Independent Positive Random Variables Research Articles

Conditional Beta Approximation: Two Applications

Suppose that X,Y are two independent positive continuous random variables. Let P=\frac{X}{X+Y} and Z=X+Y. If X, Y have gamma distributions with the same scale parameter, then P distribution will be beta and P,\ Z are independent. In the case that the distributions of these two variables are not gamma, the P distribution is well approximated by the beta distribution. However, P,\ Z are dependent. According to matching moment method, it is necessary to compute the moments of conditional distribution for beta fitting. In this paper, some new methods for computing moments of conditional distribution of P given Z are proposed. First of all, it is suggested to consider the regression method. Then Monte Carlo simulation is advised. The Bayesian posterior distribution of P is suggested. Applications of differential equations are also reviewed. These results are applied in two applications namely variance change point detection and winning percentage of gambling game are proposed. The probability of change in variance in a sequence of variables, as a leading indicator of possible change, is proposed. Similarly, the probability of winning in a sequential gambling framework is proposed. The optimal time to exit of gambling game is proposed. A game theoretic approach to problem of optimal exit time is proposed. In all cases, beta approximations are proposed. Finally, a conclusion section is also given.

Small Deviation Probabilities for Sums of Independent Positive Random Variables

We study the asymptotic behavior at zero of distributions and densities of a sum of several independent positive random variables under certain assumptions on the decay rate of their distributions at zero. We consider cases where the distributions (densities) of summable random variables are regularly or slowly varying at zero or can decrease at zero at an arbitrary rate.

Small deviation probabilities for sums of independent positive random variables

«Вестник Санкт-Петербургского университета. Математика. Механика. Астрономия» — научно-теоретический рецензируемый журнал, публикующий исследования в области математики, механики и астрономии, а также математического моделирования.

Small Deviation Probabilities of a Sum of Independent Positive Random Variables, the Common Distribution of Which Decreases at Zero Not Faster Than Exponential Function

We investigate small deviation probabilities of the cumulative sum of independent positive random variables, the common distribution of which decreases at zero not faster than exponential function.

Small Deviation Probabilities for a Weighted Sum of Independent Positive Random Variables with Common Distribution Function That Can Decrease at Zero Fast Enough

The present note puts forward the most general conditions to date under which the classical asymptotics of the small deviation probabilities for a weighted sum of independent positive random variables takes place. Regarding random variables it is assumed that their common distribution function can, in particular, have an exponential falloff at zero.

The distributions of interrecord fillings

Abstract In a sequence of independent positive random variables with the same continuous distribution function a monotonic subsequence of record values is chosen. A corresponding sequence of record times divides the initial sequence into interrecord intervals. Let α i j ( i ⩾ 1 , j = 1 , … , i ) $\alpha_i^j \ (i\geqslant 1, \, j = 1, \ldots, i)$ be the number of random variables in the interval between i-th and (i + 1)-th record moments with values between (j − 1)-th and j-th records. Explicit formulas for the joint distributions of the random variables α i j , 1 ⩽ j ⩽ i ⩽ n $\alpha_i^j,\,1\leqslant j\leqslant i\leqslant n$ , are derived, limit theorems for the distributions of α i j $\alpha_i^j$ for i − j → ∞ are proved.

Small Deviations of Probabilities for Weighted Sum of Independent Positive Random Variables with a Common Distribution That Decreases at Zero Not Faster than a Power

In the paper it is shown that the well-known asymptotics for small deviation probabilities of weighed sum of independent positive random variables with a common distribution still holds true if this distribution decreases at zero not faster than some power.

Small Deviation Probabilities for Sums of Independent Positive Random Variables with a Distribution that Slowly Varies at Zero

In the note, we study small deviation probabilities for sums of independent, identically distributed positive random variables whose distribution function is slowly varying at zero. Bibliography: 5 titles.

Comportement asymptotique de lʼestimateur non paramétrique de la fonction de renouvellement associée à des variables aléatoires positives indépendantes et non stationnaires

We estimate the renewal function, for independent and nonstationary positive random variables, by considering a sum of U-statistics or V-statistics. We study its asymptotic properties: asymptotic normality, almost sure convergence and weak convergence. This last property will be established on the space of functions defined on [0,1] that are right-continuous and have left-hand limits, provided with the Skorokhod topology.

Small deviation probabilities for sums of independent positive random variables whose density has a power decay at zero

In this note, we study small deviation probabilities of a sum of i.i.d. positive random variables whose density has a power decay at zero. Bibliography: 3 titles.

Determination of the distribution of total loss from the fractional moments of its exponential

In the insurance and banking industries, as in many branches of engineering, in many instances a model can be proposed to describe either the accumulated losses or the accumulated damage or stress. Such models are of the compound type, namely random sums of independent positive random variables, and a reliable statistical model is available for both. But computing the distribution of such random sums is a hard problem that has received much attention. Here we propose a technique based on the maximum entropy method which makes use of the fractional moments of the exponential of the accumulated loss, which is available since for many models the Laplace transform can be exactly determined. Here we provide an answer to the question of how to determine distribution of a total severity when the only information that we have consists of the Laplace transform of the compound distribution. Our technique yields fairly good reconstructions from a low number of fractional moments.

Identification of power distribution mixtures through regression of exponentials

Given two independent non-degenerate positive random variables X and Y, Lukacs (Ann Math Stat 26:319–324, 1955) proved that X/(X + Y) and X + Y are independent if and only if X and Y are gamma distributed with the same scale parameter. In this work, under the assumption X/U and U are independent, and X/U has a \({{\mathcal Be}(p,\,q)}\) distribution, we characterize the distribution of (U, X) by the condition E(h(U − X)|X) = b, where h is allowed to be a linear combination of exponential functions. Since the assumption for X and U above is equivalent to X|U being \({\mathcal{B}e(p,\,1)}\) scaled by U, hence our results can be viewed as identification of a power distribution mixture.

Lamperti-type laws

This paper explores various distributional aspects of random variables defined as the ratio of two independent positive random variables where one variable has an α-stable law, for 0 < α < 1, and the other variable has the law defined by polynomially tilting the density of an α-stable random variable by a factor θ > −α. When θ = 0, these variables equate with the ratio investigated by Lamperti [Trans. Amer. Math. Soc. 88 (1958) 380–387] which, remarkably, was shown to have a simple density. This variable arises in a variety of areas and gains importance from a close connection to the stable laws. This rationale, and connection to the PD (α, θ) distribution, motivates the investigations of its generalizations which we refer to as Lamperti-type laws. We identify and exploit links to random variables that commonly appear in a variety of applications. Namely Linnik, generalized Pareto and z-distributions. In each case we obtain new results that are of potential interest. As some highlights, we then use these results to (i) obtain integral representations and other identities for a class of generalized Mittag–Leffler functions, (ii) identify explicitly the Lévy density of the semigroup of stable continuous state branching processes (CSBP) and hence corresponding limiting distributions derived in Slack and in Zolotarev [Z. Wahrsch. Verw. Gebiete 9 (1968) 139–145, Teor. Veroyatn. Primen. 2 (1957) 256–266], which are related to the recent work by Berestycki, Berestycki and Schweinsberg, and Bertoin and LeGall [Ann. Inst. H. Poincaré Probab. Statist. 44 (2008) 214–238, Illinois J. Math. 50 (2006) 147–181] on beta coalescents. (iii) We obtain explicit results for the occupation time of generalized Bessel bridges and some interesting stochastic equations for PD (α, θ)-bridges. In particular we obtain the best known results for the density of the time spent positive of a Bessel bridge of dimension 2 − 2α.

Jackknife methods for left-truncated data

Let U * and V * be two independent positive random variables with continuous distribution functions F and G, respectively. Under random truncation, both U * and V * are observable only when U * ≥ V * . Let F ¯ n be the nonparametric maximum likelihood estimate ( Lynden-Bell, 1971) of F ¯ = 1 − F . In this paper, it is shown that the jackknife variance estimate of F ¯ n consistently estimates the limit variance. In a simulation study, we show that the jackknife works satisfactorily for moderate sample size. As an illustration we also apply our method to a real set of AIDS data.

The living Drake equation of the Tau Zero Foundation

The living Drake equation is our statistical generalization of the Drake equation such that it can take into account any number of factors. This new result opens up the possibility to enrich the equation by inserting more new factors as long as the scientific learning increases. The adjective “Living” refers just to this continuous enrichment of the Drake equation and is the goal of a new research project that the Tau Zero Foundation has entrusted to this author as the discoverer of the statistical Drake equation described hereafter. From a simple product of seven positive numbers, the Drake equation is now turned into the product of seven positive random variables. We call this “the Statistical Drake Equation”. The mathematical consequences of this transformation are then derived. The proof of our results is based on the Central Limit Theorem (CLT) of Statistics. In loose terms, the CLT states that the sum of any number of independent random variables, each of which may be arbitrarily distributed, approaches a Gaussian (i.e. normal) random variable. This is called the Lyapunov form of the CLT, or the Lindeberg form of the CLT, depending on the mathematical constraints assumed on the third moments of the various probability distributions. In conclusion, we show that: (1) The new random variable N, yielding the number of communicating civilizations in the Galaxy, follows the lognormal distribution. Then, the mean value, standard deviation, mode, median and all the moments of this lognormal N can be derived from the means and standard deviations of the seven input random variables. (2) In fact, the seven factors in the ordinary Drake equation now become seven independent positive random variables. The probability distribution of each random variable may be arbitrary. The CLT in the so-called Lyapunov or Lindeberg forms (that both do not assume the factors to be identically distributed) allows for that. In other words, the CLT “translates” into our statistical Drake equation by allowing an arbitrary probability distribution for each factor. This is both physically realistic and practically very useful, of course. (3) An application of our statistical Drake equation then follows. The (average) distance between any two neighbouring and communicating civilizations in the Galaxy may be shown to be inversely proportional to the cubic root of N. Then, this distance now becomes a new random variable. We derive the relevant probability density function, apparently previously unknown (dubbed “Maccone distribution” by Paul Davies). (4) Data Enrichment Principle. It should be noticed that any positive number of random variables in the statistical Drake equation is compatible with the CLT. So, our generalization allows for many more factors to be added in the future as long as more refined scientific knowledge about each factor will be known to the scientists. This capability to make room for more future factors in the statistical Drake equation we call the “Data Enrichment Principle”, and regard as the key to more profound, future results in Astrobiology and SETI.

Small deviations of series of independent positive random variables with weights close to exponential

Let ξ, ξ0, ξ1, ... be independent identically distributed (i.i.d.) positive random variables. The present paper is a continuation of the article [1] in which the asymptotics of probabilities of small deviations of series S = Σ =0 ∞ a(j)ξ j was studied under different assumptions on the rate of decrease of the probability ℙ(ξ < x) as x → 0, as well as of the coefficients a(j) ≥ 0 as j → ∞. We study the asymptotics of ℙ(S < x) as x → 0 under the condition that the coefficients a(j) are close to exponential. In the case when the coefficients a(j) are exponential and ℙ(ξ < x) ∼ bx α as x → 0, b > 0, a > 0, the asymptotics ℙ(S < x) is obtained in an explicit form up to the factor x o(1). Originality of the approach of the present paper consists in employing the theory of delayed differential equations. This approach differs significantly from that in [1].

Small deviation probabilities for sums of independent positive random variables

In this note, we give estimates of small deviation probabilities of the sum ∑j≥1 λj Xj, where {λj} are nonnegative numbers and {Xj} are i.i.d. positive random variables that satisfy mild assumptions at zero and infinity. Bibliography: 10 titles.

The tail behaviour of a random sum of subexponential random variables and vectors

Let \(\left\{ X,X_{i},i=1,2,...\right\} \) denote independent positive random variables having common distribution function (d.f.) F(x) and, independent of X, let ν denote an integer valued random variable. Using X 0=0, the random sum Z=∑ i=0 ν X i has d.f. \(G(x)=\sum_{n=0}^{\infty }\Pr\{\nu =n\}F^{n\ast }(x)\) where F n∗(x) denotes the n-fold convolution of F with itself. If F is subexponential, Kesten’s bound states that for each e>0 we can find a constant K such that the inequality $$ 1-F^{n\ast }(x)\leq K(1+\varepsilon )^{n}(1-F(x))\, , \qquad n\geq 1,x\geq 0 \, , $$ holds. When F is subexponential and E(1 +e) ν <∞, it is a standard result in risk theory that G(x) satisfies $$ 1 - G{\left( x \right)} \sim E{\left( \nu \right)}{\left( {1 - F{\left( x \right)}} \right)},\,\,x \to \infty \,\,{\left( * \right)} $$ In this paper, we show that (*) holds under weaker assumptions on ν and under stronger conditions on F. Stam (Adv. Appl. Prob. 5:308–327, 1973) considered the case where \( \overline{F}(x)=1-F(x)\) is regularly varying with index –α. He proved that if α>1 and \(E{\left( {\nu ^{{\alpha + \varepsilon }} } \right)} -1\), then again Eν<∞ is sufficient. The proofs of the results rely on deriving bounds for the ratio \(\overline{F^{n\ast }}(x)/\overline{F} (x)\). In the paper, we also consider (*) in the special case where X is a positive stable random variable or has a compound Poisson distribution derived from such a random variable and, in this case, we show that for n≥2, the ratio \(\overline{F^{n\ast }}(x)/\overline{F}(x)\uparrow n\) as x↑∞. In Section 3 of the paper, we briefly discuss an extension of Kesten’s inequality. In the final section of the paper, we discuss a multivariate analogue of (*).

Additive Positive Stable Frailty Models

In this article, we describe an additive stable frailty model for multivariate times to events data using a flexible baseline hazard, and assuming that the frailty component for each individual is described by additive functions of independent positive stable random variables with possibly different stability indices. Dependence properties of this frailty model are investigated. To carry out inference, the likelihood function is derived by replacing high-dimensional integration by Monte Carlo simulation. Markov chain Monte Carlo algorithms enable estimation and model checking in the Bayesian framework.

On characterizations of the gamma and generalized inverse Gaussian distributions

Given two independent non-degenerate positive random variables X and Y, Letac and Wesolowski (Ann. Probab. 28 (2000) 1371) proved that U=( X+ Y) −1 and V= X −1−( X+ Y) −1 are independent if and only if X and Y are generalized inverse Gaussian (GIG) and gamma distributed, respectively. Note that X=( U+ V) −1 and Y= U −1−( U+ V) −1. This interesting transformation between ( X, Y) and ( U, V) preserves a bivariate probability measure which is a product of GIG and gamma distributions. In this work, characterizations of the GIG and gamma distributions through the constancy of regressions of V r on U are considered.