This thesis is formed of three manuscripts (chapters) about variance estimation. Each of the chapters focuses on developing new original variance estimators. The Chapter 1 proposes a novel jackknife variance estimator for self weighted two-stage sampling. Customary jackknifes for these designs rely only on the first sampling stage. This omission may induce a bias in the variance estimation when cluster sizes vary, second stage sampling fractions are small or when there is low variability between clusters. The proposed jackknife accounts of all sampling stages via deletion of clusters and observations within clusters. It does not need join-inclusion probabilities and naturally includes finite population corrections. Its asymptotic design-consistency is shown. A simulation study show that it can be more accurate than the customary jackknife used for this kind of sampling designs (Rao, Wu and Yue, 1992). The Chapter 2 proposes a totally new replication variance estimator for any unequal-probability without-replacement sampling design. The proposed replication estimator is approximately equal to the linearisation variance estimators obtained by the Demnati and Rao (2004) approach. It is more general than the Campbell (1980); Berger and Skinner (2005) generalised jackknife. Its asymptotic design consistency is shown. A simulation study shows it is more stable than standard jackknifes (Quenouille, 1956; Tukey, 1958) with ad hoc finite population corrections and than the generalised jackknife (Campbell, 1980; Berger and Skinner, 2005). The Chapter 3 proposes a new variance estimator which accounts the item non-response under unequal-probability without-replacement sampling when estimating a change from rotating (overlapping) repeated surveys. The proposed estimator combines the original approach by Berger and Priam (2010, 2012) and the non response reverse approach for variance estimation (Fay, 1991; Shao and Steel, 1999). It gives design-consistent estimation of the variance of change when the sampling fraction is small. The proposed estimator uses random Hot-deck imputation, but it can be implemented with other imputation techniques. Further, there are two more complementary chapters. One introduces the R package called samplingVarEst which implements of some methods for variance estimation utilised for the simulations. Finally, there is a brief chapter which discusses future research work.

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Chapter 1 is a non technical introduction to the thesis. In chapter 2, Basics of Large Deviation Theory, we illustrate the basic idea of large devi- ation theory and brie‡y review the history of its development. As a preparation, some of the important theorems which we will employ in the following chapters are also introduced. In chapter 3, Asymptotic Optimality of Empirical Likelihood Tests With Weakly Dependent Data, we extend the result of Kitamura (2001) to stationary mixing data. The key thing in proving the large deviation optimality is that the empirical measure of the independently and identically distributed data will obey the large deviation principal (LDP) with rate function equal to the relative entropy, but in general the large deviation performance of empirical measure of dependent data is complicated. In this chapter we add S-mixing condition to the stationary process and we show that the rate function of the LDP of S-mixing process is indeed equal to the relative entropy, and then asymptotic optimality follows from the large deviation inequality. In chapter 4, Large Deviations of Empirical Likelihood with Nuisance Parameters, we discuss the asymptotic e¢ ciency of empirical likelihood in the presence of nuisance parameters combined with augmented moment conditions. We show that in the presence of nuisance parameters, the asymptotic e¢ ciency of the empirical likelihood estimator of the parameter of interest will increase by adding more moment conditions, in the sense of the positive semide…niteness of the di¤erence of information matrices. As a by-product, we point out a necessary condition for the asymptotic e¢ ciency to be increased when more moment condition are added. We also derive asymptotic lower bounds of the minimax risk functions for the estimator of the parameter of interest, and we show that the empirical likelihood estimator can achieve this bound. In chapter 5, Empirical Likelihood Estimation of Auction Models via Simulated Moment Conditions, we apply empirical likelihood estimation to the simplest …rst-price sealed bid auc- tion model with independent private values. Through estimation of the parameter in the distri- bution function of bidders’private values we consider a potential problem in the EL inference when the moment condition is not in an explicit form and hard to compute, or even not con- tinuous in the parameter of interest. We deal with this issue following the method of simulated moment through importance sampling. We demonstrate the convergence of the empirical likeli- hood estimator from the simulated moment condition, and found that the asymptotic variance is larger than usual which is disturbed by simulation

This three-paper thesis explores the important relationship between arbitrage and price efficiency. Chapter 3 investigates the risk-bearing capacity of arbitrageurs under varying degrees and types of risk. A novel stochastic process is introduced to the literature that is capable of jointly capturing fundamental risk factors which are absent from extant specifications. Using stochastic optimal control theory, the degree to which arbitrageurs' investment behaviour is affected by aversion to these risks is analytically characterized, as well as conditions under which arbitrageurs cut losses, effectively exacerbating pricing disequilibria. Chapter 4 explores the role of arbitrage in enforcing price parity between cross-listed securities. This work employs an overlooked mechanism by which arbitrage can maintain parity, namely pairs-trading, which is cheaper to implement than the mechanism most commonly employed in the literature on cross-listed securities. This work shows that arbitrage is successful at enforcing parity between cross-listed securities, and also documents the main limits to arbitrage in this market setting. Chapter 5 examines the extent to which arbitrage contributes to the flow of information across markets. It is shown that microscopic lead/lag relationships of the order of a few hundred milliseconds exist across three major international index futures. Importantly, these delays last long enough, and induce pricing anomalies large enough, to compensate arbitrageurs for appropriating pricing disequilibria. These results accord with the view that temporary disequilibria incentivise arbitrageurs to correct pricing anomalies.

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Nonresponse in sample surveys has been increasing over the years. This thesis covers that issue in two main parts. The first part is concerned with how to use observed data to make inference about regression coefficients in a linear regression model of cluster-level variables when some of the response variable data is missing. A naive approach estimates the regression coeffcients without considering nonresponse. We propose new methods for estimating coeffcients which incorporate information on nonresponse at the cluster level. We also extend Heckman estimators to our clustered model. The Workplace Employment Relations Survey (WERS) 2004 data and data from a prepared simulation study are used to compare the new methods with the naive approach. In the second part the generalized regression estimator (GREG) for two-stage sampling will be considered. We propose new optimum GREG estimators for stratified two-stage sampling and a simulation study is used in order to assess the performance of the new estimators.

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Since 2001, there has been a consistent year-on-year increase in the period total fertility rate for England and Wales. At the same time migration to England and Wales has accelerated from the late 1990s. It is possible that the large number of migrants of childbearing ages moving to England and Wales, larger family size norms among foreign born women and a birth timing effect among recent migrants to England and Wales have led to the increase in the total fertility rate. However, the relative influence of any timing effect among recent migrants on the total fertility rate is not known. Research on migrant fertility in France (Toulemon, 2004) and Sweden (Andersson, 2004) has identified elevated fertility among migrants in the time period immediately after the migration event. Conversely, research in England and Wales has focused on period measures of fertility rather than estimating whether there is an elevated level of fertility among the large number of recent migrants to England and Wales. The first aim of this thesis is to accurately account for non-continually resident members of the Office for National Statistics (ONS) Longitudinal Study (LS) between census dates and use these LS members in fertility analysis. The second aim of this thesis is to investigate whether migrants to England and Wales show an elevated level of fertility after migration. It is only possible to estimate the fertility of recent migrants provided the sample exposed to risk of giving birth can be identified.