Chi-squared Random Variables Research Articles

On the representation and computational aspects of the distribution of a linear combination of independent noncentral chi-squared random variables

This article presents two new results relevant to a linear combination of χ2 random variables. The first result expresses the cumulative distribution function as a simple multiple of a certain tilted density. The second result shows that in important cases the inversion integral for the density may be expressed as a sum of relatively simple real integrals which, using suitable numerical methods, are straightforward to compute quickly and accurately.

Constrained Bayesian Method for Testing Equi-Correlation Coefficient

The problem of testing the equi-correlation coefficient of a standard symmetric multivariate normal distribution is considered. Constrained Bayesian and classical Bayes methods, using the maximum likelihood estimation and Stein’s approach, are examined. For the investigation of the obtained theoretical results and choosing the best among them, different practical examples are analyzed. The simulation results showed that the constrained Bayesian method (CBM) using Stein’s approach has the advantage of making decisions with higher reliability for testing hypotheses concerning the equi-correlation coefficient than the Bayes method. Also, the use of this approach with the probability distribution of linear combinations of chi-square random variables gives better results compared to that of using the integrated probability distributions in terms of providing both the necessary precisions as well as convenience of implementation in practice. Recommendations towards the use of the proposed methods for solving practical problems are given.

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.

Distance-Based Regression Analysis for Measuring Associations

Distance-based regression model, as a nonparametric multivariate method, has been widely used to detect the association between variations in a distance or dissimilarity matrix for outcomes and predictor variables of interest in genetic association studies, genomic analyses, and many other research areas. Based on it, a pseudo-F statistic which partitions the variation in distance matrices is often constructed to achieve the aim. To the best of our knowledge, the statistical properties of the pseudo-F statistic has not yet been well established in the literature. To fill this gap, the authors study the asymptotic null distribution of the pseudo-F statistic and show that it is asymptotically equivalent to a mixture of chi-squared random variables. Given that the pseudo-F test statistic has unsatisfactory power when the correlations of the response variables are large, the authors propose a square-root F-type test statistic which replaces the similarity matrix with its square root. The asymptotic null distribution of the new test statistic and power of both tests are also investigated. Simulation studies are conducted to validate the asymptotic distributions of the tests and demonstrate that the proposed test has more robust power than the pseudo-F test. Both test statistics are exemplified with a gene expression dataset for a prostate cancer pathway.

Fluctuations of subgraph counts in graphon based random graphs

AbstractGiven a graphon $W$ and a finite simple graph $H$ , with vertex set $V(H)$ , denote by $X_n(H, W)$ the number of copies of $H$ in a $W$ -random graph on $n$ vertices. The asymptotic distribution of $X_n(H, W)$ was recently obtained by Hladký, Pelekis, and Šileikis [17] in the case where $H$ is a clique. In this paper, we extend this result to any fixed graph $H$ . Towards this we introduce a notion of $H$ -regularity of graphons and show that if the graphon $W$ is not $H$ -regular, then $X_n(H, W)$ has Gaussian fluctuations with scaling $n^{|V(H)|-\frac{1}{2}}$ . On the other hand, if $W$ is $H$ -regular, then the fluctuations are of order $n^{|V(H)|-1}$ and the limiting distribution of $X_n(H, W)$ can have both Gaussian and non-Gaussian components, where the non-Gaussian component is a (possibly) infinite weighted sum of centred chi-squared random variables with the weights determined by the spectral properties of a graphon derived from $W$ . Our proofs use the asymptotic theory of generalised $U$ -statistics developed by Janson and Nowicki [22]. We also investigate the structure of $H$ -regular graphons for which either the Gaussian or the non-Gaussian component of the limiting distribution (but not both) is degenerate. Interestingly, there are also $H$ -regular graphons $W$ for which both the Gaussian or the non-Gaussian components are degenerate, that is, $X_n(H, W)$ has a degenerate limit even under the scaling $n^{|V(H)|-1}$ . We give an example of this degeneracy with $H=K_{1, 3}$ (the 3-star) and also establish non-degeneracy in a few examples. This naturally leads to interesting open questions on higher order degeneracies.

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.

Random sections of ellipsoids and the power of random information

We study the circumradius of the intersection of an m m -dimensional ellipsoid E \mathcal E with semi-axes σ 1 ≥ ⋯ ≥ σ m \sigma _1\geq \dots \geq \sigma _m with random subspaces of codimension n n , where n n can be much smaller than m m . We find that, under certain assumptions on σ \sigma , this random radius R n = R n ( σ ) \mathcal {R}_n=\mathcal {R}_n(\sigma ) is of the same order as the minimal such radius σ n + 1 \sigma _{n+1} with high probability. In other situations R n \mathcal {R}_n is close to the maximum σ 1 \sigma _1 . The random variable R n \mathcal {R}_n naturally corresponds to the worst-case error of the best algorithm based on random information for L 2 L_2 -approximation of functions from a compactly embedded Hilbert space H H with unit ball E \mathcal E . In particular, σ k \sigma _k is the k k th largest singular value of the embedding H ↪ L 2 H\hookrightarrow L_2 . In this formulation, one can also consider the case m = ∞ m=\infty and we prove that random information behaves very differently depending on whether σ ∈ ℓ 2 \sigma \in \ell _2 or not. For σ ∉ ℓ 2 \sigma \notin \ell _2 we get E [ R n ] = σ 1 \mathbb {E}[\mathcal {R}_n] = \sigma _1 and random information is completely useless. For σ ∈ ℓ 2 \sigma \in \ell _2 the expected radius tends to zero at least at rate o ( 1 / n ) o(1/\sqrt {n}) as n → ∞ n\to \infty . In the important case \[ σ k ≍ k − α ln − β ⁡ ( k + 1 ) , \sigma _k \asymp k^{-\alpha } \ln ^{-\beta }(k+1), \] where α > 0 \alpha > 0 and β ∈ R \beta \in \mathbb {R} (which corresponds to various Sobolev embeddings), we prove E [ R n ( σ ) ] ≍ { σ 1 a m p ; if α > 1 / 2 \, or \, β ≤ α = 1 / 2 , 2 m m σ n + 1 ln ⁡ ( n + 1 ) a m p ; if β > α = 1 / 2 , 2 m m σ n + 1 a m p ; if α > 1 / 2. \begin{equation*} \mathbb E [\mathcal {R}_n(\sigma )] \asymp \left \{\begin {array}{cl} \sigma _1 & \text {if} \quad \alpha >1/2 \text {\, or \,} \beta \leq \alpha =1/2, {2mm} \\ \sigma _{n+1} \, \sqrt {\ln (n+1)} \quad & \text {if} \quad \beta >\alpha =1/2, {2mm} \\ \sigma _{n+1} & \text {if} \quad \alpha >1/2. \end{array}\right . \end{equation*} In the proofs we use a comparison result for Gaussian processes à la Gordon, exponential estimates for sums of chi-squared random variables, and estimates for the extreme singular values of (structured) Gaussian random matrices. The upper bound is constructive. It is proven for the worst case error of a least squares estimator.

Testing independence between discrete random variables

In this paper we develop a likelihood ratio test for independence between multivariate discrete random vectors whose components may be dependent or may have different marginal distributions. A special case that tests independence between two discrete random variables is also discussed. The model under the null hypothesis is constructed based on the marginal distributions of the random variables, while the model under the alternative is built by adding a common random effect which is not restricted to follow the same distribution as those of the other variables. The test statistic is shown to admit an asymptotic distribution of a weighted sum of chi-square random variables, each with one degree of freedom. In order to avoid the high computational cost of finding the parameters of the asymptotic distribution, a permutation-based procedure is introduced. It is shown in simulations that the permutation-based realization yields empirical level close to the nominal level, while achieving power comparable to or higher than that of existing competitors in the literature. The application of the test is illustrated with a real data set.

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.

Chi-Square and Student Bridge Distributions and the Behrens–Fisher Statistic

We prove that the Behrens–Fisher statistic follows a Student bridge distribution, the mixing coefficient of which depends on the two sample variances only through their ratio. To this end, it is first shown that a weighted sum of two independent normalized chi-square distributed random variables is chi-square bridge distributed, and secondly that the Behrens–Fisher statistic is based on such a variable and a standard normally distributed one that is independent of the former. In case of a known variance ratio, exact standard statistical testing and confidence estimation methods apply without the need for any additional approximations. In addition, a three pillar bridges explanation is given for the choice of degrees of freedom in Welch’s approximation to the exact distribution of the Behrens–Fisher statistic.

New Measure of the Bivariate Asymmetry

A new measure of the bivariate asymmetry of a dependence structure between two random variables is introduced based on copula characteristic function. The proposed measure is represented as the discrepancy between the rank–based distance correlation computed over two complementary order-preserved sets. General properties of the measure are established, as well as an explicit expression for the empirical version. It is shown that the proposed measure is asymptotically equivalent to a fourth–order degenerate V -statistics and that the limit distributions have representations in terms of weighted sums of an independent chi-square random variables. Under dependent random variables, the asymptotic behavior of bivariate distance covariance and variance process is demonstrated. Numerical examples illustrate the properties of the measures.

A Distribution on the Simplex Arising from Inverted Chi-square Random Variables with Odd Degrees of Freedom

We consider the random variables where are independent inverted chi-square r.v. with νi degrees of freedom. The probability density function of is obtained. When are odd, it is shown how to obtain in a fairly easy way a closed form expression for the expectation of Differentiating this expression with respect to the βi, one can find the moments of the random variables Zi. For the particular case of odd degrees of freedom, closed form expressions for the pdf of the univariate and multivariate marginal distributions of Z are also derived. The distribution of Z may be an alternative to the Dirichlet distribution for modeling compositional data.

Monochromatic subgraphs in randomly colored graphons

Let T(H,Gn) be the number of monochromatic copies of a fixed connected graph H in a uniformly random coloring of the vertices of the graph Gn. In this paper we give a complete characterization of the limiting distribution of T(H,Gn), when {Gn}n≥1 is a converging sequence of dense graphs. When the number of colors grows to infinity, depending on whether the expected value remains bounded, T(H,Gn) either converges to a finite linear combination of independent Poisson variables or a normal distribution. On the other hand, when the number of colors is fixed, T(H,Gn) converges to a (possibly infinite) linear combination of independent centered chi-squared random variables. This generalizes the classical birthday problem, which involves understanding the asymptotics of T(Ks,Kn), the number of monochromatic s-cliques in a complete graph Kn (s-matching birthdays among a group of n friends), to general monochromatic subgraphs in a network.

Evaluating Vector Multiplicative Error Models with the Hosking–Ljung–Box Portmanteau Test and Kernel-Based Test Statistics

Multivariate nonlinear time series models have experienced many developments for modeling data coming from financial applications. Several financial time series are realizations from nonnegative processes. An important class of models is composed of vector multiplicative error models (vMEM), which can describe contemporaneous correlations among innovations and the dynamic interdependencies among variables. Modeling and estimation issues have been addressed, but few diagnostic checking procedures are available. Here, new tests are proposed to check vMEM models. The asymptotic distributions of the popular Hosking–Ljung–Box (HLB) test statistics are found to converge in distribution to weighted sums of independent Chi-squared random variables under the null hypothesis of adequacy. A generalized HLB test statistic is motivated by comparing a vector spectral density estimator of the residuals with the spectral density calculated under the null hypothesis. Under general conditions, that kind of approach leads to consistent procedures and to powerful measures of lack of fit. To improve the finite sample properties, the spectral test statistics rely on the power transformation of Chen and Deo (J R Stat Soc Ser B 66:117–130, 2004). Appealing properties of the spectral procedures include the distribution-free property and the fact that they converge in distribution to convenient standard normal distributions under the null hypothesis. Simulation experiments are reported to appreciate the properties of the methods. An application using financial data previously analyzed by Cipollini et al. (J Appl Econom 28:1067–1086, 2013) illustrates the merits of our procedures.

On the limiting distribution of sample central moments

We investigate the limiting behavior of sample central moments, examining the special cases where the limiting (as the sample size tends to infinity) distribution is degenerate. Parent (non-degenerate) distributions with this property are called singular, and we show in this article that the singular distributions contain at most three supporting points. Moreover, using the delta-method, we show that the (second-order) limiting distribution of sample central moments from a singular distribution is either a multiple, or a difference of two multiples of independent Chi-square random variables with one degree of freedom. Finally, we present a new characterization of normality through the asymptotic independence of the sample mean and all sample central moments.

The modified permutation entropy-based independence test of time series

In time series, it is essential to check the independence of data by means of a proper method or an appropriate statistical test before any further analysis. Therefore, among different independence tests, a powerful and productive test has been introduced by Matilla-García and Marín via m-dimensional vectorial process, in which the value of the process at time t includes m-histories of the primary process. However, this method causes a dependency for the vectors even when the independence assumption of random variables is considered. Considering this dependency, a modified test is obtained in this article through presenting a new asymptotic distribution based on weighted chi-square random variables. Also, some other alterations to the test have been made via bootstrap method and by controlling the overlap. Compared with the primary test, it is obtained that not only the modified test is more accurate but also, it possesses higher power.

Diagnostic Checking in Multivariate ARMA Models With Dependent Errors Using Normalized Residual Autocorrelations

ABSTRACTIn this paper, we derive the asymptotic distribution of normalized residual empirical autocovariances and autocorrelations under weak assumptions on the noise. We propose new portmanteau statistics for vector autoregressive moving average models with uncorrelated but nonindependent innovations by using a self-normalization approach. We establish the asymptotic distribution of the proposed statistics. This asymptotic distribution is quite different from the usual chi-squared approximation used under the independent and identically distributed assumption on the noise, or the weighted sum of independent chi-squared random variables obtained under nonindependent innovations. A set of Monte Carlo experiments and an application to the daily returns of the CAC40 is presented. Supplementary materials for this article are available online.

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.