Exact Probability Density Function Research Articles

The distribution of the product of independent variance-gamma random variables

Let X and Y be independent variance-gamma random variables with zero location parameter; then the exact probability density function of the product XY is derived. Some basic distributional properties are also derived, including formulas for the cumulative distribution function and the characteristic function, as well as asymptotic approximations for the density, tail probabilities and the quantile function. As special cases, we deduce some key distributional properties for the product of two independent asymmetric Laplace random variables as well as the product of four jointly correlated zero mean normal random variables with a particular block diagonal covariance matrix. As a by-product of our analysis, we deduce some new reduction formulas for the Meijer G-function.

On the Exact Sum PDF and CDF of α-μ Variates

The sum of random variables (RVs) appears extensively in wireless communications, at large, both conventional and advanced, and has been subject of longstanding research. The statistical characterization of the referred sum is crucial to determine the performance of such communications systems. Although efforts have been undertaken to unveil these sum statistics, e.g., probability density function (PDF) and cumulative distribution function (CDF), no general efficient nor manageable solutions capable of evaluating the exact sum PDF and CDF are available to date. The only formulations are given in terms of either the multi-fold Brennan’s integral or the multivariate Fox <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> -function. Unfortunately, these methods are only feasible up to a certain number of RVs, meaning that when the number of RVs in the sum increases, the computation of the sum PDF and CDF is subject to stability problems, convergence issues, or inaccurate results. In this paper, we derive new, simple, exact formulations for the PDF and CDF of the sum of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> independent and identically distributed α-μ RVs. Unlike the available solutions, the computational complexity of our analytical expressions is independent of the number of summands. Capitalizing on our unprecedented findings, we analyze, in exact and asymptotic manners, the performance of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> -branch pre-detection equal-gain combining and maximal-ratio combining receivers over α-μ fading environments. The coding and diversity gains of the system for both receivers are analyzed and quantified. Moreover, numerical simulations show that the computation time reduces drastically when using our expressions, which are arguably the most efficient and manageable formulations derived so far.

An Age-Dependent Prognostic Model for Nonlinear Degrading Devices Based on Diffusion Process

Degradation modeling based remaining useful life (RUL) predicting has become a significant basis of prognotics and system health management (PHM), and has drawn much favor of both scholars and engineers in the field of reliability. Based on a purely time-dependent diffusion process, this paper presents a nonlinear degradation model with special focus on the proportional relationship between the age-dependent expectation and variance of the degradation processes of the concerned devices. Exact probability density function (PDF) and cumulative distribution function (CDF) of life time and RUL in explicit forms are derived under the concept of first hitting time (FHT), incorporating the item-to-item variability. An framework of maximizing the likelihood function has been proposed to estimate the unknown parameters utilizing condition monitoring data of the devices. Case studies of fatigue crack length data of alloy and capacity data of electrolytic capacitors have been presented to illustrate the proposed prognostics model. The results show that the proposed degradation model can be well fitted to the nonlinear degradation data and provide an accurate prediction of RUL, and thus demonstrate the effectiveness of the proposed model.

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&gt;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.

A stochastic model of edge-localized modes in magnetically confined plasmas.

Magnetically confined plasmas are far from equilibrium and pose considerable challenges in statistical analysis. We discuss a non-perturbative statistical method, namely a time-dependent probability density function (PDF) approach that is potentially useful for analysing time-varying, large, or non-Gaussian fluctuations and bursty events associated with instabilities in the low-to-high confinement transition and the H-mode. Specifically, we present a stochastic Langevin model of edge-localized modes (ELMs) by including stochastic noise terms in a previous ODE ELM model. We calculate exact time-dependent PDFs by numerically solving the Fokker-Planck equation and characterize time-varying statistical properties of ELMs for different energy fluxes and noise amplitudes. The stochastic noise is shown to introduce phase-mixing and plays a significant role in mitigating extreme bursts of large ELMs. Furthermore, based on time-dependent PDFs, we provide a path-dependent information geometric theory of the ELM dynamics and demonstrate its utility in capturing self-regulatory relaxation oscillations, bursts and a sudden change in the system. This article is part of a discussion meeting issue 'H-mode transition and pedestal studies in fusion plasmas'.

Alternative expressions of the PDF and CDF for Gamma, [formula omitted] and [formula omitted] shadowed distributions with applications in wireless communications

There are two approaches well known in the literature for reducing the complexity in averaged performance analyses of wireless communications systems (i.e., for alleviating the computational difficulty in averaged performance evaluations). The first method, known as the moment generating function MGF-based approach, aims to convert the instantaneous performance measure into an exponential integration or a certain sum of exponential functions of different scales. The second approach, which is the focus of this article, aims to convert the exact probability density function PDF and cumulative distribution function CDF of fading distributions into the sum of exponential functions of various scales. As such, we propose weighted sum of exponential functions as alternative closed-form approximations for the PDF and CDF of gamma distribution, η−μ distribution, and κ−μ shadowed distribution with integer fading parameters, and we present how these proposed expressions can be easily applied to the performance analysis of wireless communication systems operating over Nakagami-m fading channels, η−μ fading channels, and κ−μ shadowed fading channels.

Data-driven statistical nonlinearization technique based on information entropy

For a given nonlinear stochastic system, an exact stationary probability density function, which is important for system response prediction and control, is not always available. Owing to the development of various sensors and measurement systems, data acquisition and processing have become increasingly convenient. Inspired by traditional statistical nonlinearization techniques and measured data, a data-driven statistical nonlinearization technique was developed in this study based on information entropy. First, a class of appropriate equivalent systems with exact solutions was selected. Subsequently, the optimal parameters of the equivalent systems were determined based on the minimization of the Shannon information entropy and some statistical moments of the system state from the measured data. The derivation of the optimal parameters of the equivalent systems was then converted into an extremum problem of the entropy function of several variables. Finally, three typical nonlinear stochastic systems were presented to illustrate and validate the effectiveness and accuracy of the proposed procedure. The proposed data-driven statistical nonlinearization technique can be used to provide more comprehensive information on the system response, which is important for system response evaluation and control.

Performance Analysis and Optimization for Jammer-Aided Multiantenna UAV Covert Communication

Unmanned aerial vehicles (UAVs) have attracted a lot of research attention because of their high mobility and low cost in serving as temporary aerial base stations (BSs) and providing high data rates for next-generation communication networks. To protect user privacy while avoiding detection by a warden, we investigate a jammer-aided UAV covert communication system, which aims to maximize the user’s covert rate with optimized transmit and jamming power. The UAV is equipped with multi-antennas to serve multi-users simultaneously and enhance the Quality of Service. By considering the general composite fading and shadowing channel models, we derive the exact probability density (PDF) and cumulative distribution functions (CDF) of the signal-to-interference-plus-noise ratio (SINR). The obtained PDF and CDF are used to derive the closed-form expressions for detection error probability and covert rate. Furthermore, the covert rate maximization problem is formulated as a Nash bargaining game, and the Nash bargaining solution (NBS) is introduced to investigate the negotiation among users. To solve the NBS, we propose two algorithms, i.e., particle swarm optimization-based and joint two-stage power allocation algorithms, to achieve covertness and high data rates under the warden’s optimal detection threshold. All formulated problems are proven to be convex, and the complexity is analyzed. The numerical results are presented to verify the theoretical performance analysis and show the effectiveness and success of achieving the covert communication of our algorithms.

Statistical properties of partially coherent beams transmitted through nonlinear Kerr thin layers

Investigation of the propagation of partially coherent beams through nonlinear media is very difficult because of the need to deal with the fourth- or higher-order correlation function. Here a simple theoretical model is proposed to derive analytical results on the transmission of partially coherent beams through nonlinear Kerr thin layers. Considering only the self-phase modulation effect and neglecting the diffraction effect, the exact probability density function of the transmitted field at arbitrary two points has been derived and found to be not Gaussian. Statistical properties of the transmitted Gaussian Schell-model beams are studied in detail. The moment theorem is no longer valid so that the fourth-order correlation function cannot be decomposed as the sum of products of the second-order correlation functions. Numerical results are presented to see the effects of nonlinearity on the deviation from Gaussian statistics.

Autocorrelation functions and ergodicity in diffusion with stochastic resetting

Diffusion with stochastic resetting is a paradigm of resetting processes. Standard renewal or master equation approach are typically used to study steady state and other transport properties such as average, mean squared displacement etc. What remains less explored is the two time point correlation functions whose evaluation is often daunting since it requires the implementation of the exact time dependent probability density functions of the resetting processes which are unknown for most of the problems. We adopt a different approach that allows us to write a stochastic solution for a single trajectory undergoing resetting. Moments and the autocorrelation functions between any two times along the trajectory can then be computed directly using the laws of total expectation. Estimation of autocorrelation functions turns out to be pivotal for investigating the ergodic properties of various observables for this canonical model. In particular, we investigate two observables (i) sample mean which is widely used in economics and (ii) time-averaged-mean-squared-displacement (TAMSD) which is of acute interest in physics. We find that both diffusion and drift–diffusion processes with resetting are ergodic at the mean level unlike their reset-free counterparts. In contrast, resetting renders ergodicity breaking in the TAMSD while both the stochastic processes are ergodic when resetting is absent. We quantify these behaviors with detailed analytical study and corroborate with extensive numerical simulations. Our results can be verified in experimental set-ups that can track single particle trajectories and thus have strong implications in understanding the physics of resetting.

Exact Time-Dependent Solutions and Information Geometry of a Rocking Ratchet

The noise-induced transport due to spatial symmetry-breaking is a key mechanism for the generation of a uni-directional motion by a Brownian motor. By utilising an asymmetric sawtooth periodic potential and three different types of periodic forcing G(t) (sinusoidal, square and sawtooth waves) with period T and amplitude A, we investigate the performance (energetics, mean current, Stokes efficiency) of a rocking ratchet in light of thermodynamic quantities (entropy production) and the path-dependent information geometric measures. For each G(t), we calculate exact time-dependent probability density functions under different conditions by varying T, A and the strength of the stochastic noise D in an unprecedentedly wide range. Overall similar behaviours are found for different cases of G(t). In particular, in all cases, the current, Stokes efficiency and the information rate normalised by A and D exhibit one or multiple local maxima and minima as A increases. However, the dependence of the current and Stokes efficiency on A can be quite different, while the behaviour of the information rate normalised by A and D tends to resemble that of the Stokes efficiency. In comparison, the irreversibility measured by a normalised entropy production is independent of A. The results indicate the utility of the information geometry as a proxy of a motor efficiency.

Exact Statistical Distribution of the Body Mass Index (BMI): Analysis and Experimental Confirmation

Body Mass Index (BMI), defined as the ratio of individual mass (in kilograms) to the square of the associated height (in meters), is one of the most widely discussed and utilized risk factors in medicine and public health, given the increasing obesity worldwide and its relation to metabolic disease. Statistically, BMI is a composite random variable, since human weight (converted to mass) and height are themselves random variables. Much effort over the years has gone into attempts to model or approximate the BMI distribution function. This paper derives the mathematically exact BMI probability density function (PDF), as well as the exact bivariate PDF for human weight and height. Taken together, weight and height are shown to be correlated bivariate lognormal variables whose marginal distributions are each lognormal in form. The mean and variance of each marginal distribution, together with the linear correlation coefficient of the two distributions, provide 5 nonadjustable parameters for a given population that uniquely determine the corresponding BMI distribution, which is also shown to be lognormal in form. The theoretical analysis is tested experimentally by gender against a large anthropometric data base, and found to predict with near perfection the profile of the empirical BMI distribution and, to great accuracy, individual statistics including mean, variance, skewness, kurtosis, and correlation. Beyond solving a longstanding statistical problem, the significance of these findings is that, with knowledge of the exact BMI distribution functions for diverse populations, medical and public health professionals can then make better informed statistical inferences regarding BMI and public health policies to reduce obesity.

Ergodicity and invariant measures for a diffusing passive scalar advected by a random channel shear flow and the connection between the Kraichnan–Majda model and Taylor–Aris Dispersion

We study the long time behavior of an advection–diffusion equation with a random shear flow which depends on a stationary Ornstein–Uhlenbeck (OU) process in parallel-plate channels enforcing the no-flux boundary conditions. We derive a closed form formula for the long time asymptotics of the arbitrary N-point correlator using the ground state eigenvalue perturbation approach proposed in Bronski and McLaughlin (1997). In turn, appealing to the conclusion of the Hausdorff moment problem (Shohat and Tamarkin, 1943), we discover a diffusion equation with a random drift and deterministic enhanced diffusion possessing the exact same probability density function at long times. The strategy we presented is not only restricted to the parallel-plate channel domain. The same methods can derive effective equations for a straight channel with uniform arbitrary cross-section. Such equations enjoy many ergodic properties which immediately translate to ergodicity results for the original problem. In particular, we establish that the first two Aris moments using a single realization of the random field can be used to explicitly construct all ensemble averaged moments. Also, the first two ensemble averaged moments explicitly predict any long time centered Aris moment. Our formulae quantitatively depict the dependence of the deterministic effective diffusion on the interaction between spatial structure of flow and random temporal fluctuation. Further, this approximation provides many identities regarding the stationary OU process dependent time integral. We derive explicit formulae for the decaying passive scalar’s long time limiting probability density function (PDF) for different types of initial conditions (e.g. deterministic and random).

Hypothesis testing in multivariate normal models with block circular covariance structures.

In this article, we address the problem of simultaneous testing hypothesis about mean and covariance matrix for repeated measures data when both the mean vector and covariance matrix are patterned. In particular, tests about the mean vector under block circular and doubly exchangeable covariance structures have been considered. The null distributions are established for the corresponding likelihood ratio test statistics, and expressions for the exact or near-exact probability density and cumulative distribution functions are obtained. The application of the results is illustrated by both a simulation study and a real-life dataexample.

On the limitations of the Wiener path integral most probable path technique for solving nonlinear Itô stochastic differential equations

The Wiener path integral (WPI) technique for determining the solution of a given nonlinear Itô stochastic differential equation (SDE) requires the solution of an Euler–Lagrange equation for the determination of the SDE’s associated most probable path. The derived most probable path family of functions is then used for approximating the SDE’s associated probability density function (PDF). In this paper, an example of modelling a wind turbine’s power is proposed where, although multiple most probable paths solutions exist for the associated SDE, only one can be used for the purpose of determining its solution PDF, however, the classical most probable path WPI methodology is not able of determining it. It is subsequently shown that using an enhanced semi-analytical version of the WPI based solution methodology results in the determination of the SDE’s exact solution PDF effortlessly, which can also be used for choosing the aforementioned most probable path solution which leads to the exact solution in a distributional sense.

An Asymptotic Test for A Single Outlier in Linear Regression Models

It is not uncommon to find an outlier in the response variable in linear regression. Such a deviant value needs to be detected and scrutinized to find out why it is not in agreement with its fitted value. Srikantan [1] has developed a test statistic for detecting the presence of an outlier in the response variable in a multiple linear regression model. Approximate critical values of this test statistic are available and are obtained based on the first-order Bonferroni upper bound. The exact critical values are not available and a result of that, tests carried out on the basis of this approximate critical values may not be very accurate. In this paper, we obtained more accurate and precise critical values of this test statistic for large sample sizes (herein called asymptotic critical values) to improve on the tests that use these critical values. The procedure involved using the exact probability density function of this test statistic to obtain its asymptotic critical values. We then compared these asymptotic critical values with the approximate critical values obtained. An application to simulation results for linear regression models was used to examine the power of this test statistic. The asymptotic critical values obtained were found to be more accurate and precise. Also, the test performed better under these asymptotic values (the power performance of this test statistic was found to better when the asymptotic critical values were used).

Stationary distribution and density function expression for a stochastic SIQRS epidemic model with temporary immunity.

Recently, considering the temporary immunity of individuals who have recovered from certain infectious diseases, Liu et al. (Phys A Stat Mech Appl 551:124152, 2020) proposed and studied a stochastic susceptible-infected-recovered-susceptible model with logistic growth. For a more realistic situation, the effects of quarantine strategies and stochasticity should be taken into account. Hence, our paper focuses on a stochastic susceptible-infected-quarantined-recovered-susceptible epidemic model with temporary immunity. First, by means of the Khas’minskii theory and Lyapunov function approach, we construct a critical value {mathscr {R}}_0^S corresponding to the basic reproduction number {mathscr {R}}_0 of the deterministic system. Moreover, we prove that there is a unique ergodic stationary distribution if {mathscr {R}}_0^S>1. Focusing on the results of Zhou et al. (Chaos Soliton Fractals 137:109865, 2020), we develop some suitable solving theories for the general four-dimensional Fokker–Planck equation. The key aim of the present study is to obtain the explicit density function expression of the stationary distribution under {mathscr {R}}_0^S>1. It should be noted that the existence of an ergodic stationary distribution together with the unique exact probability density function can reveal all the dynamical properties of disease persistence in both epidemiological and statistical aspects. Next, some numerical simulations together with parameter analyses are shown to support our theoretical results. Last, through comparison with other articles, results are discussed and the main conclusions are highlighted.

Building a Cpk control chart – A novel and practical method for practitioners

This paper covers a hitherto unresolved problem, the search for a practical method for the Cpk^ control chart. The simplicity of the proposed method facilitates its immediate application. This control chart method reduces the complexity of the current paradigm for capability and quality control studies, while increasing sensitivity to out-of-control signals and improving the early detection aspect, as the ARL results show. The concept of distance implicit in Cpk definition is used to derive a new method based on chi-square distribution. The goodness of fit of the derived expression, which approximates the distribution of the Cpk^ statistic of the proposed method to the reference distribution of the simulated values and to the exact probability density function, was confirmed graphically, through the comparison of the histograms and the probability density functions, and analytically, using the F-test.

Random Response and Crossing Rate of Fractional Order Nonlinear System with Impact

The random response and mean crossing rate of the fractional order nonlinear system with impact are investigated through the equivalent nonlinearization technique. The random additive excitation is Gaussian white noise, while the impact is described by a phenomenological model, which is developed from the actual impact process experiments. Based on the equivalent nonlinearization technique, one class of random nonlinear system with exact probability density function (PDF) solution of response is selected. The criterion of the appropriate equivalent nonlinear system is the similarity with the original system on the damping, stiffness, and inertia. The more similar, the higher the precision. The optimal unknown parameters of the equivalent random nonlinear system in the damping and stiffness terms are determined by the rule of smallest mean-square difference. In the view of equivalent nonlinearization technique, the response of the original system is the same as that of the equivalent system with the optimal unknown parameters in analytical solution manner. Then, the mean crossing rate is derived from stationary PDF. The consistence between the results from proposed technique and Monte Carlo simulation reveals the accuracy of the proposed analytical procedure.

Evaluation of manifold representations of chemistry in stratified, swirl-stabilized flames

The validity of assuming that a simple, low-dimensional manifold can sufficiently accurately represent the chemical state in a turbulent flame is investigated. The experimental data from the Cambridge/Sandia stratified swirl burner working in nine different configurations are post-processed to explore the effects of coordinate, swirl flow ratio, and stratification factor on the conditionally-averaged reactive scalars. First, the mixture fraction and thermal progress variable are employed to construct a two-dimensional conditional manifold by using all of the data – regardless of the coordinate in the physical domain, swirl ratio, and stratification factor. Moreover, one-point, one-time measurements are utilized to compute the exact joint Probability Density Function (PDF) of the conditioning variables at each point. Having the conditional averages of temperature and mass fractions of CO2, CO, CH4, H2, and H2O in addition to the joint PDF of the conditioning variables, the low-dimensional manifold is applied to calculate the unconditional averages for each of the scalars. The mean values are also obtained by ensemble averaging all of the data available for each of the measuring points, and the discrepancies between the values calculated from the two approaches are reported in order to assess the validity of the assumptions underlying the low-dimensional chemistry representations. The results suggest that the two chosen conditioning variables are not sufficient to make the manifold independent of the real domain, so, the normalized total enthalpy is introduced as the third conditioning variable and the process repeated. Results obtained from the three-condition manifold demonstrate that the discrepancies for prediction of the reactive scalars, more significantly for CO2 and CO mass fractions, decrease when using the third condition. The discrepancies with three conditions show that a three-dimensional manifold is sufficient to assume the conditional averages are independent of swirl and stratification. A normalized accumulated discrepancy is used as a metric for each of the cases so one can see the overall effect of each assumption that is made for constructing the conditional manifold. Decoupling the manifold from the coordinate, swirl ratio, and stratification level makes it possible to obtain the filtered chemical source-terms from a static lookup table for many flames which will considerably reduce the computational cost for simulating turbulent reacting flows.