Delta method in large deviations and moderate deviations for estimators

The delta method is a popular and elementary tool for deriving limiting distributions of transformed statistics, while applications of asymptotic distributions do not allow one to obtain desirable accuracy of approximation for tail probabilities. The large and moderate deviation theory can achieve this goal. Motivated by the delta method in weak convergence, a general delta method in large deviations is proposed. The new method can be widely applied to driving the moderate deviations of estimators and is illustrated by examples including the Wilcoxon statistic, the Kaplan--Meier estimator, the empirical quantile processes and the empirical copula function. We also improve the existing moderate deviations results for $M$-estimators and $L$-statistics by the new method. Some applications of moderate deviations to statistical hypothesis testing are provided.Comment: Published in at http://dx.doi.org/10.1214/10-AOS865 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Diode laser frequency stabilization onto an optical cavity

During this thesis work, a frequency stabilization system for an External Littrow Cavity Diode Laser (ECDL) at 370 nm has been set up and tested. The goal of the frequency stabilization is to achieve a long term frequency stability of less than ±50 kHz within 8 hours, which will be used for the single Ce ion detection project in the quantum information group. The system design is centered around a Fabry-Pérot (FP) cavity which is composed of two mirrors optically contacted onto the ends of a cylindrical spacer made of Ultra-Low Expansion (ULE) glass. To first order, the cavity spacer has a zero thermal expansion coefficient around a certain temperature. The method for achieving the required frequency stability is to actively stabilize the ECDL output frequency through controlling both the ECDL driving current and the grating position by a piezoelectric actuator. Pound-Drever-Hall (PDH) locking technique [1] is used to lock the laser frequency onto one of the resonance lines of the stable FP cavity. To be able to get the desired performance each segment of the system has to be set up correctly. The work include aligning the laser beam polarization, coupling laser into a single mode polarization maintaining fiber, setting up the radio frequency resonance tank used for the Electro-Optic Modulator (EOM), putting together the vacuum chamber where the FP cavity sits inside, installing the cavity spacer into the vacuum chamber, aligning the laser beam to match the cavity modes and designing the electronic filter circuits etc. Finally, after eight months of hard work, this laser could be locked around 2 hours and gave a good start for the future work. However the locking performance has not been characterized due to the shortness of time. Considering the time plan for this thesis, the improvement for a longer-time locking is remained

Asymptotic normality of nonparametric M-estimators with applications to hypothesis testing for panel count data

In semiparametric and nonparametric statistical inference, the asymptotic normality of estimators has been widely established when they are \sqrt{n} -consistent. In many applications, nonparametric estimators are not able to achieve this rate. We have a result on the asymptotic normality of nonparametric M - estimators that can be used if the rate of convergence of an estimator is n^{-\dfrac{1}{2}} or slower. We apply this to study the asymptotic distribution of sieve estimators of functionals of a mean function from a counting process, and develop nonparametric tests for the problem of treatment comparison with panel count data. The test statistics are constructed with spline likelihood estimators instead of nonparametric likelihood estimators. The new tests have a more general and simpler structure and are easy to implement. Simulation studies show that the proposed tests perform well even for small sample sizes. We find that a new test is always powerful for all the situations considered and is thus robust. For illustration, a data analysis example is provided

New multi-sample nonparametric tests for panel count data

This paper considers the problem of multi-sample nonparametric comparison of counting processes with panel count data, which arise naturally when recurrent events are considered. Such data frequently occur in medical follow-up studies and reliability experiments, for example. For the problem considered, we construct two new classes of nonparametric test statistics based on the accumulated weighted differences between the rates of increase of the estimated mean functions of the counting processes over observation times, wherein the nonparametric maximum likelihood approach is used to estimate the mean function instead of the nonparametric maximum pseudo-likelihood. The asymptotic distributions of the proposed statistics are derived and their finite-sample properties are examined through Monte Carlo simulations. The simulation results show that the proposed methods work quite well and are more powerful than the existing test procedures. Two real data sets are analyzed and presented as illustrative examples.Comment: Published in at http://dx.doi.org/10.1214/08-AOS599 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Triple positive solutions for semipositone fractional differential equations m-point boundary value problems with singularities and p–q-order derivatives

In this paper, by means of Leggett–Williams and Guo–Krasnosel'skii fixed point theorems, together with height functions of the nonlinearity on different bounded sets, triple positive solutions are obtained for some fractional differential equations with p–q-order derivatives involved in multi-point boundary value conditions. The nonlinearity may not only take negative infinity but also may permit singularities on both the time and the space variables

Kernel meets sieve: transformed hazards models with sparse longitudinal covariates

We study the transformed hazards model with time-dependent covariates observed intermittently for the censored outcome. Existing work assumes the availability of the whole trajectory of the time-dependent covariates, which is unrealistic. We propose to combine kernel-weighted log-likelihood and sieve maximum log-likelihood estimation to conduct statistical inference. The method is robust and easy to implement. We establish the asymptotic properties of the proposed estimator and contribute to a rigorous theoretical framework for general kernel-weighted sieve M-estimators. Numerical studies corroborate our theoretical results and show that the proposed method performs favorably over existing methods. Applying to a COVID-19 study in Wuhan illustrates the practical utility of our method

Robust anomalous Hall effect in ferromagnetic metal under high pressure

Recently, the giant intrinsic anomalous Hall effect (AHE) has been observed in the materials with kagome lattice. In this study, we systematically investigate the influence of high pressure on the AHE in the ferromagnet LiMn6Sn6 with clean Mn kagome lattice. Our in-situ high-pressure Raman spectroscopy indicates that the crystal structure of LiMn6Sn6 maintains a hexagonal phase under high pressures up to 8.51 GPa. The anomalous Hall conductivity (AHC) {\sigma}xyA remains around 150 {\Omega}-1 cm-1, dominated by the intrinsic mechanism. Combined with theoretical calculations, our results indicate that the stable AHE under pressure in LiMn6Sn6 originates from the robust electronic and magnetic structure.Comment: 11 pages 5 figure

Statistical analysis of panel count data

Department of Applied Mathematic