On a crossroad of resampling plans: bootstrapping elementary symmetric polynomials

We investigate the validity of the bootstrap method for the elementary symmetric polynomials S ^{( k )} _n =( ⁿ _k )⁻¹Σ_{1≤ i 1}< ... < i _k ≤ n X _{i 1} ... X _{i k} of i.i.d. random variables X ₁, ..., X _n . For both fixed and increasing order k , as n→∞ the cases where μ=E X ₁[moe2]0, the nondegenerate case, and where μ=E X ₁=0, the degenerate case, are considered.     

Inequalities relating maximal moments to other measures of dispersion

Let X , X ₁, ..., X_k be i.i.d. random variables, and for k ∈ N let D_k ( X ) = E ( X ₁ V ... V X _{k +1}) − EX be the k th centralized maximal moment. A sharp lower bound is given for D ₁( X ) in terms of the Lévy concentration Q_l ( X ) = sup _{x ∈ R} P ( X ∈[ x , x + l ]). This inequality, which is analogous to P. Levy's concentration-variance inequality, illustrates the fact that maximal moments are a gauge of how much spread out the underlying distribution is. It is also shown that the centralized maximal moments are increased under convolution.     

Generalized densities of order statistics

Let X ₁, ... , X _n be independent identically distributed random variables with distribution F . We derive expressions for generalized joint 'densities' of order statistics of X ₁, ... , X _n , for arbitrary distributions F , in terms of Radon–Nikodym derivatives with respect to product measures based on F . We then give formulae for conditional distributions of order statistics and use them to derive results concerning Markov properties of order statistics, formulae for distributions of trimmed sums, and other useful representations. Our approach leads to simple and natural expressions which appear not to have been given before.     

On the asymptotically uniform distribution modulo 1 of extreme order statistics

Let (X_m)∞₁ be a sequence of independent and identically distributed random variables. We give sufficient conditions for the fractional part of rnax (X₁., X_n) to converge in distribution, as n ←∞ to a random variable with a uniform distribution on [0, 1).     

Selection from Logistic populations

Assume k ( k ≥ 2) independent populations π₁, π₂μ_k are given. The associated independent random variables X_i,( i = 1,2,… k ) are Logistically distributed with unknown means μ₁, μ₂, μ_k and equal variances. The goal is to select that population which has the largest mean. The procedure is to select that population which yielded the maximal sample value. Let μ₍₁₎≤μ₍₂₎≤…≤μ_(k) denote the ordered means. The probability of correct selection has been determined for the Least Favourable Configuration μ₍₁₎=μ₍₂₎==μ_{(k – 1)}=μ_(k)–δ where δ > 0. An exact formula for the probability of correct selection is given.     

A test for equality of two exponential distributions

Abstract Let X ₁., X _n1 and Y ₁., Y _n1, be two independent random samples from exponential populations. The statistical problem is to test whether or not two exponential populations are the same, based on the order statistics X _[1],. X _[r1] and Y [₁],. Y _[rs] where 1 r₁ n ₁ and 1 r₂ n ₂. A new test is given and an asymptotic optimum property of the test is proved.     

Asymptotic distributions of φ-divergences of hypothetical and observed frequencies on refined partitions

For a wide class of goodness-of-fit statistics based on φ-divergences between hypothetical cell probabilities and observed relative frequencies, the asymptotic normality is established under the assumption n / m _n →γ∈(0,∞), where n denotes sample size and m _n the number of cells. Related problems of asymptotic distributions of φ-divergence errors, and of φ-divergence deviations of histogram estimators from their expected values, are considered too.     

Some problems in actuarial finance involving sums of dependent risks

A trend in actuarial finance is to combine technical risk with interest risk. If Y_t , t = 1, 2, denotes the timevalue of money (discount factors at time t ) and X_t the stochastic payments to be made at time t , the random variable of interest is often the scalar product of these two random vectors V = X_t Y_t . The vectors X and Y are supposed to be independent, although in general they have dependent components. The current insurance practice based on the law of large numbers disregards the stochastic financial aspects of insurance. On the other hand, introduction of the variables Y ₁, Y ₂, to describe the financial aspects necessitates estimation or knowledge of their distribution function.
We investigate some statistical models for problems of insurance and finance, including Risk Based Capital/Value at Risk, Asset Liability Management, the distribution of annuities, cash flow evaluations (in the framework of pension funds, embedded value of a portfolio, Asian options) and provisions for claims incurred, but not reported (IBNR).     

A central limit theorem for sums of correlated products

Consider a sequence of random points placed on the nonnegative integers with i.i.d. geometric (1/2) interpoint spacings y _i . Let x _i denote the numbers of points placed at integer i . We prove a central limit theorem for the partial sums of the sequence x ₀ y ₀, x ₁ y ₁, . . . The problem is connected with a question concerning different bootstrap procedures.     

Diversification for general copula dependence

We generalize the extreme value analysis for Archimedean copulas (see Alink , Löwe and Wüthrich , 2003) to the non-Archimedean case: Assume we have d ≥2 exchangeable and continuously distributed risks X ₁,…, X _d . Under appropriate assumptions there is a constant q _d such that, for all large u , we have . The constant q _d describes the asymptotic dependence structure. Typically, q _d will depend on more aspects of this dependence structure than the well-known tail dependence coefficient.     

Stability and self-decomposability of semi-group valued random variables

A random variable X on IR₊ is said to be self-decomposable, dif for all c∈ (0, 1) there exists a random variable X_c on IR₊ such that X=^dcX+X_c . It is said to be stable if it is self-decomposable and X_c=^d (1 - c)X' , where X and X' are identically and independently distributed. The notions of stability and self-decomposability for infinitely divisible random variables are generalised to abelian semi-groups ( S, + ) with S having an identical involution, by using characteristic functions. The generalised definitions involve semi-groups of scaling operators T . There operators can be interpreted in a slightly different context as generalised continuous-time branching processes (with immigration). The underlying importance of the generator of the semi-groups T in the characterisation of stability and self-decomposability is stressed.     

Some remarks concerning the quotient of sample median and sample range for a sample of size 2n+1 from a normal distribution

Consider an ordered sample ₍₁₎, ₍₂₎,…, _(2n+1) of size 2 n +1 from the normal distribution with parameters μ and . We then have with probability one
₍₁₎ < ₍₂₎ < … < (2 n +1).
The random variable
_n =_(n+1)/_(2n+1)-(1)
that can be described as the quotient of the sample median and the sample range, provides us with an estimate for μ/, that is easy to calculate. To calculate the distribution of h _n is quite a different matter***. The distribution function of h₁, and the density of h₂ are given in section 1. Our results seem hardly promising for general h_n. In section 2 it is shown that h_n is asymptotically normal.
In the sequel we suppose μ= 0 and = 1, i.e. we consider only the "central" distribution. Note that h_n can be used as a test statistic replacing Student's t. In that case the central h_n is all that is needed.     

Large deviation principles and loglog laws for observation processes

Let ( X_n, n ≥ 1) be an i.i.d. sequence of positive random variables with distribution function H . Let φ _H := {(n, X_n ), n ≥ 1) be the associated observation process. We view φ h as a measure on E := [0, ∞) ∞ (0, φ] where φ_H (A) is the number of points of φ _H which lie in A . A family ( V_s, s> 0) of transformations is defined on E in such a way that for suitable H the distributions of ( V_sφ_H, S > 0) satisfy a large deviation principle and that a related Strassen-type law of the iterated logarithm also holds. Some consequent large deviation principles and loglog laws are derived for extreme values. Similar results are proved for φ _H replaced by certain planar Poisson processes.     

Estimation of a k-monotone density: characterizations, consistency and minimax lower bounds

The classes of monotone or convex (and necessarily monotone) densities on     can be viewed as special cases of the classes of k - monotone densities on     . These classes bridge the gap between the classes of monotone (1-monotone) and convex decreasing (2-monotone) densities for which asymptotic results are known, and the class of completely monotone (∞-monotone) densities on     . In this paper we consider non-parametric maximum likelihood and least squares estimators of a k -monotone density g ₀. We prove existence of the estimators and give characterizations. We also establish consistency properties, and show that the estimators are splines of degree k −1 with simple knots. We further provide asymptotic minimax risk lower bounds for estimating the derivatives     , at a fixed point x ₀ under the assumption that     .     

Uitbijternegerende schatters bij duplobepalingen

Outlyer-ignoring estimators for measurement in duplo.
By hypothesis a measurement u is the sum of two independent random variables, the normal random variable with expectation μ, and standard error σ, and a random error φ:

Basically two independent measurements u₁ and u₂ over u are to give the estimate x=1/2(u₁+ u₂) over μ.
However, to reduce the effect of the error φ on a final estimate of μ, one adds, according to a common practice, a third or even a fourth measurement u₃, u₄, in the case that the basic pair differs by more than a number A. For this extended set of measurements two outlyer-ignoring estimator y and z of μ are defined, and investigated against three specifications fo the error φ. Also an outlyer-ignoring estimate of σ is considered, and its application is illustrated by an example.     

First-Order Integer-Valued Autoregressive (INAR (1)) Process: Distributional and Regression Properties

Some properties of a first-order integer-valued autoregressive process (INAR)) are investigated. The approach begins with discussing the self-decomposability and unimodality of the 1-dimensional marginals of the process {X_n} generated according to the scheme X_n=α° X _n-i +e_n, where α° X _n-1 denotes a sum of X_{n - 1}, independent 0 - 1 random variables Y^(n-1), independent of X _n-1 with Pr -( y ^{(n - 1)}= 1) = 1 - Pr ( y ^(n-i)= 0) =α. The distribution of the innovation process ( e _n) is obtained when the marginal distribution of the process ( X _n) is geometric. Regression behavior of the INAR(1) process shows that the linear regression property in the backward direction is true only for the Poisson INAR(1) process.     

Het verdelen van steekproeven over subpopulaties bij accountantscontroles

Summary  "Stratificationprocedures for a typical auditing problem".
During the past ten years, much experience was gained in The Netherlands in using random sampling methods for typical auditing problems. Especially, a method suggested by VAN. HEERDEN [2] turned out to be very fruitful. In this method a register of entries is considered to be a population of T guilders, if all entries total up to T guilders. The sample size n ₀ is determined in such a way that the probability β not to find any mistake in the sample, if a fraction p ₀ or more of T is incorrect, is smaller than a preassigned value β₀. So n ₀ should satisfy (l- p )ⁿ⁰≤β₀ for p ≥ p ₀. A complication arises if it is not possible to postpone sampling until the whole population T is available. One then wants to take samples from a population which is growing up to T . Suppose one is going to take samples n _i from e.g. r subpopulations

Using the minimax procedure, it is shown, that in this case one should choose the sizes n _i equal to ( T _i/ T ) n ₀. The minimax-value of the probability not to find any incorrect guilder in the r samples, taken together is equal to β₀.     

Superpositie van twee frequentieverdelingen

Summary  (Superposition of two frequency distributions)
Notation:
n: number of observations
M: arithmetic mean
: standard deviation
μ_r: r^th moment coefficient
β₁: coefficient of skewness
β₂: coefficient of kurtosis.
The suffixes a and b apply to the component distributions. The suffix t applies to the resulting distributions.

The problem: Given the first r moments of two frequency distributions (to begin with μ₀). Find the first r moments of the distribution resulting from superposition of the two components ( r ≥ 5 ).
Formulae [1]. … [ 5 ] (§ 3 ) give the results in their most general form up to μ₄.
Some special cases are treated in § 4, and eight different cases of superposition of two normal distributions in § 5.
In § 6 some remarks are made about the reverse situation, i.e. the splitting into two normal components of a combined frequency distribution.     

A property of Stein's estimator for the mean of a multivariate normal distribution

A uniform bound on the risk (under squared error loss) of Stein's estimator Ψ₁ for the mean of the multivariate normal distribution is given. Using the bound, the asymptotic behaviour of the risk of Ψ₁ under a Bayesian assumption is obtained.     

The ascending ladder height distribution for a certain class of dependent random walks

A random walk { S_n } with S_n = (X_l - Y_l) +…+ ( X_n - Y_n ) is considered where the X_n Y_n are non-negative random variables, the Y_n are exponentially distributed with rate δ and the X_n have common distribution function B . It is shown that the expression δ(1 - S (x)) for the density of the ascending ladder height distribution of (S_n), which is well-known for i.i.d. X_n , holds also when the X_n form a stationary sequence of not necessarily independent random variables.