Estimating the J function without edge correction

The interaction between points in a spatial point process can be measured by its empty space function F , its nearest-neighbour distance distribution function G , and by combinations such as the J function J = (1 G )/(1 F ). The estimation of these functions is hampered by edge effects: the uncorrected, empirical distributions of distances observed in a bounded sampling window W give severely biased estimates of F and G . However, in this paper we show that the corresponding uncorrected estimator of the function J = (1 G )/(1 F ) is approximately unbiased for the Poisson case, and is useful as a summary statistic. Specifically, consider the estimate _W of J computed from uncorrected estimates of F and G . The function J _W ( r ), estimated by _W , possesses similar properties to the J function, for example J _W ( r ) is identically 1 for Poisson processes. This enables direct interpretation of uncorrected estimates of J , something not possible with uncorrected estimates of either F , G or K . We propose a Monte Carlo test for complete spatial randomness based on testing whether J _W ( r ) 1. Computer simulations suggest this test is at least as powerful as tests based on edge corrected estimators of J .     

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.     

On piecewise linear density estimators

We study piecewise linear density estimators from the L ₁ point of view: the frequency polygons investigated by S cott (1985) and J ones et al. (1997), and a new piecewise linear histogram. In contrast to the earlier proposals, a unique multivariate generalization of the new piecewise linear histogram is available. All these estimators are shown to be universally L ₁ strongly consistent. We derive large deviation inequalities. For twice differentiable densities with compact support their expected L ₁ error is shown to have the same rate of convergence as have kernel density estimators. Some simulated examples are presented.     

Vergelijking van een tweetal toetsen tegen verloop voor een aantal kansen

Two test statistics t and b , testing equality of probabilities p_i of success in k different series against the hypothesis of trend with given numbers g_i (weights) of the series, are compared. The first teststatistic, due to C. van Ee den en J, Hemelrijk [1] is

with n_i1 equal to the number of successes in n_i trials and

Defining trend by

it appears that the teststatistic t gives rise to a test consistent for the complete set of alternatives τ≠ 0.
The other teststatistic is

b gives rise to a test which is not consistent for the general set of alternatives τ≠ 0 but for a rather important subset of these alternatives, i.e. those alternatives which show a lineair trend.
Neither of the tests in necessarily unbiased. The asymptotic relative efficiency of test t with respect to b is equal to or lower than unity. (Equal in case
n₁= n₂=…= n_k with b = t;
in this case b is also consistent for the set of alternatives τ≠ 0). The variances of the teststatistics can be estimated with

Both tests are based on the approximately normal distribution of the teststatistics. To judge this approximation the 3rd and 4th cumulants of the distributions of the statistics are evaluated in terms of number of elements of a binomial distribution.
It is concluded that in case of possible non-lineair relationships the teststatistic t is preferable as it gives rise to a consistent, designfree test. In case a lineair relationship has to be tested against the hypothesis of no trend the teststatistic b has to be prefierred, especially if the number of trials in the series are very different. An example is discussed.     

De omvang van de steekproef bij een enkelvoudig steekproefsysteem

Summary  (Sample size for a single sampling scheme).
The operating characteristic of a sampling scheme may be specified by the producers 1 in 20 risk point ( p ₁), at which the probability of rejecting a batch is 0.05, and the consumers 1 in 20 risk point ( p ₂) at which the probability of accepting a batch of that quality is also 0.05.
A nomogram is given (fig. 2) to determine for single sampling schemes and for given values of p₁ and p ₂ the necessary sample size ( n ) and the allowable number of defectives in the sample ( c ).
The nomogram may reversedly be used to determine the producers and consumers 1 in 20 risk points for a given single sampling scheme.
The curves in this nomogram were computed from a table of percentage points of the χ² distribution. For v > 30 Wilson and Hilferty's approximation to the χ² distribution was used.     

Substituut-variabelen in correlatieberekingen

The recently repeated assertion that in correlation analysis it makes little difference whether one variable (x₂) is used instead of another one (x₃), provided the coefficient of correlation (r₂₃) between x₂ and x₃ is high, is scrutinized.
To that purpose the ranges of coefficients of correlation with respect to the substitute variable are expressed in formula 3. Moreover, by way of example, extreme values of coefficients of simple correlation (r₁₃ and r₃₄), of multiple correlation (R_1.34 and R_3.14) and of regression (α₁₃ and α₁₄, α₃₁ and α₃₄) relating to the substitute variable, are calculated on the basis of empirical values of coefficients of simple correlation relating to the substituted and the remaining variables.
The outcome of those calculations are summarized in the tables 1 and 3, and in the graph.
Table 1 presents ranges of r₁₃ for given values of r₁₂ and r₂₃, table 3 shows extreme values of coefficients of single and multiple correlation and regression in case an additional variable x₄ is introduced and r₁₂, r₁₄, r₂₄ and r₂₃ are given. The graph shows an ellipse as the boundary of the inner closed domain of compatible values of r₁₃ and r₃₄.
Those results clearly indicate the need for caution in substituting one variable by another.     

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.     

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.     

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.     

Estimation of the intensity of a Poisson point process by means of nearest neighbor distances

A new unbiased consistent asymptotically normal estimator U _k of the intensity λ of a stationary multivariate Poisson point process is exhibited. This estimate is based on a combination of the j -th nearest neighbor (possibly non Euclidean) distances ( j =1, ..., k ) to a single fixed site x . A simple closed form containing logarithmic terms is obtained for E ( U ^l _k )(0< l < k ).     

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 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.     

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).     

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.     

Time Series Analysis of Repeated Surveys: The State–space Approach

Cross sectional estimates from repeated surveys form a time series { y_t }. These estimates can be viewed as the sum y _t = Y _t + e _t of two processes, { Y _t }, the population process and { e _t }, the survey error process. Serial correlations in the latter series are usually present, mainly due to sample overlap. Other sources of data such as censuses, administrative records and demographic population counts are also available. The state–space modelling approach to the analysis of repeated surveys allows combining information from different sources, incorporating benchmarking constraints in a natural way. Results from these methods seem to compare favourably with those from X-11-ARIMA in filtering out survey errors.     

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.     

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.     

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.     

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.     

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.