Independent Standard Normal Random Variables Research Articles

Tests Based on Kurtosis for Multivariate Normality

In this paper, we first transform a multivariate normal random vector into a random vector with elements that are approximately independent standard normal random variables. Then we propose the multivariate version generalized from the univariate normality test based on kurtosis from the literature. Power is investigated through the Monte Carlo Simulation with different significance level, dimension, and sample size. To assess the validity and accuracy of the new tests, we carry a comparative study with several other existing tests by selecting certain types of symmetric and asymmetric alternative distributions.

On the norming constants for normal maxima

Given n independent standard normal random variables, it is well known that their maxima Mn can be normalized such that their distribution converges to the Gumbel law. In a remarkable study, Hall proved that the Kolmogorov distance dn between the normalized Mn and its associated limit distribution is less than 3/log⁡n. In the present study, we propose a different set of norming constants that allow this upper bound to be decreased with dn≤C(m)/log⁡n for n≥m≥5. Furthermore, the function C(m) is computed explicitly, which satisfies C(m)≤1 and limm→∞⁡C(m)=1/3. As a consequence, some new and effective norming constants are provided using the asymptotic expansion of a Lambert W type function.

Stochastic Perturbation Analysis of Deteriorated Structures

The paper addresses a generalized perturbation method for the uncertainty analysis of concrete structures subject to reinforcement corrosion. The initiation phase is modelled by the well known Ficks 2ndlaw of diffusion. Deterioration models derived from experimental studies and a non-linear finite element (FE) model are used to simulate the structural behaviour of beams from the serviceability to the ultimate conditions. The time of corrosion initiation and the failure load are expressed by a Taylor series expansion about the origin of the space of independent standard normal random variables.

Distribution of the Ratio of Normal and Rice Random Variables

The ratio of independent random variables arises in many applied problems. The distribution of the ratio |X/Y| is studied when X and Y are independent Normal and Rice random variables, respectively. Ratios of such random variables have extensive applications in the analysis of noises in communication systems. The exact forms of probability density function (PDF), cumulative distribution function (CDF) and the existing moments are derived in terms of several special functions. As a special case, the PDF and CDF of the ratio of independent standard Normal and Rayleigh random variables have been obtained. Tabulations of associated percentage points and a computer program for generating tabulations are also given.

Sequential multi-sensor change-point detection

We develop a mixture procedure to monitor parallel streams of data for a change-point that affects only a subset of them, without assuming a spatial structure relating the data streams to one another. Observations are assumed initially to be independent standard normal random variables. After a change-point the observations in a subset of the streams of data have nonzero mean values. The subset and the post-change means are unknown. The procedure we study uses stream specific generalized likelihood ratio statistics, which are combined to form an overall detection statistic in a mixture model that hypothesizes an assumed fraction $p_0$ of affected data streams. An analytic expression is obtained for the average run length (ARL) when there is no change and is shown by simulations to be very accurate. Similarly, an approximation for the expected detection delay (EDD) after a change-point is also obtained. Numerical examples are given to compare the suggested procedure to other procedures for unstructured problems and in one case where the problem is assumed to have a well-defined geometric structure. Finally we discuss sensitivity of the procedure to the assumed value of $p_0$ and suggest a generalization.

Complex random energy model: zeros and fluctuations

The partition function of the random energy model at inverse temperature $$\beta $$ is a sum of random exponentials $$ \mathcal{Z }_N(\beta )=\sum _{k=1}^N \exp (\beta \sqrt{n} X_k)$$ , where $$X_1,X_2,\ldots $$ are independent real standard normal random variables (=random energies), and $$n=\log N$$ . We study the large N limit of the partition function viewed as an analytic function of the complex variable $$\beta $$ . We identify the asymptotic structure of complex zeros of the partition function confirming and extending predictions made in the theoretical physics literature. We prove limit theorems for the random partition function at complex $$\beta $$ , both on the logarithmic scale and on the level of limiting distributions. Our results cover also the case of the sums of independent identically distributed random exponentials with any given correlations between the real and imaginary parts of the random exponent.

Asymptotics of the variance of the number of real roots of random trigonometric polynomials

Let a n , n ⩾ 1 be a sequence of independent standard normal random variables. Consider the random trigonometric polynomial T n (θ) = Σ =1 a j cos(jθ), 0 ⩽ θ ⩽ 2π and let N n be the number of real roots of T n (θ) in (0, 2π). In this paper it is proved that $\lim _{n \to \infty } \frac{{Var(N_n )}} {n} = c_0 $ where 0 < c 0 < ∞.

Representation of Gaussian fields in series with independent coefficients

The numerical discretization of problems with stochastic data or stochastic parameters generally involves the introduction of coordinates that describe the stochastic behavior, such as coefficients in a series expansion or values at discrete points. The series expansion of a Gaussian field with respect to any orthonormal basis of its Cameron–Martin space has independent standard normal coefficients. A standard choice for numerical simulations is the Karhunen–Loeve series, which is based on eigenfunctions of the covariance operator. We suggest an alternative, the hierarchic discrete spectral expansion, which can be constructed directly from the covariance kernel. The resulting basis functions are often well localized, and the convergence of the series expansion seems to be comparable to that of the Karhunen–Loeve series. We provide explicit formulas for particular cases, and general numerical methods for computing exact representations of such bases. Finally, we relate our approach to numerical discretizations based on replacing a random field by its values on a finite set. Introduction The numerical discretization of problems with stochastic data or stochastic parameters requires that the random inputs are approximated by finite quantities. This is generally done in one of two ways. Either the random data is expanded in a series, which can be truncated for numerical computations, or it is replaced by a finite dimensional random variable, describing for example the value of a random field on a discrete set of points. A standard approach, falling strictly into the first category, is to expand a random field into its Karhunen–Loeve series. For Gaussian fields, the coefficients in this series are independent standard normal random variables. The independence of these coefficients is crucial to many numerical methods. For example, in Monte Carlo simulation, coefficient sequences can be generated by independent draws of pseudorandom numbers. The construction of polynomial chaos bases as tensor products of orthonormal bases with respect to the distributions of the coefficients also requires that these are independent. Similarly, in collocation and quasi Monte Carlo methods, constructions of collocation points make use of the product structure of the joint distribution of the coefficients. Nevertheless, the Karhunen–Loeve series is often ill-suited for numerical computations, as it requires eigenfunctions of the covariance operator. These are usually not known exactly, and are expensive to approximate numerically. Furthermore, the eigenfunctions generally have global supports. We suggest an alternative to the Karhunen–Loeve series for general continuous Gaussian fields on bounded domains, which we call the hierarchic discrete spectral expansion. Assuming that the covariance kernel is given, the basis functions in our series expansion can be constructed exactly. As these form an orthonormal basis of Date: September 29, 2010. Research supported in part by the Swiss National Science Foundation grant No. 200021120290/1. The author wishes to thank Hans-Jorg Starkloff for his insightful suggestions. 1

Goodness of fit and homogeneity tests on the basis of N-distances

In this paper we propose an application of N-distance theory [Klebanov, L.B., 2005. N-distances and their applications. Karolinum, Prague] for testing simple hypotheses of goodness of fit and homogeneity. The asymptotic null distribution of test statistics is established and coincides with the distribution of infinite quadratic form of independent standard normal random variables. A construction of multivariate free-of-distribution homogeneity test is considered. The power of proposed criteria is compared with classical tests using Monte-Carlo simulations.

Extremes of Shepp statistics for Gaussian random walk

Let (xi(i), i >= 1) be a sequence of independent standard normal random variables and let S-k = Sigma(k)(i=1)xi(i) be the corresponding random walk. We study the renormalized Shepp statistic M-T((N)) = 1/root N (1 u) when u, N and T -> infinity in a synchronized way. There are three types of relations between u and N that give different asymptotic behavior. For these three cases we establish the limiting Gumbel distribution of M-T((N)) when T, N -> infinity and present corresponding normalization sequences.

Asymptotic inference for a nonstationary double AR(1) model

We investigate the nonstationary double AR(1) model, y t = Φy t-1 + η t √ (ω + αy 2 t-1 ), where ω > 0, a > 0, the η t are independent standard normal random variables and E log |Φ+ η t √α | ≥ 0. We show that the maximum likelihood estimator of (Φ, α) is consistent and asymptotically normal. Combination of this result with that in Ling (2004) for the stationary case gives the asymptotic normality of the maximum likelihood estimator of Φ for any Φ in the real line, with a root-n rate of convergence. This is in contrast to the results for the classical AR(1) model, corresponding to a = 0.

A Logistic Regression Mixture Model for Interval Mapping of Genetic Trait Loci Affecting Binary Phenotypes

Often in genetic research, presence or absence of a disease is affected by not only the trait locus genotypes but also some covariates. The finite logistic regression mixture models and the methods under the models are developed for detection of a binary trait locus (BTL) through an interval-mapping procedure. The maximum-likelihood estimates (MLEs) of the logistic regression parameters are asymptotically unbiased. The null asymptotic distributions of the likelihood-ratio test (LRT) statistics for detection of a BTL are found to be given by the supremum of a chi2-process. The limiting null distributions are free of the null model parameters and are determined explicitly through only four (backcross case) or nine (intercross case) independent standard normal random variables. Therefore a threshold for detecting a BTL in a flanking marker interval can be approximated easily by using a Monte Carlo method. It is pointed out that use of a threshold incorrectly determined by reading off a chi2-probability table can result in an excessive false BTL detection rate much more severely than many researchers might anticipate. Simulation results show that the BTL detection procedures based on the thresholds determined by the limiting distributions perform quite well when the sample sizes are moderately large.

Some properties of inferences in misspecified linear models

Let Y denote an n × 1 vector of observations such that Y = μ + σε where μ is an unknown n × 1 vector, σ > 0 is an unknown parameter, and ε is an n × 1 vector of independent standard normal random variables. A linear regression analysis is often based on a model for μ such as μ = Xβ where X is a known n × p matrix of independent variables and β is a p × 1 vector of unknown parameters. When the assumption that μ = Xβ for some β holds, the results of the analysis can be interpreted as applying to μ, the mean of Y. In this paper, the properties of interferences based on the model hold, although with respect to μ ∗ , the vector of form Xβ closest to μ, rather than with respect to μ. Hence, the results of a linear regression analysis have a certain type of validity that applies whether or not the model is correctly specified.

Diagnostics in Linear Discriminant Analysis

Abstract Some new diagnostic measures in discriminant analysis are proposed. They can be expressed in terms of the two fundamental influence statistics in discriminant analysis: d i 2 and ψ i . A theorem on the asymptotic distributions of the fundamental statistics is derived. Based on the theorem, the proposed measures can be shown to be asymptotically distributed as functions of independent chi-squared and standard normal random variables. Critical values and expected quantiles of the measures can then be constructed. Hence influential observations are detected using Q-Q plots and significance tests. Two measures have analogous forms in regression. The theorem is also useful for getting the asymptotic distributions of existing measures that are functions of d i 2 and ψ i . A comparison of the diagnostics in linear discriminant analysis, linear regression, and linear logistic regression (discriminant) analysis is made. Although discriminant coefficients can be determined under a regression model, regression diagnostic measures are shown to be inappropriate for detecting influential observations in linear discriminant analysis. The temptation of applying regression diagnostic measures in linear discriminant analysis must be resisted.

The complex zeros of random polynomials

Mark Kac gave an explicit formula for the expectation of the number, ν n ( Ω ) {\nu _n}(\Omega ) , of zeros of a random polynomial, \[ P n ( z ) = ∑ j = 0 n − 1 η j z j , {P_n}(z) = \sum \limits _{j = 0}^{n - 1} {{\eta _j}{z^j}} , \] in any measurable subset Ω \Omega of the reals. Here, η 0 , … , η n − 1 {\eta _0}, \ldots ,{\eta _{n - 1}} are independent standard normal random variables. In fact, for each n &gt; 1 n &gt; 1 , he obtained an explicit intensity function g n {g_n} for which \[ E ν n ( Ω ) = ∫ Ω g n ( x ) d x . {\mathbf {E}}{\nu _n}(\Omega ) = \int _\Omega {{g_n}(x)\,dx.} \] Here, we extend this formula to obtain an explicit formula for the expected number of zeros in any measurable subset Ω \Omega of the complex plane C \mathbb {C} . Namely, we show that \[ E ν n ( Ω ) = ∫ Ω h n ( x , y ) d x d y + ∫ Ω ∩ R g n ( x ) d x , {\mathbf {E}}{\nu _n}(\Omega ) = \int _\Omega {{h_n}(x,y)\,dxdy + \int _{\Omega \cap \mathbb {R}} {{g_n}(x)\,dx,} } \] where h n {h_n} is an explicit intensity function. We also study the asymptotics of h n {h_n} showing that for large n n its mass lies close to, and is uniformly distributed around, the unit circle.

A unified method to calculate the moments of the least squares estimators in nonlinear regression

A unified method and a set of regularity conditions are presented in this paper to calculate the moments of the least squares estimators, residuals, and fitting errors in nonlinear regression. The results given by Box, Clarke and other authors have been greatly improved and developed. The key point of the method is that we find a series of stochastic expansions related to the estimators. These expansions consist of curvature measures and independent standard normal random variables; therefore they are easy to understand and to deal with from statistical and geometrical point of view.

On the asymptotic distributions of maxima of trigonometric polynomials with random coefficients

Let {ε t, t = 1, 2, ···, n} be a sequence of mutually independent standard normal random variables. Let Xn(λ) and Yn(λ) be respectively the real and imaginary parts of exp iλ t, and let . It is shown that as n tends to∞, the distribution functions of the normalized maxima of the processes {Xn(λ)}, (Yn(λ)}, {In(λ)} over the interval λ∈ [0,π] each converge to the extremal distribution function exp [–e–x], —∞ &lt; x &lt;∞.It is also shown that these results can be extended to the case where {ε t} is a stationary Gaussian sequence with a moving-average representation.

On the asymptotic distributions of maxima of trigonometric polynomials with random coefficients

Let {ε t, t = 1, 2, ···, n} be a sequence of mutually independent standard normal random variables. Let X n(λ) and Y n(λ) be respectively the real and imaginary parts of exp iλ t, and let . It is shown that as n tends to∞, the distribution functions of the normalized maxima of the processes {X n(λ)}, (Y n(λ)}, {I n(λ)} over the interval λ∈ [0,π] each converge to the extremal distribution function exp [–e–x ], —∞ &amp;lt; x &amp;lt;∞. It is also shown that these results can be extended to the case where {ε t} is a stationary Gaussian sequence with a moving-average representation.