The flower intersection problem for Kirkman triple systems

The flower at a point x in a Steiner triple system is the set of all triples containing x. Denote by I_R^*[r] the set of all integers k such that there exists a pair of KTS(2r+1) having k+r triples in common, r of them being the triples of a common flower. In this article we determine the set I_R^*[r] for any positive integer r≡1 (mod 3) (only nine cases are left undecided for r=7,13,16,19), and establish that I_R^*[r]=J[r] for r≡1 (mod 3) and r22 where J[r]={0,1,…,2r(r−1)/3−6,2r(r−1)/3−4,2r(r−1)/3}.     

Adaptive wavelet empirical Bayes estimation of a location or a scale parameter

Assume that in independent two-dimensional random vectors (X₁,θ₁),…,(X_n,θ_n), each θ_i is distributed according to some unknown prior density function g. Also, given θ_i=θ, X_i has the conditional density function q(x−θ), x,θ(−∞,∞) (a location parameter case), or θ⁻¹q(x/θ), x,θ(0,∞) (a scale parameter case). In each pair the first component is observable, but the second is not. After the (n+1)th pair (X_n+1,θ_n+1) is obtained, the objective is to construct an empirical Bayes (EB) estimator of θ. In this paper we derive the EB estimators of θ based on a wavelet approximation with Meyer-type wavelets. We show that these estimators provide adaptation not only in the case when g belongs to the Sobolev space H with an unknown , but also when g is supersmooth.     

Nonparametric estimation of the lower tail dependence λL in bivariate copulas

The lower tail dependence λ_L is a measure that characterizes the tendency of extreme co-movements in the lower tails of a bivariate distribution. It is invariant with respect to strictly increasing transformations of the marginal distribution and is therefore a function of the copula of the bivariate distribution. λ_L plays an important role in modelling aggregate financial risk with copulas. This paper introduces three non-parametric estimators for λ_L. They are weakly consistent under mild regularity conditions on the copula and under the assumption that the number k = k(n) of observations in the lower tail, used for estimation, is asymptotically k ≈ √n. The finite sample properties of the estimators are investigated using a Monte Carlo simulation in special cases. It turns out that these estimators are biased, where amount and sign of the bias depend on the underlying copula, on the sample size n, on k, and on the true value of λ_L.     

On the non-existence of generalised Hadamard matrices

The determinant of a generalized Hadamard matrix over its group ring factored out by the relation Σ_gεG G = 0 is shown to have certain number theoretic properties. These are exploited to prove the non-existence of many generalised Hadamard matrices for groups whose orders are divisible by 3, 5 or 7. For example the GH(15, C₁₅), GH(15, C₃) and GH(15, C₅) do not exist. Also for certain n and G we find the set of determinants of the GH(n, G) matrices.     

A critical look at Lo's modified R/S statistic

We report on an empirical investigation of the modified rescaled adjusted range or R/S statistic that was proposed by Lo, 1991. Econometrica 59, 1279–1313, as a test for long-range dependence with good robustness properties under ‘extra’ short-range dependence. In contrast to the classical R/S statistic that uses the standard deviation S to normalize the rescaled range R, Lo's modified R/S-statistic V_q is normalized by a modified standard deviation S_q which takes into account the covariances of the first q lags, so as to discount the influence of the short-range dependence structure that might be present in the data. Depending on the value of the resulting test-statistic V_q, the null hypothesis of no long-range dependence is either rejected or accepted. By performing Monte-Carlo simulations with ‘truly’ long-range- and short-range dependent time series, we study the behavior of V_q, as a function of q, and uncover a number of serious drawbacks to using Lo's method in practice. For example, we show that as the truncation lag q increases, the test statistic V_q has a strong bias toward accepting the null hypothesis (i.e., no long-range dependence), even in ideal situations of ‘purely’ long-range dependent data.     

Optimal partially balanced nested row-column designs

A partially balanced nested row-column design, referred to as PBNRC, is defined as an arrangement of v treatments in b p × q blocks for which, with the convention that p q, the information matrix for the estimation of treatment parameters is equal to that of the column component design which is itself a partially balanced incomplete block design. In this paper, previously known optimal incomplete block designs, and row-column and nested row-column designs are utilized to develop some methods of constructing optimal PBNRC designs. In particular, it is shown that an optimal group divisible PBNRC design for v = mn kn treatments in p × q blocks can be constructed whenever a balanced incomplete block design for m treatments in blocks of size k each and a group divisible PBNRC design for kn treatments in p × q blocks exist. A simple sufficient condition is given under which a group divisible PBNRC is Ψ_f-better for all f> 0 than the corresponding balanced nested row-column designs having binary blocks. It is also shown that the construction techniques developed particularly for group divisible designs can be generalized to obtain PBNRC designs based on rectangular association schemes.     

Self-affine time series: measures of weak and strong persistence

In this paper, we examine self-affine time series and their persistence. Time series are defined to be self-affine if their power-spectral density scales as a power of their frequency. Persistence can be classified in terms of range, short or long range, and in terms of strength, weak or strong. Self-affine time series are scale-invariant, thus they always exhibit long-range persistence. Synthetic self-affine time series are generated using the Fourier power-spectral method. We generate fractional Gaussian noises (fGns), −1β1, where β is the power-spectral exponent. These are summed to give fractional Brownian motions (fBms), 1β3, where the series are self-affine fractals with fractal dimension 1D2; β=2 is a Brownian motion. With β>1, the time series are non-stationary and moments of the time series depend upon its length; with β<1 the time series are stationary. We define self-affine time series with β>1 to have strong persistence and with β<1 to have weak persistence. We use a variety of techniques to quantify the strength of persistence of synthetic self-affine time series with −3β5. These techniques are effective in the following ranges: (1) semivariograms, 1β3, (2) rescaled-range (R/S) analyses, −1β1, (3) Fourier spectral techniques, all values of β, and (4) wavelet variance analyses, all values of β. Wavelet variance analyses lack many of the inherent problems that are found in Fourier power-spectral analysis.     

Search with small sets in presence of a liar

One unknown element of an n-element set is sought by asking if it is contained in given subsets. It is supposed that the question sets are of size at most k and all the questions are decided in advance, the choice of the next question cannot depend on previous answers. At most l of the answers can be incorrect. The minimum number of such questions is determined when the order of magnitude of k is n with <1. The problem can be formulated as determination of the maximum sized l-error-correcting code (of length n) in which the number of ones in a given position is at most k.     

A class of BIB designs with repeated blocks

Balanced incomplete block design (BIBD) with repeated blocks is studied in detail. Methods of construction of BIB designs with repeated blocks are developed so as to distinguish the usual BIBD and BIBD with repeated blocks. One additional parameter, say d, is considered here, where d denotes the number of distinct blocks present in the BIB design with repeated blocks. Further, a class of BIB design with parameters: v = 7, b = 28, r = 12, k = 3, λ= 4, has been constructed where, out of 15, 14 BIB designs have repeated blocks. These 15 BIB designs, which have the same parameters, are compared on the basis of number of distinct blocks (d) and the multiplicities of variance of elementary contrasts of the block effect.     

Testing equality of proportions with incomplete correlated data

Let (ψ_i,φ_i) be independent, identically distributed pairs of zero-one random variables with (possible) dependence of ψ_i and φ_i within the pair. For n pairs, both variables are observed, but for m₁ additional pairs only ψ_i is observed and for m₂ others φ_i is observed. If π_1· = P{ψ_i = 1} and π_·1=P{φ_i, the problem is to test π_1·=π_·1. Maximum likelihood estimates of π_1· and π_·1 are obtained via the EM algorithm. A test statistic is developed whose null distribution is asymptotically chi-square with one degree of freedom (as n and either m₁ or m₂ tend to infinity). If m₁ = m₂ = 0 the statistic reduces to that of McNemar's test; if n = 0, it is equivalent to the statistic for testing equality of two independent proportions. This test is compared with other tests by means of Pitman efficiency. Examples are presented.     

Linearity of regression for non-adjacent record values

Let X₁,X₂,… be a sequence of iid random variables having a continuous distribution; by R₁,R₂,… denote the corresponding record values. All the distributions allowing linearity of regressions either E(R_m+k|R_m) or E(R_m|R_m+k) are identified.     

Pairwise orthogonal F-rectangle designs

The concept of pairwise orthogonal Latin square design is applied to r row by c column experiment designs which are called pairwise orthogonal F-rectangle designs. These designs are useful in designing successive and/or simulataneous experiments on the same set of rc experimental units, in constructing codes, and in constructing orthogonal arrays. A pair of orthogonal F-rectangle designs exists for any set of v treatment (symbols), whereas no pair of orthogonal Latin square designs of order two and six exists; one of the two construction methods presented does not rely on any previous knowledge about the existence of a pair of orthogonal Latin square designs, whereas the second one does. It is shown how to extend the methods to r=pv row by c=qv column designs and how to obtain t pairwise orthogonal F-rectangle design. When the maximum possible number of pairwise orthogonal F-rectangle designs is attained the set is said to be complete. Complete sets are obtained for all v for which v is a prime power. The construction method makes use of the existence of a complete set of pairwise orthogonal Latin square designs and of an orthogonal array with vⁿ columns, (vⁿ−1)/(v−1) rows, v symbols, and of strength two.     

On Berry-Esseen theorem for some functions of U-statistics

The rate of convergence in the central limit theorem and in the random central limit theorem for some functions of U-statistics are established. The theorems refer to the asymptotic behaviour of the sequence {g(U_n),n≥1}, where g belongs to the class of all differentiable functions g such that g′εL(δ) and U_n is a U-statistics.     

A new approach to use multivariate auxiliary information in sample surveys

A new approach to form multivariate difference estimator is suggested which does not require the knowledge of unknown population parameters as such. It gives minimum variance among the class of multivariate difference estimators. The performance of this estimator with respect to Des Raj's (J. Amer. Statist. Assoc. 60 (1965), 270–277) multivariate difference estimator is illustrated. Using the information on two auxiliary variates, the robustness of Des Raj's estimator y_d is studied empirically. Two new estimators to estimate population mean/total are developed on the same lines as that of y_d. The performance of these estimators is studied for a wide variety of populations.     

Likelihood ratio ordering of order statistics

This paper provides an improvement on the work of Bapat and Kochar (1994, Linear Algebra Appl., 199, 281–291) and strengthens the literature on the likelihood ratio ordering of order statistics. For independent (but possibly nonidentically distributed) absolutely continuous random variables X₁,…,X_n, it is shown under some weak conditions that

X_1:n^lrlrX_n:n,

where ^lr stands for the likelihood ratio ordering and X_k:n represents the kth-order statistic.     

Optimal spacing of the selected sample quantiles for the joint estimation of the location and scale parameters of a symmetric distribution

We are considering the ABLUE’s – asymptotic best linear unbiased estimators – of the location parameter μ and the scale parameter σ of the population jointly based on a set of selected k sample quantiles, when the population distribution has the density of the form

where the standardized function f(u) being of a known functional form.A set of selected sample quantiles with a designated spacing

or in terms of u=(x−μ)/σ

where

λ_i=∫_−∞^u_if(t) dt, i=1,2,…,k

are given by

x(n₁)<x(n₂)<<x(n_k),

where

Asymptotic distribution of the k sample quantiles when n is very large is given by

h(x(n₁),x(n₂),…,x(n_k);μ,σ)=(2πσ²)^−k/2[λ₁(λ₂−λ₁)(λ_k−λ_k−1)(1−λ_k)]^−1/2n^k/2 exp(−nS/2σ²),

where

f_i=f(u_i), i=0,1,…,k,k+1,

f₀=f_k+1=0, λ₀=0, λ_k+1=1.

The relative efficiency of the joint estimation is given by

where

and κ being independent of the spacing . The optimal spacing is the spacing which maximizes the relative efficiency η(μ,σ).We will prove the following rather remarkable theorem. Theorem. The optimal spacing for the joint estimation is symmetric, i.e.

λ_i+λ_k−i+1=1,

or

u_i+u_k−i+1=0, i=1,2,…,k,

if the standardized density f(u) of the population is differentiable infinitely many times and symmetric

f(−u)=f(u), f′(−u)=−f′(u).

     

Estimators for a Poisson parameter using ranked set sampling

Using ranked set sampling, a viable BLUE estimator is obtained for estimating the mean of a Poisson distribution. Its properties, such as efficiency relative to the ranked set sample mean and to the maximum likelihood estimator, have been calculated for different sample sizes and values of the Poisson parameter. The estimator (termed the normal modified r.s.s. estimator is more efficient than both the ranked set sample mean and the MLE. It is recommended as a reasonable estimator of the Poisson mean ( λ) to be used in a ranked set sampling environment.     

Variants of the graph dependent model in extreme value theory

With a set X₁, X₂, .... X_n n random variables, a graph is associated whose vertices are the integers 1,2,..., n and whose edges represent those pairs i and j for which the events {X_i>X} and {X_j>X} do not become “almost independent” for “large X”. With a variety of assumption on the edge set of the graph, the asymptotic distribution of the extremes of the X_j, when properly normalized, is determined. This refines the earlier result of the present author on this kind of dependence, and extends and unifies several known dependent extreme value models.     

An approach to evaluating sensitivity in Bayesian regression analyses

This paper presents a method for assessing the sensitivity of predictions in Bayesian regression analyses. In parametric Bayesian analyses there is a family s₀ of regression functions, parametrized by a finite-dimensional vector B. The family s₀ is a subset of R, the set of all possible regression functions. A prior π₀ on B induces a prior on R. This paper assesses sensitivity by computing bounds on the predictive probability of a fixed set K over a class of priors, Γ, induced by a class of families of regression functions, Γ_s, and a class of priors, Γ_π. This paper is divided into three parts which (1) define Γ, (2) describe an algorithm for finding accurate bounds on predictive probabilities over Γ and (3) illustrate the method with two examples. It is found that sensitivity to the family of regression functions can be much more important than sensitivity to π₀.