Noncentral Chi-square Random Variables Research Articles

Analytically Computing the Moments of a Conic Combination of Independent Noncentral Chi-Square Random Variables and Its Application for the Extended Cox–Ingersoll–Ross Process with Time-Varying Dimension

This paper focuses mainly on the problem of computing the γth, γ>0, moment of a random variable Yn:=∑i=1nαiXi in which the αi’s are positive real numbers and the Xi’s are independent and distributed according to noncentral chi-square distributions. Finding an analytical approach for solving such a problem has remained a challenge due to the lack of understanding of the probability distribution of Yn, especially when not all αi’s are equal. We analytically solve this problem by showing that the γth moment of Yn can be expressed in terms of generalized hypergeometric functions. Additionally, we extend our result to computing the γth moment of Yn when Xi is a combination of statistically independent Zi2 and Gi in which the Zi’s are distributed according to normal or Maxwell–Boltzmann distributions and the Gi’s are distributed according to gamma, Erlang, or exponential distributions. Our paper has an immediate application in interest rate modeling, where we can explicitly provide the exact transition probability density function of the extended Cox–Ingersoll–Ross (ECIR) process with time-varying dimension as well as the corresponding γth conditional moment. Finally, we conduct Monte Carlo simulations to demonstrate the accuracy and efficiency of our explicit formulas through several numerical tests.

Valuation of volatility derivatives with time-varying volatility: An analytical probabilistic approach using a mixture distribution for pricing nonlinear payoff volatility derivatives in discrete observation case

In this paper, we present an analytical probabilistic approach for pricing nonlinear payoff volatility derivatives with discrete sampling by assuming that the underlying asset price evolves according to the Black–Scholes model with time-varying volatility. A major difficulty to solve the pricing problem analytically is the volatility of the underlying asset price is time-varying, resulting the realized variance is distributed according to a mixture distribution, the probability density function of which is unknown. By utilizing the properties of a linear combination of noncentral chi-square random variables, we can calculate the expectation of square root of the realized variance analytically and provide formulas for pricing volatility swaps, volatility options, and variance options, including put–call parity relationships. Furthermore, we demonstrate an interesting application of our formulas by constructing simple closed-form approximate formulas for pricing the volatility derivatives for the constant elasticity of variance model. Finally, Monte Carlo simulations are conducted to illustrate the performance our approach, and effects of price volatility on the fair strike prices of the volatility derivatives are investigated and analyzed through several numerical experiments.

Bayesian optimization for a multiple-component system with target values

Bayesian optimization (BO) that employs the Gaussian process (GP) as a surrogate model has recently gained much attention in optimization of expensive black-box functions. In BO, the number of experiments necessary to optimize a function can be considerably reduced by sequentially selecting next design points that are optimal with respect to some sampling criterion. However, little research has been done to address the optimization of a multiple-component system where each component has a certain target value to meet. In this paper, we aim to find an optimal design parameter in the sense that the response function is close to the target value for every component. To this end, the squared errors from the targets are aggregated to produce an objective function. Instead of modeling this objective using GP as in the standard BO formulation, we place the GP prior over the response function. As a result, the distribution over the objective function follows that of the weighted sum of non-central chi-squared random variables (WSNC) due to the inter-dependency between responses. When components of the system are changed, the standard BO suffers inefficiency; however, our formulation enables us to retain a learned model, resulting in better efficiency. We compare the rates of convergence of different BO methods and other black-box optimization baselines using several test functions. The performance of our model is comparable to the standard BO when there is no change in the system, but the superiority of our method becomes clear when changes in the components occur.

Exponential Tail Bounds for Chisquared Random Variables

The paper derives some exponential tail bounds for central and non-central chisquared random variables. The bounds are simple and can easily be applied in statistical analysis. Especially relevant are the tail bounds for non-central chisquares, which are different from some of the other exponential bounds available in the literature, for example the one given in [1].

Beamforming Through Reconfigurable Intelligent Surfaces in Single-User MIMO Systems: SNR Distribution and Scaling Laws in the Presence of Channel Fading and Phase Noise

We consider a fading channel in which a multi-antenna transmitter communicates with a multi-antenna receiver through a reconfigurable intelligent surface (RIS) that is made of ${N}$ reconfigurable passive scatterers impaired by phase noise. The beamforming vector at the transmitter, the combining vector at the receiver, and the phase shifts of the ${N}$ scatterers are optimized in order to maximize the signal-to-noise-ratio (SNR) at the receiver. By assuming Rayleigh fading (or line-of-sight propagation) on the transmitter-RIS link and Rayleigh fading on the RIS-receiver link, we prove that the SNR is a random variable that is equivalent in distribution to the product of three (or two) independent random variables whose distributions are approximated by two (or one) gamma random variables and the sum of two scaled non-central chi-square random variables. The proposed analytical framework allows us to quantify the robustness of RIS-aided transmission to fading channels. For example, we prove that the amount of fading experienced on the transmitter-RIS-receiver channel linearly decreases with ${N}~\gg ~1.$ This proves that RISs of large size can be effectively employed to make fading less severe and wireless channels more reliable.

On the distribution of extended CIR model

We study the probability distribution of the interest rate in the extended Cox–Ingersoll–Ross model, where all the parameters are time-varying. We show that the distribution can be represented as that of a convergent series of weighted independent central and noncentral chi-square random variables. Simulation algorithms and their applications to finance have been discussed.

Statistics of Point Vortex Turbulence in Non-neutral Flows and in Flows with Translational and Rotational Symmetries

A theory (Esler and Ashbee in J Fluid Mech 779:275–308, 2015) describing the statistics of N freely-evolving point vortices in a bounded two-dimensional domain is extended. First, the case of a non-neutral vortex gas is addressed, and it is shown that the density of states function can be identified with the probability density function of an infinite sum of independent non-central chi-squared random variables, the details of which depend only on the shape of the domain. Equations for the equilibrium energy spectrum and other statistical quantities follow, the validity of which are verified against direct numerical simulations of the equations of motion. Second, domains with additional conserved quantities associated with a symmetry (e.g., circle, periodic channel) are investigated, and it is shown that the treatment of the non-neutral case can be modified to account for the additional constraint.

Characterizations of Noncentral Chi-Squared-Generating Covariance Structures for a Normally Distributed Random Vector

Let \({\mathbf {y}} \sim N_{n}\left ({\boldsymbol {\mu }}, {\mathbf {V}} \right )\), where y is a n×1 random vector and V is a n×n covariance matrix. We explicitly characterize the general form of the covariance structure V for which the family of quadratic forms \(\left \{{\mathbf {y}}^{\prime } {\mathbf {A}}_{i}{\mathbf {y}} \right \}^{k}_{i=1}\) for \(i \in \left \{1,...,k \right \}\), 2≤k≤n, is distributed as multiples of mutually independent non-central chi-squared random variables. We consider the case when the Ai’s and V are both nonnegative definite, including several cases where the Ai’s have special properties, and the case where the Ai’s are symmetric and V is positive definite. Our results generalize the work of Pavur (Sankhyā 51, 382–389, 1989), Baldessari (Comm. Statist. - Theory Meth. 16, 785–803, 1987), and Chaganty and Vaish (Linear Algebra Appl. 264, 421–437, 1997).

Holonomic gradient method for distribution function of a weighted sum of noncentral chi-square random variables

We apply the holonomic gradient method to compute the distribution function of a weighted sum of independent noncentral chi-square random variables. It is the distribution function of the squared length of a multivariate normal random vector. We treat this distribution as an integral of the normalizing constant of the Fisher–Bingham distribution on the unit sphere and make use of the partial differential equations for the Fisher–Bingham distribution.

Approximate ellipsoidal tolerance regions for multivariate normal populations

For the approximate computation of an ellipsoidal tolerance region for multivariate normal distributions, an approximation due to Liu et al. (2008) for the cdf of a positive linear combination of independent non-central chisquare random variables is used, along with Monte Carlo simulation. Such a computational approach is originally due to Krishnamoorthy and Mondal (2006, 2008), and the present work provides an improvement.

On the Capacity of Decode-and-Forward Relaying over Rician Fading Channels

In this letter, we derive the probability density function (PDF) and cumulative distribution function (CDF) of the minimum of two non-central Chi-square random variables with two degrees of freedom in terms of power series. With the help of the derived PDF and CDF, we obtain the exact ergodic capacity of the following adaptive protocols in a {decode-and-forward} (DF) cooperative system over dissimilar {Rician} fading channels: (i) constant power with optimal rate adaptation; (ii) optimal simultaneous power and rate adaptation; (iii) channel inversion with fixed rate. By using the analytical expressions of the capacity, it is observed that the optimal power and rate adaptation provides better capacity than the optimal rate adaptation with constant power from low to moderate signal-to-noise ratio values over dissimilar Rician fading channels. Despite low complexity, the channel inversion based adaptive transmission is shown to suffer from significant loss in capacity as compared to the other adaptive transmission based techniques over DF Rician channels.

A CURE for Noisy Magnetic Resonance Images: Chi-Square Unbiased Risk Estimation

n this article we derive an unbiased expression for the expected mean-squared error associated with continuously differentiable estimators of the noncentrality parameter of a chisquare random variable. We then consider the task of denoising squared-magnitude magnetic resonance image data, which are well modeled as independent noncentral chi-square random variables on two degrees of freedom. We consider two broad classes of linearly parameterized shrinkage estimators that can be optimized using our risk estimate, one in the general context of undecimated filterbank transforms, and another in the specific case of the unnormalized Haar wavelet transform. The resultant algorithms are computationally tractable and improve upon most state-of-the-art methods for both simulated and actual magnetic resonance image data.

A novel trace test for the mean parameters in a multivariate growth curve model

A trace test for the mean parameters of the growth curve model is proposed. It is constructed using the restricted maximum likelihood followed by an estimated likelihood ratio approach. The statistic reduces to the Lawley–Hotelling trace test for the Multivariate Analysis of Variance (MANOVA) models. Our test statistic is, therefore, a natural extension of the classical trace test to GMANOVA models. We show that the distribution of the test under the null hypothesis does not depend on the unknown covariance matrix Σ . We also show that the distributions under the null and alternative hypotheses can be represented as sums of weighted central and non-central chi-square random variables, respectively. Under the null hypothesis, the Satterthwaite approximation is used to get an approximate critical point. A novel Satterthwaite type approximation is proposed to obtain an approximate power. A simulation study is performed to evaluate the performance of our proposed test and numerical examples are provided as illustrations.

ANALYTIC PRICING OF CONTINGENT CLAIMS UNDER THE REAL-WORLD MEASURE

This article derives a series of analytic formulae for various contingent claims under the real-world probability measure using the stylised minimal market model (SMMM). This model provides realistic dynamics for the growth optimal portfolio (GOP) as a well-diversified equity index. It captures both leptokurtic returns with correct tail properties and the leverage effect. Under the SMMM, the discounted GOP takes the form of a time-transformed squared Bessel process of dimension four. From this property, one finds that the SMMM possesses a special and interesting relationship to non-central chi-square random variables with zero degrees of freedom. The analytic formulae derived under the SMMM include options on the GOP, options on exchange prices and options on zero-coupon bonds. For options on zero-coupon bonds, analytic prices facilitate efficient calculation of interest rate caps and floors.

Probability density of nonlinear phase noise

The probability density of nonlinear phase noise, often called the Gordon–Mollenauer effect, is derived analytically. The nonlinear phase noise can be accurately modeled as the summation of a Gaussian random variable and a noncentral chi-square random variable with two degrees of freedom. Using the received intensity to correct for the phase noise, the residual nonlinear phase noise can be modeled as the summation of a Gaussian random variable and the difference of two noncentral chi-square random variables with two degrees of freedom. The residual nonlinear phase noise can be approximated by Gaussian distribution better than the nonlinear phase noise without correction.

On the trace of a wishart

A derivation based on spectral decomposition allows specifying the characteristic function of the trace of a singular or nonsingular, central or noncentral, true or pseudo-Wishart. The trace equals a weighted sum of noncentral chi-squared random variables and constants. We describe computational methods.

A unified method for computing probabilities of misclassification when the covariance matricesare known

The Qos and Qm are two leading The Linear and Quadratic discriminant functions are shown to be expressible as linear combinations of independent non-central chi-square random variables when the covariance matrices are known. Hence the probabilityof misclassification associated with them could be computed using numerical integration procedures.

Estimators with nondecreasing risk: application of a chi-squared identity

By using an apparently little known fact about concave functions together with a new expectation identity for noncentral chi-squared random variables, a characterization of risk functions of Stein-type estimators is obtained. In particular, concavity of the function appearing in the shrinkage factor is related to the estimator's risk function being nondecreasing.

A simple way of computing the inverse moments of a non-central chi-square random variable

Theorems are given which permit recurrence formulas to be used in the evaluation of the inverse moments on non-central chi-square random variables.

The effect of sample design on the comparison of associations

SUMMARY A family of measures of association designed to compare the inherent association in two r x c contingency tables was proposed by Altham (1970). Under simple random sampling the measures are asymptotically distributed as noncentral chi-squared random variables. In this paper we show that under complex sampling this distribution changes to a mixture and that ignoring the design may lead to misleading results. We derive measures of the misestimation and measures of association that are adequate for most standard designs and which have a simple distribution theory. Also, simple corrections for the simple random sampling statistic are given. Two cases of stratified sampling and clustering are discussed.