On the convergence of the chi square and noncentral chi square distributions to the normal distribution

A simple and novel asymptotic bound for the maximum error resulting from the use of the central limit theorem to approximate the distribution of chi square and noncentral chi square random variables is derived. The bound enables the quick calculation of the number of degrees of freedom required to ensure a given approximation error, and is significantly tighter than bounds derived using the Berry-Esseen theorem. An application to widely-used approximations for the decision probabilities of energy detectors is also provided

Inequalities for noncentral chi-square distributions

An upper and lower bound are presented for the difference between the distribution functions of noncentral chi-square variables with the same degrees of freedom and different noncentralities. The inequalities are applied in a comparison of two approximations to the power of Pearson's chi-square test

스태거드 페르미온을 이용한 격자 양자색역학에서 파이온 붕괴 상수와 초표준모형 B 파라미터의 계산

학위논문 (박사)-- 서울대학교 대학원 : 물리·천문학부(물리학전공), 2013. 2. 이원종.In part I, we calculate the next-to-leading order corrections to pion decay constants for the taste non-Goldstone pions using staggered chiral perturbation theory. This is a generalization of the calculation for the taste Goldstone case. New low-energy couplings are limited to analytic corrections that vanish in the continuum limitthe chiral logarithms contain no new couplings. We report results for quenched, fully dynamical, and partially quenched cases of interest in the chiral SU(3) and SU(2) theories. The results can be used to refine existing determinations of decay constants and low energy constants. In part II, we calculate the beyond the standard model B-parameters using HYP-smeared improved staggered fermions on the MILC asqtad lattices with Nf = 2 + 1 flavors. We use three different lattice spacings (a ? 0.045, 0.06 and 0.09 fm) to obtain the continuum results. Operator matching is done using one-loop perturbative matching, and results are run to 2 and 3 GeV in the MS scheme. For the chiral and continuum extrapolations, we use SU(2) staggered chiral perturbation theory. We present preliminary results with only statistical errors. In part III, we give a detailed introduction to the data anlysis including basic probability theory, error anlalysis techniques and least chi-square fitting method. We also explain how to analyse highly correlated data by applying a number of prescriptions such as diagonal approximation, singular value decomposition (SVD) method and Bayesian method. We propose a brand new method, the eigenmode shift method which allows a full covariance fitting without modifying the covariance matrix.1. Introduction 1.1. Quantum chromodynamics 1.2. Lattice QCD 1.3. Recent progress of the lattice calculation 1.4. Summary of this thesis 1.4.1. Decay constants in staggered chiral perturbation theory 1.4.2. Kaon mixing matrix elements from BSM operators 1.4.3. Art of data analysis 2. QCD on the Lattice 2.1. Gluons on the lattice 2.2. Fermions on the lattice 2.2.1. Fermion doubling 2.2.2. Wilson fermions 2.2.3. Staggered fermions 3. Chiral Perturbation Theory 3.1. Introduction to chiral perturbation theory 3.1.1. Chiral Effective Lagrangian 3.2. Staggered chiral perturbation theory 3.2.1. Chiral Lagrangian for staggered quarks 3.2.2. Propagators 4. Decay Constants in Staggered Chiral Perturbation Theory 4.1. Chiral Lagrangian that contribute to the decay constants at NLO 4.2. Decay constants of flavor-charged pseudo-goldstone bosons 4.2.1. Wavefunction renormalization correction 4.2.2. Current correction 4.2.3. Next-to-leading order analytic contributions 4.3. Results 4.3.1. SU(3) chiral perturbation theory 4.3.1.1. Fully dynamical case 4.3.1.2. Partially quenched case 4.3.1.3. Quenched case 4.3.2. SU(2) chiral perturbation theory 4.3.2.1. Fully dynamical case 4.3.2.2. Partially quenched case 4.4. Conclusion 5. Introduction to the Kaon Mixing Matrix Elements from BSM Operators 5.1. Kaon mixing matrix elements from the Standard Model 5.2. Kaon mixing matrix elements from beyond the Standard Model 6. Numerical Study of Kaon Mixing Matrix Elements from BSM Operators 6.1. Computation of BSM B-parameters 6.2. SU(2) fitting 6.3. RG evolution 6.4. Continuum extrapolation 6.5. Conclusion 7. Basic Probability Theory 7.1. Mean and variance 7.1.1. Probability and probability distribution 7.1.2. Mean and variance 7.1.3. Sample mean and sample variance 7.1.4. Fundamental theorems of probability 7.2. Special distributions 7.2.1. Normal distribution 7.2.2. chi-square-distribution and noncentral chi-square-distribution 8. Error Analysis 8.1. Propagation of error 8.2. Resampling methods 8.2.1. Bootstrap method 8.2.2. Jackknife method 8.3. Calculating error of error 8.4. Dealing with Jackknife samples 8.4.1. From jackknife samples to original samples 8.4.2. From jackknife results to bootstrap results 9. Least chi-square Fitting 9.1. Theory of least chi-square fitting 9.1.1. Uncorrelated chi-square 9.1.2. Correlated chi-square 9.1.3. Quality of the fit 9.1.4. Uncertainty of fitting parameters 9.2. Constrained fitting 9.3. Finding fitting parameters 9.3.1. Fitting data to linear functions 9.3.2. Fitting data to nonlinear functions 10.Covariance Fitting of Highly Correlated Data 10.1. Trouble with correlated data fitting 10.2. Prescriptions 10.2.1. Diagonal approximation 10.2.2. Cutoff method 10.2.3. Eigenmode shift method 10.2.3.1. Equivalence of cutoff method and unconstrained ES method 10.2.4. Bayesian method 10.2.5. Probability distribution of minimized chi-square 10.2.5.1. Distribution of chi-square for the full covariance fitting 10.2.5.2. Distribution of chi-square for the cutoff method 10.2.5.3. Distribution of chi-square for the ES method 10.2.6. An example of fitting with random data 11.Multidimensional Function Minimizer 11.1. Amoeba method 11.2. Conjugate gradient algorithm 11.2.1. Calculation of α(i) 11.2.2. Calculation of β(i+1) 11.2.3. Convergence 11.2.4. Practical implementation 11.2.5. Variants 11.3. Function minimization using CG 11.3.1. Minimization of quadratic functions 11.3.2. Outline of minimization for general functions 11.3.3. Calculation of β(i+1) 11.3.4. Calculation of α(i) 11.3.5. Limits 11.3.6. Practical implementation 11.4. Function minimization using Newton method 11.4.1. Outline of Newton method A. Noether current B. Gamma function C. A Derivation of the Probability Distribution Function of chi-square distribution C.1. chi-square distribution with one degrees of freedom C.2. chi-square distribution with two degrees of freedom C.3. chi-square distribution with k degrees of freedom D. Error of Jackknife Estimation for Variance of Mean BibliographyDocto

Computing Bayesian predictive distributions: The K-square and K-prime distributions

The computation of two Bayesian predictive distributions which are discrete mixtures of incomplete beta functions is considered. The number of iterations can easily become large for these distributions and thus, the accuracy of the result can be questionable. Therefore, existing algorithms for that class of mixtures are improved by introducing round-off error calculation into the stopping rule. A further simple modification is proposed to deal with possible underflows that may prevent recurrence to work properly

Analysing energy detector diversity receivers for spectrum sensing

The analysis of energy detector systems is a well studied topic in the literature: numerous models have been derived describing the behaviour of single and multiple antenna architectures operating in a variety of radio environments. However, in many cases of interest, these models are not in a closed form and so their evaluation requires the use of numerical methods. In general, these are computationally expensive, which can cause difficulties in certain scenarios, such as in the optimisation of device parameters on low cost hardware. The problem becomes acute in situations where the signal to noise ratio is small and reliable detection is to be ensured or where the number of samples of the received signal is large. Furthermore, due to the analytic complexity of the models, further insight into the behaviour of various system parameters of interest is not readily apparent. In this thesis, an approximation based approach is taken towards the analysis of such systems. By focusing on the situations where exact analyses become complicated, and making a small number of astute simplifications to the underlying mathematical models, it is possible to derive novel, accurate and compact descriptions of system behaviour. Approximations are derived for the analysis of energy detectors with single and multiple antennae operating on additive white Gaussian noise (AWGN) and independent and identically distributed Rayleigh, Nakagami-m and Rice channels; in the multiple antenna case, approximations are derived for systems with maximal ratio combiner (MRC), equal gain combiner (EGC) and square law combiner (SLC) diversity. In each case, error bounds are derived describing the maximum error resulting from the use of the approximations. In addition, it is demonstrated that the derived approximations require fewer computations of simple functions than any of the exact models available in the literature. Consequently, the regions of applicability of the approximations directly complement the regions of applicability of the available exact models. Further novel approximations for other system parameters of interest, such as sample complexity, minimum detectable signal to noise ratio and diversity gain, are also derived. In the course of the analysis, a novel theorem describing the convergence of the chi square, noncentral chi square and gamma distributions towards the normal distribution is derived. The theorem describes a tight upper bound on the error resulting from the application of the central limit theorem to random variables of the aforementioned distributions and gives a much better description of the resulting error than existing Berry-Esseen type bounds. A second novel theorem, providing an upper bound on the maximum error resulting from the use of the central limit theorem to approximate the noncentral chi square distribution where the noncentrality parameter is a multiple of the number of degrees of freedom, is also derived

A low-bias simulation scheme for the SABR stochastic volatility model

The Stochastic Alpha Beta Rho Stochastic Volatility (SABR-SV) model is widely used in the financial industry for the pricing of fixed income instruments. In this paper we develop an lowbias simulation scheme for the SABR-SV model, which deals efficiently with (undesired) possible negative values, the martingale property of the discrete scheme and the discretization bias of commonly used Euler discretization schemes. The proposed algorithm is based the analytic properties of the governing distribution. Experiments with realistic model parameters show that this scheme is robust for interest rate valuation

On the use of the noncentral chi-square density function for the distribution of helicopter spectral estimates

A probability density function for the variability of ensemble averaged spectral estimates from helicopter acoustic signals in Gaussian background noise was evaluated. Numerical methods for calculating the density function and for determining confidence limits were explored. Density functions were predicted for both synthesized and experimental data and compared with observed spectral estimate variability

Testing for Weak Instruments in Linear IV Regression

Weak instruments can produce biased IV estimators and hypothesis tests with large size distortions. But what, precisely, are weak instruments, and how does one detect them in practice? This paper proposes quantitative definitions of weak instruments based on the maximum IV estimator bias, or the maximum Wald test size distortion, when there are multiple endogenous regressors. We tabulate critical values that enable using the first-stage F-statistic (or, when there are multiple endogenous regressors, the Cragg-Donald (1993) statistic) to test whether given instruments are weak. A technical contribution is to justify sequential asymptotic approximations for IV statistics with many weak instruments.

The distribution of McKay's approximation for the coefficient of variation

McKay's approximation for the coefficient of variation is type II noncentral beta distributed and asymptotically normal with mean n - 1 and variance smaller than 2(n - 1)