Noncentral Chi-square Distribution Research Articles

Profile Inferences on Restricted Additive Partially Linear EV Models

We consider the testing problem for the parameter and restricted estimator for the nonparametric component in the additive partially linear errors-in-variables (EV) models under additional restricted condition. We propose a profile Lagrange multiplier test statistic based on modified profile least-squares method and two-stage restricted estimator for the nonparametric component. We derive two important results. One is that, without requiring the undersmoothing of the nonparametric components, the proposed test statistic is proved asymptotically to be a standard chi-square distribution under the null hypothesis and a noncentral chi-square distribution under the alternative hypothesis. These results are the same as the results derived by Wei and Wang (2012) for their adjusted test statistic. But our method does not need an adjustment and is easier to implement especially for the unknown covariance of measurement error. The other is that asymptotic distribution of proposed two-stage restricted estimator of the nonparametric component is asymptotically normal and has an oracle property in the sense that, though the other component is unknown, the estimator performs well as if it was known. Some simulation studies are carried out to illustrate relevant performances with a finite sample. The asymptotic distribution of the restricted corrected-profile least-squares estimator, which has not been considered by Wei and Wang (2012), is also investigated.

Contaminated Chi-Square Modeling and Large-Scale ANOVA Testing

We propose a convenient moment-based procedure for testing the omnibus null hypothesis of no contamination of a central chi-square distribution by a non-central chi-square distribution. In sharp contrast with likelihood ratio tests for mixture models, there is no need for re-sampling or random field theory to obtain critical values. Rather, critical values are available from an asymptotic normal distribution, and there is excellent agreement between nominal and actual significance levels. This procedure may be used to model numerous chi-square statistics, obtained via monotonic transformations of F statistics, from large-scale ANOVA testing, such as that encountered in microarray data analysis. In that context, modeling chi-square statistics instead of p-values may improve detection of differential gene expression, as we demonstrate through simulation studies, while also reducing false declarations of the same, as we illustrate in a case study on aging and cognition. Our procedure may also be incorporated into a gene filtration process, which may reduce type II errors on genewise null hypotheses by justifying lighter controls for Type I errors.

Computing the noncentral gamma distribution, its inverse and the noncentrality parameter

The noncentral gamma distribution can be viewed as a generalization of the noncentral chi-squared distribution and it can be expressed as a mixture of a Poisson density function with a incomplete g...

A new fourth-order numerical scheme for option pricing under the CEV model

The empirically observed negative relationship between a stock price and its return volatility can be captured by the constant elasticity of variance option pricing model. For European options, closed form expressions involve the non-central chi-square distribution whose computation can be slow when the elasticity factor is close to one, volatility is low or time to maturity is small. We present a fast numerical scheme based on a high-order compact discretisation which accurately computes the option price. Various numerical examples indicate that for comparable computational times, the option price computed with the scheme has higher accuracy than the Crank–Nicolson numerical solution. The scheme accurately computes the hedging parameters and is stable for strongly negative values of the elasticity factor.

Contributions to the Weighted Non-Central Chi-Square Distribution

Contributions to the Weighted Non-Central Chi-Square Distribution

Percentiles of von Mises Stress from Combined Random Vibration and Static Loading by Approximate Noncentral Chi Square Distribution

Firstly, a calculation for percentiles of von Mises stress in linear structures subjected to Gaussian random loads is extended to the case of Gaussian random loads having nonzero mean values, i.e.,the inclusion of static loads. The development is restricted to the case of plane stress. The method includes calculation of a given percentile of von Mises stress to any desired accuracy, a rapid estimate of the percentile, and upper and lower bounds on the von Mises stress. The calculation expands the cumulative distribution function of the von Mises stress as a series of noncentral chi-square distributions. Summation of a sufficient number of terms of the series calculates the percentile to the desired accuracy. The rapid estimate of the percentile interpolates the distribution of the von Mises stress in a small number of inverse noncentral chi-square2 distribution functions. The upper and lower bounds on the percentiles take advantage of the noncentral chi-square distribution of summations of normally distributed stress components. Second and third calculation methods arise from approximations of the distribution of quadratic forms of noncentral normal variables, or equivalently, linear combinations of noncentral chi-square variables. These methods provide rapid estimates of percentiles of von Mises stress in linear structures under random loads having nonzero mean values. The accuracy and computational efficiency of the methods are reviewed and compared. The methods are expected to have wide application in design of and prognostics for components subjected to constant structural loads coupled with random loading arising from vibrations caused by wind, waves, seismic events, engines, turbulence, acoustic noise, etc.

TESTING UNDER WEAK IDENTIFICATION WITH CONDITIONAL MOMENT RESTRICTIONS

AbstractWe propose a semiparametric test for the value of coefficients in models with conditional moment restrictions that has correct size regardless of identification strength. The test is in essence an Anderson-Rubin (AR) test using nonparametrically estimated instruments to which we apply a standard error correction. We show that the test is (1) always size-correct, (2) consistent when identification is not too weak, and (3) asymptotically equivalent to an infeasible AR test when identification is sufficiently strong. We moreover prove that under homoskedasticity and strong identification our test has a limiting noncentral chi-square distribution under a sequence of local alternatives, where the noncentrality parameter is given by a quadratic form of the inverse of the semiparametric efficiency bound.

Reliability Analysis by Mean-Value Second-Order Expansion

In this paper, two second-order methods are proposed for reliability analysis. First, general random variables are transformed to standard normal random variables. Then, the limit-state function is additively decomposed into one-dimensional functions, which are then expanded at the mean-value point to second-order terms. The approximated limit-state function becomes the sum of independent variables following noncentral chi-square distributions or normal distributions. The first method computes the probability of failure by the saddle-point approximation. If a saddle-point does not exist, the second method is then used. The second method approximates the limit-state function by a quadratic function with independent variables following normal distributions with the same variances. This treatment leads to a quadratic function that follows a noncentral chi-square distribution. These methods generally produce more accurate reliability approximations than the first-order reliability method (FORM) with 2n + 1 function evaluations, where n is the dimension of the problem. The effectiveness of the proposed methods is demonstrated with three examples, and the proposed methods are compared with the first- and second-order reliability methods (SROMs).

Hypothesis testing analysis and unknown parameter estimation of GPS signal detection

Hypothesis testing analysis and unknown parameter estimation of both the intermediate frequency (IF) and baseband GPS signal detection are given by using the generalized likelihood ratio test (GLRT) approach, applying the model of GPS signal in white Gaussian noise. It is proved that the test statistic follows central or noncentral F distribution. It is also pointed out that the test statistic is nearly identical to central or noncentral chi-squared distribution because the processing samples are large enough to be considered as infinite in GPS acquisition problem. It is also proved that the probability of false alarm, the probability of detection and the threshold are affected largely when the hypothesis testing refers to the full pseudorandom noise (PRN) code phase and Doppler frequency search space cells instead of each individual cell. The performance of the test statistic is also given with combining the noncoherent integration.

Simulation of the CEV process and the local martingale property

We consider the constant elasticity of variance (CEV) process, reviewing the relationships between its transition density and that of the non-central chi-squared distribution. When the CEV parameter exceeds one, the forward price process is a strictly local martingale, and the price of a plain vanilla European call option reflects this fact. We develop techniques for Monte Carlo simulation of the CEV process, for all parameter regimes, and compare the results against the analytic expressions for plain vanilla European option prices. Using these techniques, we also verify the local martingale property.

GPS Signal Detection Using Hypothesis Testing Analysis

GPS signal detection using hypothesis testing analysis are given by using the generalized likelihood ratio test (GLRT) approach, applying the model of intermediate frequency (IF) GPS signal of one satellite in white Gaussian noise. The test statistic follows central or noncentral F distribution and is nearly identical to central or noncentral chi-squared distribution because the processing samples are large enough to be considered as infinite in GPS acquisition algorithms. The probability of false alarm, the probability of detection and the threshold are affected largely when the hypothesis testing refers to the full PRN code phase and Doppler frequency search space cells instead of to each individual cell. The performance of the test statistic is also given with combining the noncoherent integration. Given the probability of false alarm to achieve a desired probability of detection, examples are illustrated to determine the relations among the threshold, the coherent integration time, the number of noncoherent integration and signal to noise ratio.

Power of the likelihood ratio test for models of DNA base substitution

The goal of this work is to study the properties of the likelihood ratio (LR) tests comparing base substitution models. These are the most widely used hypothesis tests. With mild regularity conditions, we show that the asymptotic distribution of the LR statistic test, under the alternative hypothesis, is a non-central chi-square distribution. The asymptotic normal distribution of the LR test is proved when the sequence length S goes to infinity. We also propose a consistent estimator for the non-centrality parameter D. Through asymptotic theory and based on this consistent estimator for D, we propose a low computational cost estimator for the power of the LR test. The methodology is applied to 17 different gene sequences of the ECP–EDN family in primates.

Beam Selection Gain Versus Antenna Selection Gain

We consider beam selection using a fixed beamforming network (FBN) at a base station with <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> array antennas. In our setting, a Butler matrix is deployed at the RF stage to form <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> beams, and then the best beam is selected for transmission. We introduce some properties of the noncentral chi-square distribution and prove the resulting properties of the beam selection gain verifying that beam selection is superior to antenna selection in Rician channels with any <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> -factors. Furthermore, we find asymptotically tight stochastic bounds of the beam selection gain, which yield approximate closed form expressions of the expected selection gain and the ergodic capacity. Beam selection has the order of growth of the ergodic capacity Θ(log <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> ) regardless of user location in contrast to Θ(log(log <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</i> )) for antenna selection.

Novel Representations for the Equicorrelated Multivariate Non-Central Chi-Square Distribution and Applications to MIMO Systems in Correlated Rician Fading

Novel representations for the multivariate probability density function and cumulative distribution function of the non-central chi-square distribution are derived. The solutions are expressed in terms of well known functions which are available in common mathematical software. When the number of degrees of freedom, m, is even, the new representation has the advantage that only a single-integral computation is needed to compute a N-dimensional distribution. The new representations are used to numerically evaluate the outage probability of multiple input multiple output systems with receiver antenna selection, operating in correlated Rician fading. The well known equal correlation model, which is considered to be a useful model for a closely placed set of antennas, is used for the analysis.

Process Capability Evaluation for Square Bumps with Mean Shift

Abstract Process capability indices have been widely used in manufacturing industries to provide numerical measures of process potential and performance. In in-plant applications, some inevitable process mean shift may be undetected when the statistical process control charts are applied, particularly, in the bumping process for square bumps. Bothe in 2002 provided a statistical reason for considering such a shift in the process mean to accommodate the undetected mean shifts for normally distributed processes when evaluating process capability. In this paper, we consider a bumping process for square bumps in which the data can be formulated as a non-central chi-square distribution, a class of non-normal distributions. To evaluate the popular yield-based process capability Cpk more accurately, we accommodate the magnitudes of undetected mean shifts using the modified capability evaluation formula and tabulate the detection powers under various subgroup sizes and non-central chi-square parameters. Based on the modified process capability evaluation, we can provide a more reliable process capability evaluation of capability index Cpk for square bumps and make more correct decisions. For illustration purpose, a real application in a bumping factory which is located on the Science-based Industrial Park in Hsinchu, Taiwan, is presented.

Asymptotic optimal inference for multivariate branching-Markov processes via martingale estimating functions and mixed normality

Multivariate tree-indexed Markov processes are discussed with applications. A Galton–Watson super-critical branching process is used to model the random tree-indexed process. Martingale estimating functions are used as a basic framework to discuss asymptotic properties and optimality of estimators and tests. The limit distributions of the estimators turn out to be mixtures of normals rather than normal. Also, the non-null limit distributions of standard test statistics such as Wald, Rao’s score, and likelihood ratio statistics are shown to have mixtures of non-central chi-square distributions. The models discussed in this paper belong to the local asymptotic mixed normal family. Consequently, non-standard limit results are obtained.

On MP Test and the MVUEs in a &lt;Var&gt;N(θ, cθ)&lt;/Var&gt; Distribution with &lt;Var&gt;θ&lt;/Var&gt; Unknown: Illustrations and Applications

Consider a sequence of independent observations X1, . . . , Xn from a N(θ, cθ) distribution with 0 0) is known. We begin with the problem of testing H0 : θ = θ0 against H1 : θ = θ1 where θ0, θ1(θ0 = θ1) are specified values of θ. The most powerful (MP) level α test depends upon ∑n i=1 X 2 i , a complete and sufficient statistic for θ, which has a multiple of a non-central chi-square distribution with its non-centrality parameter involving n and the true parameter value θ under H0, H1. We first target type-I and type-II error probabilities α and β respectively, with α > 0, β > 0, α + β < 1. We set out to determine the required exact sample size which will control these error probabilities and provide two useful large-sample approximations for the sample size. The three methods provide nearly the same required sample size whether n is small, moderate or large. We also show how one may derive the minimum variance unbiased estimators (MVUEs) for a number of interesting and useful functionals of θ by combining some previous work from Mukhopadhyay and Cicconetti (2004) and Mukhopadhyay and Bhattacharjee (2010). All methodologies are illustrated with both simulated data and real data.

Corrections to “Unified Laguerre Polynomial-Series-Based Distribution of Small-Scale Fading Envelopes” [Oct 09 3988-3999

In this correspondence, we point out two typographical errors in Chai and Tjhung's paper, and we offer the correct formula of the unified Laguerre polynomial-series-based cumulative distribution function (cdf) for small-scale fading distributions. A Laguerre polynomial-series-based cdf formula for noncentral chi-square distribution is also provided as a special case of our unified cdf result.

The bivariate noncentral chi-square distribution – A compound distribution approach

This paper proposes the bivariate noncentral chi-square (BNC) distribution by compounding the Poisson probabilities with the bivariate central chi-square distribution. The probability density and cumulative distribution functions of the joint distribution of the two noncentral chi-square variables are derived for arbitrary values of the correlation coefficient, degrees of freedom(s), and noncentrality parameters. Computational procedures to calculate the upper tail probabilities as well as the percentile points for selected values of the parameters, for both equal and unequal degrees of freedom, are discussed. The graphical representation of the distribution for different values of the parameters are provided. Some applications of the distribution are outlined.

Random variate generation by numerical inversion when only the density is known

We present a numerical inversion method for generating random variates from continuous distributions when only the density function is given. The algorithm is based on polynomial interpolation of the inverse CDF and Gauss-Lobatto integration. The user can select the required precision, which may be close to machine precision for smooth, bounded densities; the necessary tables have moderate size. Our computational experiments with the classical standard distributions (normal, beta, gamma, t-distributions) and with the noncentral chi-square, hyperbolic, generalized hyperbolic, and stable distributions showed that our algorithm always reaches the required precision. The setup time is moderate and the marginal execution time is very fast and nearly the same for all distributions. Thus for the case that large samples with fixed parameters are required the proposed algorithm is the fastest inversion method known. Speed-up factors up to 1000 are obtained when compared to inversion algorithms developed for the specific distributions. This makes our algorithm especially attractive for the simulation of copulas and for quasi--Monte Carlo applications.