Ratio Of Random Variables Research Articles

On The Product and Ratio of Pareto and Erlang Random Variables

The distributions of products and ratios of random variables are of interest in many areas of the sciences.In this paper, we find analytically the probability distributions of the product XY and the ratio X/Y ,when X and Y are two independent random variables following Pareto and Erlang distributions, respectively

State-dependent importance sampling for estimating expectations of functionals of sums of independent random variables

Estimating the expectations of functionals applied to sums of random variables (RVs) is a well-known problem encountered in many challenging applications. Generally, closed-form expressions of these quantities are out of reach. A naive Monte Carlo simulation is an alternative approach. However, this method requires numerous samples for rare event problems. Therefore, it is paramount to use variance reduction techniques to develop fast and efficient estimation methods. In this work, we use importance sampling (IS), known for its efficiency in requiring fewer computations to achieve the same accuracy requirements. We propose a state-dependent IS scheme based on a stochastic optimal control formulation, where the control is dependent on state and time. We aim to calculate rare event quantities that could be written as an expectation of a functional of the sums of independent RVs. The proposed algorithm is generic and can be applied without restrictions on the univariate distributions of RVs or the functional applied to the sum. We apply this approach to the log-normal distribution to compute the left tail and cumulative distribution of the ratio of independent RVs. For each case, we numerically demonstrate that the proposed state-dependent IS algorithm compares favorably to most well-known estimators dealing with similar problems.

Environmental concentrations as ratios of random variables

Human-induced environmental change increasingly threatens the stability of socio-ecological systems. Careful statistical characterization of environmental concentrations is critical to quantify and predict the consequences of such changes on human and ecosystems conditions. However, while concentrations are naturally defined as the ratio between solute mass and solvent volume, they have rarely been treated as such, typically limiting the analysis to familiar distributions generically used for any other environmental variable. To address this gap, we propose a more general framework that leverages their definition explicitly as ratios of random variables. We show that the resulting models accurately describe the behavior of nitrate plus nitrite in US rivers and salt concentration in estuaries in the Everglades by accounting for heavy tails potentially emerging when the water volume fluctuates around low values. Models that preclude the presence of heavy tails and the related high probability of extreme concentrations could significantly undermine the accuracy of diagnostic frameworks and the effectiveness of mitigation interventions, especially for soil contamination characterized by a water volume (i.e. soil moisture) frequently approaching zero.

Dynamic Panel GMM with Near Unity

Limit theory is developed for the dynamic panel GMM estimator in the presence of an autoregressive root near unity. In the unit root case, Anderson-Hsiao lagged variable instruments satisfy orthogonality conditions but are well-known to be irrelevant. For a fixed time series sample size (T) GMM is inconsistent and approaches a shifted Cauchy-distributed random variate as the cross section sample size n → ∞. But when T → ∞, either for fixed n or as n → ∞, GMM is √T consistent and its limit distribution is a ratio of random variables that converges to twice a standard Cauchy as n → ∞. In this case, the usual instruments are uncorrelated with the regressor but irrelevance does not prevent consistent estimation. The same Cauchy limit theory holds sequentially and jointly as (n,T) → ∞ with no restriction on the divergence rates of n and T. When the common autoregressive root ρ = 1 c/√T the panel comprises a collection of mildly integrated time series. In this case, the GMM estimator is √n consistent for fixed T and √(nT) consistent with limit distribution N(0,4) when n,T → ∞ sequentially or jointly. These results are robust for common roots of the form ρ = 1 c/T^{γ} for all γ ∈ (0,1) and joint convergence holds. Limit normality holds but the variance changes when γ = 1. When γ > 1 joint convergence fails and sequential limits differ with different rates of convergence. These findings reveal the fragility of conventional Gaussian GMM asymptotics to persistence in dynamic panel regressions.

A Universal Splitting Estimator for the Performance Evaluation of Wireless Communications Systems

We propose a unified rare-event estimator for the performance evaluation of wireless communication systems. The estimator is derived from the well-known multilevel splitting algorithm. In its original form, the splitting algorithm cannot be applied to the simulation and estimation of time-independent problems, because splitting requires an underlying continuous-time Markov process whose trajectories can be split. We tackle this problem by embedding the static problem of interest within a continuous-time Markov process, so that the target time-independent distribution becomes the distribution of the Markov process at a given time instant. The main feature of the proposed multilevel splitting algorithm is its large scope of applicability. For illustration, we show how the same algorithm can be applied to the problem of estimating the cumulative distribution function (CDF) of sums of random variables (RVs), the CDF of partial sums of ordered RVs, the CDF of ratios of RVs, and the CDF of weighted sums of Poisson RVs. We investigate the computational efficiency of the proposed estimator via a number of simulation studies and find that it compares favorably with existing estimators.

Statistics of ratios of random variables arising in analysis of wireless Poisson networks

In this study, the author analyses statistics of some typical ratios of random variables (RVs) occurring in analysis of large-scale wireless networks where node locations can be represented by Poisson point processes. In such networks, fading and path-loss effects are the most important factors affecting the signal strength. The author obtains formulas for the probability density, cumulative distribution and moment generating functions of ratios of RVs occurring under many typical scenarios in cellular-like, spectrum-sharing, and cooperative wireless networks. All obtained results are valid for arbitrary fading models of propagation paths.

On exact strong laws of large numbers for ratios of random variables and their applications

We study the almost sure convergence of weighted sums of ratios of independent random variables satisfying some general, mild conditions. The obtained results are applied to exact laws for order statistics. An exact law for independent random variables which are nonidentically distributed is also proved and applied to ratios of adjacent order statistics for a sample of uniformly distributed random variables.

Statistics of spatial averages and optimal averaging in the presence of missing data

We estimate statistics of spatial averages defined by linearly weighting available observations over a spatial domain, and derive an account of bias and variance in the presence of missing observations. With missing observations, the spatial average is a ratio of random variables, and estimators are derived by truncating Taylor series of functions of the ratio followed by taking suitable expectations (’delta method’). The resulting estimators are approximate, and perform well when compared with simulations. Previous authors have examined “optimal averaging” strategies for minimizing mean squared error (MSE) of a spatial average, and we extend the analysis to the case of missing observations. It is shown that to minimize variance primarily requires higher weights where local variance and covariance are small, whereas minimizing bias requires higher weights where observations lie nearer to true spatial average. Missing data increases variance and contributes to bias, and reducing both effects involves emphasizing locations with observations nearer to the spatial average. The framework is applied to spatially averaged rainfall over India. We estimate standard error of all-India rainfall as the combined effect of measurement uncertainty and bias, in the case where weights are chosen to minimize MSE. We also discuss different special cases of the estimators, and applications thereof.

Non-parametric estimation of low-concentration benzene metabolism

Two apparently contradictory findings in the literature on low-dose human metabolism of benzene are as follows. First, metabolism is approximately linear at low concentrations, e.g., below 10 ppm. This is consistent with decades of quantitative modeling of benzene pharmacokinetics and dose-dependent metabolism. Second, measured benzene exposure and metabolite concentrations for occupationally exposed benzene workers in Tianjin, China show that dose-specific metabolism (DSM) ratios of metabolite concentrations per ppm of benzene in air decrease steadily with benzene concentration, with the steepest decreases below 3 ppm. This has been interpreted as indicating that metabolism at low concentrations of benzene is highly nonlinear. We reexamine the data using non-parametric methods. Our main conclusion is that both findings are correct; they are not contradictory. Low-concentration metabolism can be linear, with metabolite concentrations proportional to benzene concentrations in air, and yet DSM ratios can still decrease with benzene concentrations. This is because a ratio of random variables can be negatively correlated with its own denominator even if the mean of the numerator is proportional to the denominator. Interpreting DSM ratios that decrease with air benzene concentrations as evidence of nonlinear metabolism is therefore unwarranted when plots of metabolite concentrations against benzene ppm in air show approximately straight-line relationships between them, as in the Tianjin data. Thus, an apparent contradiction that has fueled heated discussions in the recent literature can be resolved by recognizing that highly nonlinear, decreasing DSM ratios are consistent with linear metabolism.

On distributions of ratios

A large number of exact inferential procedures in statistics and econometrics involve the sampling distribution of ratios of random variables. If the denominator variable is positive, then tail probabilities of the ratio can be expressed as those of a suitably defined difference of random variables. If in addition, the joint characteristic function of numerator and denominator is known, then standard Fourier inversion techniques can be used to reconstruct the distribution function from it. Most research in this field has been based on this correspondence, but which breaks down when both numerator and denominator are supported on the entire real line. The present manuscript derives inversion formulae and saddlepoint approximations that remain valid in this case, and reduce to known results when the denominator is almost surely positive. Applications include the IV est imator of a structural parameter in a just identified equation.

DYNAMIC PANEL ANDERSON-HSIAO ESTIMATION WITH ROOTS NEAR UNITY

Limit theory is developed for the dynamic panel IV estimator in the presence of an autoregressive root near unity. In the unit root case, Anderson–Hsiao lagged variable instruments satisfy orthogonality conditions but are well known to be irrelevant. For a fixed time series sample size (T) IV is inconsistent and approaches a shifted Cauchy-distributed random variate as the cross-section sample sizen→ ∞. But whenT→ ∞, either for fixednor asn→ ∞, IV is$\sqrt T$consistent and its limit distribution is a ratio of random variables that converges to twice a standard Cauchy asn→ ∞. In this case, the usual instruments are uncorrelated with the regressor but irrelevance does not prevent consistent estimation. The same Cauchy limit theory holds sequentially and jointly as (n,T) → ∞ with no restriction on the divergence rates ofnandT.When the common autoregressive root$\rho = 1 + c/\sqrt T$the panel comprises a collection of mildly integrated time series. In this case, the IV estimator is$\sqrt n$consistent for fixedTand$\sqrt {nT}$consistent with limit distributionN(0, 4) when (n,T) → ∞ sequentially or jointly. These results are robust for common roots of the formρ= 1+c/Tγfor allγ∈ (0, 1) and joint convergence holds. Limit normality holds but the variance changes whenγ= 1. Whenγ> 1 joint convergence fails and sequential limits differ with different rates of convergence. These findings reveal the fragility of conventional Gaussian IV asymptotics to persistence in dynamic panel regressions.

Distribution of products of independently distributed pathway random variables

A lot of work has been done on products and ratios of random variables by Provost and his co-workers, see, for example, Provost [S.B. Provost, The exact distribution of the ratio of a linear combination of chi-square variables over the root of a product of chi-square variables, Canad. J. Statist. 14 (1986), pp. 61–67; S.B. Provost, The distribution function of a statistic for testing the equality of scale parameters in two gamma populations, Metrika 36 (1989), pp. 337–345]. Here, we extend this idea by introducing a pathway model. The exact density functions of the products of pathway random variables are obtained using the Mellin transform technique. Their computable series forms are derived. The particular cases of the derived results are shown to be associated with the thermonuclear functions and reaction rate probability integral in the theory of nuclear reaction rate, Krätzel integral in applied analyses and inverse Gaussian density in stochastic processes. Graphical representations of the density functions of the product of random variables for the different values of the pathway parameters are shown. The new probability model is fitted to revenue data.

Statistical Analysis of Ratio of Random Variables and Its Application in Performance Analysis of Multihop Wireless Transmissions

The distributions of random variables are of interest in many areas of science. In this paper, the probability density function (PDF) and cumulative distribution function (CDF) of ratio of products of two random variables and random variable are derived. Random variables are described with Rayleigh, Nakagami‐m, Weibull, and α‐μ distributions. An application of obtained results in performance analysis of multihop wireless communication systems in different transmission environments described in detail. The proposed mathematical analysis is also complemented by various graphically presented numerical results.

Confidence Bounds for Multiplicative Comparisons

Statements that are inherently multiplicative have historically been justified using ratios of random variables. Although recent work on ratios has extended the classical theory to produce confidence bounds conditioned on a positive denominator, this current article offers a novel perspective that eliminates the need for such a condition. Although seemingly trivial, this new perspective leads to improved lower confidence bounds to support multiplicative statements. This perspective is also more satisfying as it allows comparisons that are inherently multiplicative in nature to be properly analyzed as such.

SOME ALGEBRA FOR PEARSON TYPE VII RANDOM VARIABLES

The distributions of products and ratios of random variables are of interest in many areas of the sciences. In this paper, the exact distributions of the product |XY| and the ratio |X/Y| are derived when X and Y are independent Pearson type VII random variables.

Evaluation for moments of a ratio with application to regression estimation

Ratios of random variables often appear in probability and statistical applications. We aim to approximate the moments of such ratios under several dependence assumptions. Extending the ideas in Collomb [C. R. Acad. Sci. Paris 285 (1977) 289--292], we propose sharper bounds for the moments of randomly weighted sums and for the $L^p$-deviations from the asymptotic normal law when the central limit theorem holds. We indicate suitable applications in finance and censored data analysis and focus on the applications in the field of functional estimation.

The linear combination, product and ratio of Laplace random variables

The distributions of linear combinations, products and ratios of random variables arise in many areas of engineering. In this paper, the exact distributions of the linear combination α X+β Y, the product |X Y| and the ratio |X/Y| are derived when X and Y are independent Laplace random variables. The Laplace distribution, being the oldest model for continuous data, has been one of the most popular models for measurement errors in engineering.

Financial Pareto ratios

Ratios of random variables arise most frequently in accounting. There are many financial indices that take the form of ratios. Some of the most commonly known examples are current ratio defined by ...

Asymptotics for Ratios with Applications to Reinsurance

This paper provides an overview of results pertaining to moment convergence for certain ratios of random variables involving sums, order statistics and extreme terms in the sense of modulus. Most of the literature on this matter originates from Darling (1952) who gave a criterion for the convergence in probability to 1 of the ratio of the maximum to the sum in case of nonnegative random variables.

New Tests for Null Hypothesis of Non Unity Ratio of Proportions

Testing for noninferiority and equivalence between an experimental therapy and a standard therapy in terms of the ratio of binomial proportions is considered. New tests based on the Fieller–Hinkley distribution of the ratio of random variables are proposed. Restricted maximum likelihood estimates of the null variances are used to derive the tests. Sample size determination is discussed. The proposed test procedure is extended to multiple tables. The tests are applied to numerical examples.