Independent Normal Random Variables Research Articles

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.

Proportion of Non-Zero Normal Means: Universal Oracle Equivalences and Uniformly Consistent Estimators

SummarySince James and Stein's seminal work, the problem of estimating n normal means has received plenty of enthusiasm in the statistics community. Recently, driven by the fast expansion of the field of large-scale multiple testing, there has been a resurgence of research interest in the n normal means problem. The new interest, however, is more or less concentrated on testing n normal means: to determine simultaneously which means are 0 and which are not. In this setting, the proportion of the non-zero means plays a key role. Motivated by examples in genomics and astronomy, we are particularly interested in estimating the proportion of non-zero means, i.e. given n independent normal random variables with individual means Xj∼N(μj,1), j=1,…,n, to estimate the proportion ɛn=(1/n) #{j:μj /= 0}. We propose a general approach to construct the universal oracle equivalence of the proportion. The construction is based on the underlying characteristic function. The oracle equivalence reduces the problem of estimating the proportion to the problem of estimating the oracle, which is relatively easier to handle. In fact, the oracle equivalence naturally yields a family of estimators for the proportion, which are consistent under mild conditions, uniformly across a wide class of parameters. The approach compares favourably with recent works by Meinshausen and Rice, and Genovese and Wasserman. In particular, the consistency is proved for an unprecedentedly broad class of situations; the class is almost the largest that can be hoped for without further constraints on the model. We also discuss various extensions of the approach, report results on simulation experiments and make connections between the approach and several recent procedures in large-scale multiple testing, including the false discovery rate approach and the local false discovery rate approach.

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.

Total-Variance Reduction Via Thresholding: Application to Cepstral Analysis

We consider a vector of independent normal random variables with unknown means but known variances. Our problem is to reduce the total variance of these random variables by exploiting the a priori information that a significant proportion of them have small means. We show that thresholding is an effective means of solving this problem and propose two schemes for threshold selection: one based on a uniformly most powerful unbiased test and another on a Bayesian information criterion selection rule. Reduction of the total variance of estimated spectra, obtained by cepstral analysis, can be cast in the above mathematical framework, and we use it as an example application throughout this paper. We show via numerical simulation that the use of the simple thresholding method, proposed here, in cepstral analysis can achieve significant reductions of total variance

Asymptotic properties of computationally efficient alternative estimators for a class of multivariate normal models

Parameters of Gaussian multivariate models are often estimated using the maximum likelihood approach. In spite of its merits, this methodology is not practical when the sample size is very large, as, for example, in the case of massive georeferenced data sets. In this paper, we study the asymptotic properties of the estimators that minimize three alternatives to the likelihood function, designed to increase the computational efficiency. This is achieved by applying the information sandwich technique to expansions of the pseudo-likelihood functions as quadratic forms of independent normal random variables. Theoretical calculations are given for a first-order autoregressive time series and then extended to a two-dimensional autoregressive process on a lattice. We compare the efficiency of the three estimators to that of the maximum likelihood estimator as well as among themselves, using numerical calculations of the theoretical results and simulations.

Statistical interconnect metrics for physical-design optimization

In this paper, statistical models for the efficient analysis of interconnect delay and crosstalk noise in the presence of back-end process variations are developed. The proposed models enable closed-form computation of means and variances of interconnect-delay, crosstalk-noise peak, and coupling-induced-delay change for given magnitudes of variation in relevant process parameters, such as linewidth, metal thickness, metal spacing, and interlayer dielectric (ILD) thickness. The proposed approach is based on the observation that if the variations in different physical dimensions are assumed to be independent normal random variables, then the interconnect behavior also tends to have a Gaussian distribution. In the proposed statistical models, delay and noise are expressed directly as functions of changes in physical parameters. This formulation allows us to preserve all correlations and can be very useful in evaluating delay and noise sensitivities due to changes in various physical dimensions. For interconnect-delay computation, the authors express the resistance and capacitance of a line as a linear function of random variables and then use these to compute circuit moments. They show that ignoring higher order terms in the resulting variational moments does not result in a loss of accuracy. Finally, these variability-aware moments are used in known closed-form delay and slew metrics to compute interconnect-delay probability density functions (pdfs). Similarly for coupling noise and dynamic-delay analysis, the authors rely on the linearity (Gaussian) assumption, allowing us to truncate nonlinear terms and express noise and dynamic-delay pdfs as linear functions of variations in relevant geometric dimensions. They compare their approach to SPICE-based Monte Carlo simulations and report the error in mean and standard deviation of interconnect delay to be 1% and 4% on average, respectively

Comments onA new model for statistical error analysis in XAS: about the distribution function of the absorption coefficientby E. Curis & S. Bénazeth (2001).J. Synchrotron Rad.8, 264–266

The recent paper by Curis & Benazeth (2001) considers modeling of the experimental distribution of the absorption coefficient. As a preamble to this, the authors consider the exact law of the quotient of two independent normal random variables. They claim to have derived the exact law, it being a ‘quite long and complex’ expression not given in the paper. In fact, the problem of the quotient of two independent or dependent normal random variables was considered in the 1960s by Marsaglia (1965) and Hinkley (1969). Both these papers provide neat and compact expressions for the exact law of the ratio. So, we are somewhat surprised by the claim by Curis & Benazeth (2001). For completeness, we would like to add that Yatchew (1986) has recently extended the work of Marsaglia (1965) and Hinkley (1969) to the multivariate case. The purpose of this correspondence is not just to correct the mistake. We feel the references mentioned above can help the readers and authors of this journal in making appropriate choices with regard to similar modeling problems. It will also help to prevent similar mistakes in the future.

Real zeros of random algebraic polynomials with binomial elements

This paper provides an asymptotic estimate for the expected number of real zeros of a random algebraic polynomial a0+a1x+a2x2+…+an−1xn−1. The coefficients aj(j=0,1,2,…,n−1) are assumed to be independent normal random variables with mean zero. For integers m and k=O(log⁡n)2 the variances of the coefficients are assumed to have nonidentical value var⁡(aj)=(k−1j−ik), where n=k⋅m and i=0,1,2,…,m−1. Previous results are mainly for identically distributed coefficients or when var⁡(aj)=(nj). We show that the latter is a special case of our general theorem.

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.

Accurate and efficient power calculations for 2 ×m tables in unmatched case-control designs

The aim of this article was to derive an approximation of the distribution of Pearson's statistic for 2 x m contingency tables to calculate power for case-control studies accurately and efficiently in terms of computer time. We first prove that, rather than a non-central chi-square distribution, Pearson's statistic is asymptotically equivalent with the sum of squares of m independent normal random variables whose variances are typically different under the alternative hypothesis. Based on this asymptotically equivalent (AE) we derive a cost-effective approximation (CE). Numerical results show that CE was almost as precise as AE, but computationally more efficient. Although somewhat less costly in terms of CPU time, we show that a commonly used approximation using the non-central chi-square distribution can be very inaccurate and overestimate power in some scenarios and underestimate it in others. Simulations and exact distributions, on the other hand, are accurate but computationally very intensive compared to our CE. The CE reached its lowest accuracy under the null hypothesis where it has a chi-square distribution with m - 1 degree of freedom. This suggests that CE can be used to calculate power for tables where researchers would feel comfortable using Pearson's chi-square test.

Inference for Change-Point and Post-Change Mean with Possible Change in Variance

For a sequence of independent normal random variables, we consider the estimation of the change-point and the post-change mean after a change in the mean is detected by a CUSUM procedure, subject to a possible change in variance. Conditional on the event that a change is detected and it occurred far away from the starting point and the threshold is large, the (absolute) bias of the maximum likelihood estimator for the change-point (obtained at the reference value) is found. The first-order biases for the post-change mean and variance estimators are also obtained by using Wald's Likelihood Ratio Identity and the renewal theorem. In the local case when the reference value and the post-change mean are both small, accurate approximations are derived. A confidence interval for post-change mean based on a corrected normal pivot is then discussed.

New Approximate Inferential Methods for the Reliability Parameter in a Stress–Strength Model: The Normal Case

This article considers the problem of testing and interval estimation of the reliability parameter P(Y 1 > Y 2), where Y 1 and Y 2 are independent normal random variables with unknown means and variances. A new approximate method is proposed, and is compared with two existing approximate methods and a generalized variable method. Simulation studies indicate that the sizes of the existing approximate tests exceed the nominal level considerably in some situations while the new method controls the sizes satisfactorily in all situations considered. The studies also indicated that the generalized variable test is too conservative for small samples. A confidence limit for the reliability parameter based on the new approximate test is also given. The results are extended to the case where Y 1 and Y 2 depend on some explanatory variables. The methods are illustrated using three examples, and some recommendations are made regarding what method should be used for practical applications.

Cycle-transitive comparison of independent random variables

The discrete dice model, previously introduced by the present authors, essentially amounts to the pairwise comparison of a collection of independent discrete random variables that are uniformly distributed on finite integer multisets. This pairwise comparison results in a probabilistic relation that exhibits a particular type of transitivity, called dice-transitivity. In this paper, the discrete dice model is generalized with the purpose of pairwisely comparing independent discrete or continuous random variables with arbitrary probability distributions. It is shown that the probabilistic relation generated by a collection of arbitrary independent random variables is still dice-transitive. Interestingly, this probabilistic relation can be seen as a graded alternative to the concept of stochastic dominance. Furthermore, when the marginal distributions of the random variables belong to the same parametric family of distributions, the probabilistic relation exhibits interesting types of isostochastic transitivity, such as multiplicative transitivity. Finally, the probabilistic relation generated by a collection of independent normal random variables is proven to be moderately stochastic transitive.

The Mean of the Inverse of a Punctured Normal Distribution and Its Application

AbstractThe fundamental properties of a punctured normal distribution are studied. The results are applied to three issues concerning X/Y where X and Y are independent normal random variables with means μX and μY respectively. First, estimation of μX/μY as a surrogate for E(X/Y) is justified, then the reason for preference of a weighted average, over an arithmetic average, as an estimator of μX/μY is given. Finally, an approximate confidence interval for μX/μY is provided. A grain yield data set is used to illustrate the results. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Algebraic polynomials with non-identical random coefficients

There are many known asymptotic estimates for the expected number of real zeros of a random algebraic polynomial a 0 + a 1 x + a 2 x 2 + ⋯ + a n − 1 x n − 1 . a_0 +a_1 x+ a_2 x^2+ \cdots +a_{n-1}x^{n-1}. The coefficients a j a_j ( j = 0 , 1 , 2 , … , n − 1 ) (j=0, 1, 2, \dotsc , n-1) are mostly assumed to be independent identical normal random variables with mean zero and variance unity. In this case, for all n n sufficiently large, the above expected value is shown to be O ( log ⁡ n ) O(\log n) . Also, it is known that if the a j a_j have non-identical variance ( n − 1 j ) \binom {n-1}{j} , then the expected number of real zeros increases to O ( n ) O(\sqrt {n}) . It is, therefore, natural to assume that for other classes of distributions of the coefficients in which the variance of the coefficients is picked at the middle term, we would also expect a greater number of zeros than O ( log ⁡ n ) O(\log n) . In this work for two different choices of variance for the coefficients we show that this is not the case. Although we show asymptotically that there is some increase in the number of real zeros, they still remain O ( log ⁡ n ) O(\log n) . In fact, so far the case of var ( a j ) = ( n − 1 j ) \mbox {var}(a_j)={\binom {n-1}{j}} is the only case that can significantly increase the expected number of real zeros.

Zeros of an algebraic polynomial with nonequal means randomcoefficients

This paper provides an asymptotic estimate for the expected number of real zeros of a random algebraic polynomial a0 + a1x + a2x2 + ⋯+an − 1xn−1. The coefficients aj(j = 0, 1, 2, …, n − 1) are assumed to be independent normal random variables with nonidentical means. Previous results are mainly for identically distributed coefficients. Our result remains valid when the means of the coefficients are divided into many groups of equal sizes. We show that the behaviour of the random polynomial is dictated by the mean of the first group of the coefficients in the interval (−1, 1) and the mean of the last group in (−∞, −1)∪(1, ∞).

Nonparametric analysis of covariance for censored data

The fully nonparametric model for nonlinear analysis of covariance, proposed in Akritas et al. (2000), is considered in the context of censored observations. Under this model, the distributions for each factor level combination and covariate value are not restricted to comply to any parametric or semiparametric model. The data can be continuous or ordinal categorical. The possibility of different shapes of covariate effect in different factor level combinations is also allowed. This generality is useful whenever modelling. assumptions such as additive risks, proportional hazards or proportional odds appear suspect. Test statistics are obtained for the nonparametric hypotheses of no main effect and of no interaction effect which adjusts for the presence of a covariate. They are quadratic forms based on averages over-the covariate values of Beran estimators of the conditional distribution of the survival time given each covariate value, The derivation of the asymptotic chi(2) distribution of the test statistics uses a recently-obtained asymptotic representation of the Bekan estimator as average of independent random variables. A real-data set is analysed and results of simulation studies are reported.

Functional local law of the iterated logarithm for geometrically weighted random series

The paper proves a functional local law of the iterated logarithm and a moderate deviation principle for properly normalized geometrically weighted random series of centered independent normal real random variables with variances satisfying Kolmogorov's conditions. The methodology used here allows an unified treatment, extends and gives the exact rate of convergence in the pointwise laws previously proved by Zhang (Ann. Probab. 25 (1997) 1621) and Bovier and Picco (Ann. Probab. 21 (1993) 168).

EQUILIBRIUM DISTRIBUTION OF ZEROS OF RANDOM POLYNOMIALS

We consider ensembles of random polynomials of the form p(z) = P N=1 ajPj where {aj} are independent complex normal random variables and where {Pj} are the orthonormal polynomials on the boundary of a bounded simply connected analytic plane domain ⊂ C relative to an analytic weight �(z)|dz| . In the simplest case where is the unit disk and � = 1, so that Pj(z) = z j , it is known that the average distribution of zeros is the uniform measure on S 1 . We show that for any analytic (,�), the zeros of random polynomials almost surely become equidistributed relative to the equilibrium measure on @ as N → ∞. We further show that on the length scale of 1/N, the correlations have a universal scaling limit independent of (,�).

Strong Approximation Theorems for Sums of Random Variables when Extreme Terms are Excluded

Let {X n ;n≥1} be a sequence of i.i.d. random variables and let $$ X^{{{\left( r \right)}}}_{n} = X_{j} $$ if |X j | is the r-th maximum of |X 1|, ..., |X n |. Let S n = X 1+⋯+X n and $$ {}^{{{\left( r \right)}}}S_{n} = S_{n} - {\left( {X^{{{\left( 1 \right)}}}_{n} + \cdots + X^{{{\left( r \right)}}}_{n} } \right)}. $$ Sufficient and necessary conditions for (r) S n approximating to sums of independent normal random variables are obtained. Via approximation results, the convergence rates of the strong law of large numbers for (r) S n are studied.