On the Efficient Simulation of the Left-Tail of the Sum of Correlated Log-normal Variates

The sum of Log-normal variates is encountered in many challenging applications such as in performance analysis of wireless communication systems and in financial engineering. Several approximation methods have been developed in the literature, the accuracy of which is not ensured in the tail regions. These regions are of primordial interest wherein small probability values have to be evaluated with high precision. Variance reduction techniques are known to yield accurate, yet efficient, estimates of small probability values. Most of the existing approaches, however, have considered the problem of estimating the right-tail of the sum of Log-normal random variables (RVS). In the present work, we consider instead the estimation of the left-tail of the sum of correlated Log-normal variates with Gaussian copula under a mild assumption on the covariance matrix. We propose an estimator combining an existing mean-shifting importance sampling approach with a control variate technique. The main result is that the proposed estimator has an asymptotically vanishing relative error which represents a major finding in the context of the left-tail simulation of the sum of Log-normal RVs. Finally, we assess by various simulation results the performances of the proposed estimator compared to existing estimators

Optimal calibration in immunoassay and inference on the coefficient of variation

This thesis examines and develops statistical methods for design and analysis with applications in immunoassay and other analytical techniques. In immunoassay, concentrations of components in clinical samples are measured using antibodies. The responses obtained are related to the concentrations in the samples. The relationship between response and concentration is established by fitting a calibration curve to responses of samples with known concentrations, called calibrators or standards. The concentrations in the clinical samples are estimated, through the calibration curve, by inverse prediction. The optimal choice of calibrator concentrations is dependent on the true relationship between response and concentration. A locally optimal design is conditioned on a given true relationship. This thesis presents a novel method that accounts for the variation in the true relationships by considering unconditional variances and expected values. For immunoassay, it is suggested that the average coefficient of variation in inverse predictions be minimised. In immunoassay, the coefficient of variation is the most common measure of variability. Several clinical samples or calibrators may share the same coefficient of variation, although they have different expected values. It is shown here that this phenomenon can be a consequence of a random variation in the dispensed volumes, and that inverse regression is appropriate when the random variation is in concentration rather than in response. An estimator of a common coefficient of variation that is shared by several clinical samples is proposed, and inferential methods are developed for common coefficients of variation in normally distributed data. These methods are based on McKay's chi-square approximation for the coefficient of variation. This study proves that McKay's approximation is noncentral beta distributed, and that it is asymptotically normal with mean n - 1 and variance slightly smaller than 2(n - 1)