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.     

CENTRAL LIMIT THEORMS IN C[0,1] FOR A CLASS OF ESTIMATORS OF A DISTRIBUTION FUNCTION

As non–parametric estimates of an unknown distribution function (d.f.) F based on i.i.d. observations X ₁ X_n with this d.f.

are used, where H _n is a sequence of d.f.'s converging weakly to the unit mass at zero. Under regularity conditions on F and the sequence ( H _n) it is shown that √_n( F _n– F ) and √_n( R _n – F ) in C [0,1] converge in distribution to a process G with G( t ) = W° ( F ( t )), where W ° is a Brownian bridge in C [0,1]. Further the a.s. uniform convergence of R., is considered and some examples are given.     

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.     

Fourier inversion and the Hausdorff distance

This paper continues research done by F.H. Ruymgaart and the author. For a function f on R ^d we consider its Fourier transform F f and the functions f_M (M>0) derived from F f by the formula f_M(x) =( F( ε_M · F f ))(− x );, where the ε_M are suitable integrable functions tending to 1 pointwise as M →∞. It was shown earlier that, relative to a metric d _H , analogous to the Hausdorff distance between closed sets, one has d _H (f_M, f) = O( M ^−½) for all f in a certain class. We now show that, for such f , the estimate O( M ^−½) is optimal if and only if f has a discontinuity point.     

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.     

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.     

The Revenue Potential of Site Value Tax: Extension and Update of General Equilibrium Model With Recent Empirical Estimates of Several Key Parameters

A bstract . Although appealing on the consideration of efficiency, the site ( land ) value tax has been dismissed by some economists as an unviable alternative to the local real estate tax on the ground that it cannot generate sufficient revenue. From earlier work based on a general equilibrium model, however, a switch from a real estate to an equal yield site value tax could result in an increase in equilibrium land prices (and hence the site value tax base). In particular, equilibrium land prices will rise with a site value relative to a real estate tax if: (L+K/L) > e_x. (f_L+ f_k)/f_k. s_x+ e_x. f_L Critical to that theoretical result are the magnitudes of several parameters including the percent land constitutes of total real estate value , (L + K/L), the elasticity of substitution, s_x, the elasticity of demand for real estate e_x, and the output elasticities, f_k and f_L. Based on recent empirical estimates of those parameters, the above stated condition holds.     

Functional laws of the iterated logarithm for small increments of empirical processes

Let F , denote the uniform empirical distribution based on the first n ≥ 1 observations from an i.i.d. sequence of uniform (0, 1) random variables. We describe the almost sure limiting behavior of the sets of increment functions {F_n(t + h_n.) - F_n(t): 0 ≤ t ≤ 1 - h_n}, when {h_n: n ≥ 1) is a nonincreasing sequence of constants such that nh_n /log n ← 0.     

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     .     

On the generalized Poisson distribution

We use Euler's difference lemma to prove that, for θ > 0 and 0 ≤λ < 1, the function P _n defined on the non-negative integers by
P _n (θ, λ) = [θ(θ + n λ) ^{n −1}/ n !]e^{− n λ−θ}
defines a probability distribution, known as the Generalized Poisson Distribution.     

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.     

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.     

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.     

Steekproeven uit de halve cauchy verdeling

Summary  Let x₁…, x_n be a sample from a distribution with infinite expectation, then for n →∞ the sample average _n tends to +∞ with probability 1 (see [4]).
Sometimes _n contains high jumps due to large observations. In this paper we consider samples from the "absolute Cauchy" distribution. In practice, on may consider the logarithm of the observations as a sample from a normal distribution. So we found in our simulation. After rejecting the log-normality assumption, one will be tempted to regard the extreme observations as outliers. It is shown that the discarding of the outlying observations gives an underestimation of the expectation, variance and 99 percentile of the actual distribution.     

On divergence and convergence of sums of nonnegative random variables

Abstract  If X ₁, X ₂,… are exponentially distributed random variables thenσ^∞_{k= 1} X_k=∞ with probability 1 iff σ^∞_{k= 1} EX_k=∞. This result, which is basic for a criterion in the theory of Markov jump processes for ruling out explosions (infinitely many transitions within a finite time) is usually proved under the assumption of independence (see FREEDMAN (1971), p. 153–154 or BREI-MAN (1968), p. 337–338), but is shown in this note to hold without any assumption on the joint distribution. More generally, it is investigated when sums of nonnegative random variables with given marginal distributions converge or diverge whatever are their joint distributions.     

The effect of rounding off in samples

Dit artikel bevat een elementaire behandeling van afrondingseffecten in steekproeven.
Zoals men verwacht, neemt de variantie van het steekproefgermddelde door af ronding der waarnemingen toe met een factor 1 + b ²/12σ _x ², waarbij b het afrondings- interval en σ _x ² de stamvariantie is, in analogie met de bekende correcties van Sheppard. Verder wordt het effect van afronding op de variantie van de steek-proefvariantie onderzocht. Er wordt op gewezen, dat de afrondingsfout ε in één steekproef gecorreleerd is met de afwijking van de nauwkeurige — d.w.z. niet afgeronde — stochastische variabele x van zijn gemiddelde μ, hoewel deze correlatie gemiddeld nul is. Het gevolg is, dat de variantie van de steekproefvariantie toeneemt. Bij "gewone" waarschijnlijkheidsverdelingen, die continue naar nul gaan bij x →, is deze toename gelijk aan
1/ n (1/3 b ²σ _x ²+ 1/180 b ⁴)
waarbij n de grootte van de steekproef is. Bij normale verdelingen is het effect ongeveer twee maal zo groot als de toename volgens de Sheppard's correctie van de afgeronde variantie.     

Characteristic properties of order statistics based on random sample size from an exponential distribution

Suppose X₁, X₂, X_m is a random sample of size m from a population with probability density function f (x), x > 0), and let X_{1, m}< × 2, m <… < X_{m, m} be the corresponding order statistics.
We assume m is an integer-valued random variable with P( m = k ) = p (1- p )^k-1, k = 1,2,… and 0 < p < 1. Two characterizations of the exponential distribution are given based on the distributional properties of X_{l, m}.     

On a simple sequential decision problem

We consider a simple sequential problem that generalizes the gamble with a fair coin. A sequence [ X _n] ot {0,1} r.v.'s is observed, and at each step the gambler can bet on either 0 or 1. The sequence ( X _n) is assumed exchangeable. Except for the case of i.i.d. r.v.'s with even probability (the fair coin), there exists a strategy such that the cumulated expected gain diverges to +∞.     

A Simulation Approach to Nonparametric Empirical Bayes Analysis

We deal with general mixture of hierarchical models of the form m(x) = f_øf(x |θ) g (θ)dθ , where g(θ) and m(x) are called mixing and mixed or compound densities respectively, and θ is called the mixing parameter. The usual statistical application of these models emerges when we have data x_i, i = 1,…,n with densities f(x_i|θ_i) for given θ_i, and the θ₁ are independent with common density g(θ) . For a certain well known class of densities f(x |θ) , we present a sample-based approach to reconstruct g(θ) . We first provide theoretical results and then we use, in an empirical Bayes spirit, the first four moments of the data to estimate the first four moments of g(θ) . By using sampling techniques we proceed in a fully Bayesian fashion to obtain any posterior summaries of interest. Simulations which investigate the operating characteristics of our proposed methodology are presented. We illustrate our approach using data from mixed Poisson and mixed exponential densities.     

A characterization of gamma mixtures of stable laws motivated by limit theorems

This pape; is concerned with distributional solutions of X₁+…+ X_m^d= U(X₁+…+ X_m+n) where the X's are iid and independent of U which takes values in [0,1]. When U is a constant the only possible non-trivial solutions lie in the class of semi-stable laws, and they are stable under a simple regularity condition. This material is reviewed. A unified account is given of some results known for the case where U has a beta (α, 1) law, apparently the only other case allowing explicit identification of all possible solutions.