risk and the instrument we want to use is Value-at-Risk (VaR). VaR is the maximum loss over a given period of time at a given confidence level. Many definitions of VaR exist and some will be introduced throughout this thesis. There two main ways to measure risk and VaR: through volatility and through percentiles. Large volatility in financial returns implies greater probability of large losses, but also larger probability of large profits. Percentiles describe tail behaviour. The estimation of VaR is a complex task. It is important to know the main characteristics of financial data to choose the best model. The existing literature is very wide, maybe controversial, but helpful in drawing a picture of the problem. It is commonly recognised that financial data are characterised by heavy tails, time-varying volatility, asymmetric response to bad and good news, and skewness. Ignoring any of these features can lead to underestimating VaR with a possible ultimate consequence being the default of the protagonist (firm, bank or investor). In recent years, skewness has attracted special attention. An open problem is the detection and modelling of time-varying skewness. Is skewness constant or there is some significant variability which in turn can affect the estimation of VaR? This thesis aims to answer this question and to open the way to a new approach to model simultaneously time-varying volatility (conditional variance) and skewness. The new tools are modifications of the Generalised Lambda Distributions (GLDs). They are four-parameter distributions, which allow the first four moments to be modelled nearly independently: in particular we are interested in what we will call para-moments, i.e., mean, variance, skewness and kurtosis. The GLDs will be used in two different ways. Firstly, semi-parametrically, we consider a moving window to estimate the parameters and calculate the percentiles of the GLDs. Secondly, parametrically, we attempt to extend the GLDs to include time-varying dependence in the parameters. We used the local linear regression to estimate semi-parametrically conditional mean and conditional variance. The method is not efficient enough to capture all the dependence structure in the three indices —ASX 200, S&P 500 and FT 30—, however it provides an idea of the DGP underlying the process and helps choosing a good technique to model the data. We find that GLDs suggest that moments up to the fourth order do not always exist, there existence appears to vary over time. This is a very important finding, considering that past papers (see for example Bali et al., 2008; Hashmi and Tay, 2007; Lanne and Pentti, 2007) modelled time-varying skewness, implicitly assuming the existence of the third moment. However, the GLDs suggest that mean, variance, skewness and in general the conditional distribution vary over time, as already suggested by the existing literature. The GLDs give good results in estimating VaR on three real indices, ASX 200, S&P 500 and FT 30, with results very similar to the results provided by historical simulation.

Asset health prediction using condition indicators and operating environment indicators : a case study

Prognostics and asset life prediction is one of research potentials in engineering asset health management. We previously developed the Explicit Hazard Model (EHM) to effectively and explicitly predict asset life using three types of information: population characteristics; condition indicators; and operating environment indicators. We have formerly studied the application of both the semi-parametric EHM and non-parametric EHM to the survival probability estimation in the reliability field. The survival time in these models is dependent not only upon the age of the asset monitored, but also upon the condition and operating environment information obtained. This paper is a further study of the semi-parametric and non-parametric EHMs to the hazard and residual life prediction of a set of resistance elements. The resistance elements were used as corrosion sensors for measuring the atmospheric corrosion rate in a laboratory experiment. In this paper, the estimated hazard of the resistance element using the semi-parametric EHM and the non-parametric EHM is compared to the traditional Weibull model and the Aalen Linear Regression Model (ALRM), respectively. Due to assuming a Weibull distribution in the baseline hazard of the semi-parametric EHM, the estimated hazard using this model is compared to the traditional Weibull model. The estimated hazard using the non-parametric EHM is compared to ALRM which is a well-known non-parametric covariate-based hazard model. At last, the predicted residual life of the resistance element using both EHMs is compared to the actual life data.

Asset health prediction using the explicit hazard model

The ability to estimate the asset reliability and the probability of failure is critical to reducing maintenance costs, operation downtime, and safety hazards. Predicting the survival time and the probability of failure in future time is an indispensable requirement in prognostics and asset health management. In traditional reliability models, the lifetime of an asset is estimated using failure event data, alone; however, statistically sufficient failure event data are often difficult to attain in real-life situations due to poor data management, effective preventive maintenance, and the small population of identical assets in use. Condition indicators and operating environment indicators are two types of covariate data that are normally obtained in addition to failure event and suspended data. These data contain significant information about the state and health of an asset. Condition indicators reflect the level of degradation of assets while operating environment indicators accelerate or decelerate the lifetime of assets. When these data are available, an alternative approach to the traditional reliability analysis is the modelling of condition indicators and operating environment indicators and their failure-generating mechanisms using a covariate-based hazard model. The literature review indicates that a number of covariate-based hazard models have been developed. All of these existing covariate-based hazard models were developed based on the principle theory of the Proportional Hazard Model (PHM). However, most of these models have not attracted much attention in the field of machinery prognostics. Moreover, due to the prominence of PHM, attempts at developing alternative models, to some extent, have been stifled, although a number of alternative models to PHM have been suggested. The existing covariate-based hazard models neglect to fully utilise three types of asset health information (including failure event data (i.e. observed and/or suspended), condition data, and operating environment data) into a model to have more effective hazard and reliability predictions. In addition, current research shows that condition indicators and operating environment indicators have different characteristics and they are non-homogeneous covariate data. Condition indicators act as response variables (or dependent variables) whereas operating environment indicators act as explanatory variables (or independent variables). However, these non-homogenous covariate data were modelled in the same way for hazard prediction in the existing covariate-based hazard models. The related and yet more imperative question is how both of these indicators should be effectively modelled and integrated into the covariate-based hazard model. This work presents a new approach for addressing the aforementioned challenges. The new covariate-based hazard model, which termed as Explicit Hazard Model (EHM), explicitly and effectively incorporates all three available asset health information into the modelling of hazard and reliability predictions and also drives the relationship between actual asset health and condition measurements as well as operating environment measurements. The theoretical development of the model and its parameter estimation method are demonstrated in this work. EHM assumes that the baseline hazard is a function of the both time and condition indicators. Condition indicators provide information about the health condition of an asset; therefore they update and reform the baseline hazard of EHM according to the health state of asset at given time t. Some examples of condition indicators are the vibration of rotating machinery, the level of metal particles in engine oil analysis, and wear in a component, to name but a few. Operating environment indicators in this model are failure accelerators and/or decelerators that are included in the covariate function of EHM and may increase or decrease the value of the hazard from the baseline hazard. These indicators caused by the environment in which an asset operates, and that have not been explicitly identified by the condition indicators (e.g. Loads, environmental stresses, and other dynamically changing environment factors). While the effects of operating environment indicators could be nought in EHM; condition indicators could emerge because these indicators are observed and measured as long as an asset is operational and survived. EHM has several advantages over the existing covariate-based hazard models. One is this model utilises three different sources of asset health data (i.e. population characteristics, condition indicators, and operating environment indicators) to effectively predict hazard and reliability. Another is that EHM explicitly investigates the relationship between condition and operating environment indicators associated with the hazard of an asset. Furthermore, the proportionality assumption, which most of the covariate-based hazard models suffer from it, does not exist in EHM. According to the sample size of failure/suspension times, EHM is extended into two forms: semi-parametric and non-parametric. The semi-parametric EHM assumes a specified lifetime distribution (i.e. Weibull distribution) in the form of the baseline hazard. However, for more industry applications, due to sparse failure event data of assets, the analysis of such data often involves complex distributional shapes about which little is known. Therefore, to avoid the restrictive assumption of the semi-parametric EHM about assuming a specified lifetime distribution for failure event histories, the non-parametric EHM, which is a distribution free model, has been developed. The development of EHM into two forms is another merit of the model. A case study was conducted using laboratory experiment data to validate the practicality of the both semi-parametric and non-parametric EHMs. The performance of the newly-developed models is appraised using the comparison amongst the estimated results of these models and the other existing covariate-based hazard models. The comparison results demonstrated that both the semi-parametric and non-parametric EHMs outperform the existing covariate-based hazard models. Future research directions regarding to the new parameter estimation method in the case of time-dependent effects of covariates and missing data, application of EHM in both repairable and non-repairable systems using field data, and a decision support model in which linked to the estimated reliability results, are also identified.

Spatial variation of particle number concentration in school microscale environments and its impact on exposure assessment

It has not yet been established whether the spatial variation of particle number concentration (PNC) within a microscale environment can have an effect on exposure estimation results. In general, the degree of spatial variation within microscale environments remains unclear, since previous studies have only focused on spatial variation within macroscale environments. The aims of this study were to determine the spatial variation of PNC within microscale school environments, in order to assess the importance of the number of monitoring sites on exposure estimation. Furthermore, this paper aims to identify which parameters have the largest influence on spatial variation, as well as the relationship between those parameters and spatial variation. Air quality measurements were conducted for two consecutive weeks at each of the 25 schools across Brisbane, Australia. PNC was measured at three sites within the grounds of each school, along with the measurement of meteorological and several other air quality parameters. Traffic density was recorded for the busiest road adjacent to the school. Spatial variation at each school was quantified using coefficient of variation (CV). The portion of CV associated with instrument uncertainty was found to be 0.3 and therefore, CV was corrected so that only non-instrument uncertainty was analysed in the data. The median corrected CV (CVc) ranged from 0 to 0.35 across the schools, with 12 schools found to exhibit spatial variation. The study determined the number of required monitoring sites at schools with spatial variability and tested the deviation in exposure estimation arising from using only a single site. Nine schools required two measurement sites and three schools required three sites. Overall, the deviation in exposure estimation from using only one monitoring site was as much as one order of magnitude. The study also tested the association of spatial variation with wind speed/direction and traffic density, using partial correlation coefficients to identify sources of variation and non-parametric function estimation to quantify the level of variability. Traffic density and road to school wind direction were found to have a positive effect on CVc, and therefore, also on spatial variation. Wind speed was found to have a decreasing effect on spatial variation when it exceeded a threshold of 1.5 (m/s), while it had no effect below this threshold. Traffic density had a positive effect on spatial variation and its effect increased until it reached a density of 70 vehicles per five minutes, at which point its effect plateaued and did not increase further as a result of increasing traffic density.

The relationship between hip abductor muscle strength and magnitude of pelvic drop following a 3 week strengthening protocol in non-specific low back pain patients.

Purpose: To examine the relationship between hip abductor muscle (HABD) strength and the magnitude of pelvic drop (MPD) for patients with non-specific low back pain (NSLBP) and controls (CON) prior to and following a 3-week HABD strengthening protocol. At baseline, we hypothesized that NSLBP patients would exhibit reduced HABD strength and greater MPD compared to CON. Following the protocol, we hypothesized that strength would increase and MPD would decrease. Relevance: The Trendelenburg test (TT) is a common clinical test used to examine the ability of the HABD to maintain horizontal pelvic position during single limb stance. However, no study has specifically tested this theory. Moreover, no study has investigated the relationship between HABD strength and pelvic motion during walking or tested whether increased HABD strength would reduce the MPD. Methods: Quasi-experimental with 3-week exercise intervention. Fifteen NSLBP patients (32.5yrs,range 21-51yrs; VAS baseline: 5.3cm) and 10 CON (29.5yrs,range 22-47yrs) were recruited. Isometric HABD strength was measured using a force dynamometer and the average of three maximal voluntary contractions were normalized to body mass (N/kg). Two-dimensional MPD (degrees) was measured using a 60 Hz camera and was derived from two retroreflective-markers placed on the posterior superior iliac spines. MPD was measured while performing the static TT and while walking and averaged over 10 consecutive footfalls. NSLBP patients completed a 3-week HABD strengthening protocol consisting of 2 open-kinetic-chain exercises then all measures were repeated. Non-parametric analysis was used for group comparisons and correlation analysis. Results: At baseline, the NSLBP patients demonstrated 31% reduced HABD strength (mean=6.6 N/kg) compared to CON (mean=9.5 N/kg: p=0.03) and no significant differences in maximal pelvic frontal plane excursion while walking (NSLBP:mean=8.1°, CON:mean=7.1°: p=0.72). No significant correlations were measured between left HABD strength and right MPD (r=-0.37, p=0.11), or between right HABD strength and left MPD (r=-0.04, p=0.84) while performing the static TT. Following the 3-week strengthening protocol, NSLBP patients demonstrated a 12% improvement in strength (Post:mean=7.4 N/kg: p=0.02), a reduction in pain (VAS followup: 2.8cm), but no significant decreases in MPD while walking (p=0.92). Conclusions: NSLBP patients demonstrated reduced HABD strength at baseline and were able to increase strength and reduce pain in a 3-week period. However, despite increases in HABD strength, the NSLBP group exhibited similar MPD motion during the static TT and while walking compared to baseline and controls. Implications: The results suggest that the HABD alone may not be primarily responsible for controlling a horizontal pelvic position during static and dynamic conditions. Increasing the strength of the hip abductors resulted in a reduction of pain in NSLBP patients providing evidence for further research to identify specific musculature responsible for controlling pelvic motion.

The individual and interactive relationship between long chain omega-3 polyunsaturated fatty acids and physical activity as predictors of cognition in cognitively impaired and non-impaired older adults

Alterations in cognitive function are characteristic of the aging process in humans and other animals. However, the nature of these age related changes in cognition is complex and is likely to be influenced by interactions between genetic predispositions and environmental factors resulting in dynamic fluctuations within and between individuals. These inter and intra-individual fluctuations are evident in both so-called normal cognitive aging and at the onset of cognitive pathology. Mild Cognitive Impairment (MCI), thought to be a prodromal phase of dementia, represents perhaps the final opportunity to mitigate cognitive declines that may lead to terminal conditions such as dementia. The prognosis for people with MCI is mixed with the evidence suggesting that many will remain stable within 10-years of diagnosis, many will improve, and many will transition to dementia. If the characteristics of people who do not progress to dementia from MCI can be identified and replicated in others it may be possible to reduce or delay dementia onset, thus reducing a growing personal and public health burden. Furthermore, if MCI onset can be prevented or delayed, the burden of cognitive decline in aging populations worldwide may be reduced. A cognitive domain that is sensitive to the effects of advancing age, and declines in which have been shown to presage the onset of dementia in MCI patients, is executive function. Moreover, environmental factors such as diet and physical activity have been shown to affect performance on tests of executive function. For example, improvements in executive function have been demonstrated as a result of increased aerobic and anaerobic physical activity and, although the evidence is not as strong, findings from dietary interventions suggest certain nutrients may preserve or improve executive functions in old age. These encouraging findings have been demonstrated in older adults with MCI and their non-impaired peers. However, there are some gaps in the literature that need to be addressed. For example, little is known about the effect on cognition of an interaction between diet and physical activity. Both are important contributors to health and wellbeing, and a growing body of evidence attests to their importance in mental and cognitive health in aging individuals. Yet physical activity and diet are rarely considered together in the context of cognitive function. There is also little known about potential underlying biological mechanisms that might explain the physical activity/diet/cognition relationship. The first aim of this program of research was to examine the individual and interactive role of physical activity and diet, specifically long chain polyunsaturated fatty acid consumption(LCn3) as predictors of MCI status. The second aim is to examine executive function in MCI in the context of the individual and interactive effects of physical activity and LCn3.. A third aim was to explore the role of immune and endocrine system biomarkers as possible mediators in the relationship between LCn3, physical activity and cognition. Study 1a was a cross-sectional analysis of MCI status as a function of erythrocyte proportions of an interaction between physical activity and LCn3. The marine based LCn3s eicosapentaenoic acid (EPA) and docosahexaenoic acid (DHA) have both received support in the literature as having cognitive benefits, although comparisons of the relative benefits of EPA or DHA, particularly in relation to the aetiology of MCI, are rare. Furthermore, a limited amount of research has examined the cognitive benefits of physical activity in terms of MCI onset. No studies have examined the potential interactive benefits of physical activity and either EPA or DHA. Eighty-four male and female adults aged 65 to 87 years, 50 with MCI and 34 without, participated in Study 1a. A logistic binary regression was conducted with MCI status as a dependent variable, and the individual and interactive relationships between physical activity and either EPA or DHA as predictors. Physical activity was measured using a questionnaire and specific physical activity categories were weighted according to the metabolic equivalents (METs) of each activity to create a physical activity intensity index (PAI). A significant relationship was identified between MCI outcome and the interaction between the PAI and EPA; participants with a higher PAI and higher erythrocyte proportions of EPA were more likely to be classified as non-MCI than their less active peers with less EPA. Study 1b was a randomised control trial using the participants from Study 1a who were identified with MCI. Given the importance of executive function as a determinant of progression to more severe forms of cognitive impairment and dementia, Study 1b aimed to examine the individual and interactive effect of physical activity and supplementation with either EPA or DHA on executive function in a sample of older adults with MCI. Fifty male and female participants were randomly allocated to supplementation groups to receive 6-months of supplementation with EPA, or DHA, or linoleic acid (LA), a long chain polyunsaturated omega-6 fatty acid not known for its cognitive enhancing properties. Physical activity was measured using the PAI from Study 1a at baseline and follow-up. Executive function was measured using five tests thought to measure different executive function domains. Erythrocyte proportions of EPA and DHA were higher at follow-up; however, PAI was not significantly different. There was also a significant improvement in three of the five executive function tests at follow-up. However, regression analyses revealed that none of the variance in executive function at follow-up was predicted by EPA, DHA, PAI, the EPA by PAI interaction, or the DHA by PAI interaction. The absence of an effect may be due to a small sample resulting in limited power to find an effect, the lack of change in physical activity over time in terms of volume and/or intensity, or a combination of both reduced power and no change in physical activity. Study 2a was a cross-sectional study using cognitively unimpaired older adults to examine the individual and interactive effects of LCn3 and PAI on executive function. Several possible explanations for the absence of an effect were identified. From this consideration of alternative explanations it was hypothesised that post-onset interventions with LCn3 either alone or in interation with self-reported physical activity may not be beneficial in MCI. Thus executive function responses to the individual and interactive effects of physical activity and LCn3 were examined in a sample of older male and female adults without cognitive impairment (n = 50). A further aim of study 2a was to operationalise executive function using principal components analysis (PCA) of several executive function tests. This approach was used firstly as a data reduction technique to overcome the task impurity problem, and secondly to examine the executive function structure of the sample for evidence of de-differentiation. Two executive function components were identified as a result of the PCA (EF 1 and EF 2). However, EPA, DHA, the PAI, or the EPA by PAI or DHA by PAI interactions did not account for any variance in the executive function components in subsequent hierarchical multiple regressions. Study 2b was an exploratory correlational study designed to explore the possibility that immune and endocrine system biomarkers may act as mediators of the relationship between LCn3, PAI, the interaction between LCn3 and PAI, and executive functions. Insulin-like growth factor-1 (IGF-1), an endocrine system growth hormone, and interleukin-6 (IL-6) an immune system cytokine involved in the acute inflammatory response, have both been shown to affect cognition including executive functions. Moreover, IGF-1 and IL-6 have been shown to be antithetical in so far as chronically increased IL-6 has been associated with reduced IGF-1 levels, a relationship that has been linked to age related morbidity. Further, physical activity and LCn3 have been shown to modulate levels of both IGF-1 and IL-6. Thus, it is possible that the cognitive enhancing effects of LCn3, physical activity or their interaction are mediated by changes in the balance between IL-6 and IGF-1. Partial and non-parametric correlations were conducted in a subsample of participants from Study 2a (n = 13) to explore these relationships. Correlations of interest did not reach significance; however, the coefficients were quite large for several relationships suggesting studies with larger samples may be warranted. In summary, the current program of research found some evidence supporting an interaction between EPA, not DHA, and higher energy expenditure via physical activity in differentiating between older adults with and without MCI. However, a RCT examining executive function in older adults with MCI found no support for increasing EPA or DHA while maintaining current levels of energy expenditure. Furthermore, a cross-sectional study examining executive function in older adults without MCI found no support for better executive function performance as a function of increased EPA or DHA consumption, greater energy expenditure via physical activity or an interaction between physical activity and either EPA or DHA. Finally, an examination of endocrine and immune system biomarkers revealed promising relationships in terms of executive function in non-MCI older adults particularly with respect to LCn3 and physical activity. Taken together, these findings demonstrate a potential benefit of increasing physical activity and LCn3 consumption, particularly EPA, in mitigating the risk of developing MCI. In contrast, no support was found for a benefit to executive function as a result of increased physical activity, LCn3 consumption or an interaction between physical activity and LCn3, in participants with and without MCI. These results are discussed with reference to previous findings in the literature including possible limitations and opportunities for future research.