On a conjecture of Seneta on regular variation of truncated moments

We prove that h?(x) = ??x0 y??1F?(y)dy is regularly varying with index ? [0, ?) if and only if V?(x) = ?[0,x] y?dF(y) is regularly varying with the same index, where ? > 0, F(x) is a distribution function of a nonnegative random variable, and F?(x) = 1?F(x). This contains at ? = 0, ?= 1 a result of Rogozin [8] on relative stability, and at ? = 0, ? = 2 a new, equivalent characterization of the domain of attraction of the normal law. For ? = 0 and ? > 0 our result implies a recent conjecture by Seneta [9].

Some Useful Integral Representations for Information-Theoretic Analyses

This work is an extension of our earlier article, where a well-known integral representation of the logarithmic function was explored and was accompanied with demonstrations of its usefulness in obtaining compact, easily-calculable, exact formulas for quantities that involve expectations of the logarithm of a positive random variable. Here, in the same spirit, we derive an exact integral representation (in one or two dimensions) of the moment of a nonnegative random variable, or the sum of such independent random variables, where the moment order is a general positive non-integer real (also known as fractional moments). The proposed formula is applied to a variety of examples with an information-theoretic motivation, and it is shown how it facilitates their numerical evaluations. In particular, when applied to the calculation of a moment of the sum of a large number, n, of nonnegative random variables, it is clear that integration over one or two dimensions, as suggested by our proposed integral representation, is significantly easier than the alternative of integrating over n dimensions, as needed in the direct calculation of the desired moment.

On the total length of external branches for beta-coalescents

In this paper we consider the beta(2 − α, α)-coalescents with 1 < α < 2 and study the moments of external branches, in particular, the total external branch lengthof an initial sample ofnindividuals. For this class of coalescents, it has been proved thatnα-1T(n)→DT, whereT(n)is the length of an external branch chosen at random andTis a known nonnegative random variable. For beta(2 − α, α)-coalescents with 1 < α < 2, we obtain limn→+∞n3α-5𝔼(Lext(n)−n2-α𝔼T)2= ((α − 1)Γ(α + 1))2Γ(4 − α) / ((3 − α)Γ(4 − 2α)).

On the total length of external branches for beta-coalescents

In this paper we consider the beta(2 − α, α)-coalescents with 1 < α < 2 and study the moments of external branches, in particular, the total external branch length of an initial sample of n individuals. For this class of coalescents, it has been proved that nα-1T(n) →DT, where T(n) is the length of an external branch chosen at random and T is a known nonnegative random variable. For beta(2 − α, α)-coalescents with 1 < α < 2, we obtain limn→+∞n3α-5 𝔼(Lext(n) − n2-α𝔼T)2 = ((α − 1)Γ(α + 1))2Γ(4 − α) / ((3 − α)Γ(4 − 2α)).

Sample and population exponents of generalized Taylor’s law

Taylor’s law (TL) states that the variance V of a nonnegative random variable is a power function of its mean M; i.e., V=aMb. TL has been verified extensively in ecology, where it applies to population abundance, physics, and other natural sciences. Its ubiquitous empirical verification suggests a context-independent mechanism. Sample exponents b measured empirically via the scaling of sample mean and variance typically cluster around the value b=2. Some theoretical models of population growth, however, predict a broad range of values for the population exponent b pertaining to the mean and variance of population density, depending on details of the growth process. Is the widely reported sample exponent b≃2 the result of ecological processes or could it be a statistical artifact? Here, we apply large deviations theory and finite-sample arguments to show exactly that in a broad class of growth models the sample exponent is b≃2 regardless of the underlying population exponent. We derive a generalized TL in terms of sample and population exponents bjk for the scaling of the kth vs. the jth cumulants. The sample exponent bjk depends predictably on the number of samples and for finite samples we obtain bjk≃k/j asymptotically in time, a prediction that we verify in two empirical examples. Thus, the sample exponent b≃2 may indeed be a statistical artifact and not dependent on population dynamics under conditions that we specify exactly. Given the broad class of models investigated, our results apply to many fields where TL is used although inadequately understood.

SHARP TWO-SIDED BOUNDS FOR DISTRIBUTIONS UNDER A HAZARD RATE CONSTRAINT

Consider a continuous nonnegative random variable X with mean μ and hazard function h. Assume further that a≤h(t)≤b for all t≥0. Under these constraints, we obtain sharp two-sided bounds for $\bar{F}(t) = \hbox{Pr}(X \gt t)$. An application to birth and death processes is discussed.

APLIKASI METODE KERNEL DALAM ESTIMASI FUNGSI HAZARD

Let T be a nonnegative random variable representing the lifetimes of individuals in some population. Let f(t) denote the probability density function of T and F(t) denote the distribution function of T, the hazard function of T defined as F(t) - 1 S(t) whereS(t) f(t) h(t)   If equation (1) integrated we have cumulative hazard function H (t). This paper describes application of kernel method for estimation of hazard function h (.) based censoring data. And then we will show that the hazard estimator is unbiased asymptotically, consistent, and normal asymptotically. Key word: kernel methods, estimation hazard function.