On Christensen's conjecture

In this paper we give solution to a conjecture appearing in Christensen (1997, p. 360) in relation to the definitions of standardized residuals in loglinear models. Work done when he was visiting as the Distinguished Lukacs Professor on leave from the Complutense University of Madrid. An erratum to this article is available at .     

On a conjecture concerning the least favorable configuration of a two-stage selection procedure

A two-stage procedure 𝓅with screening in the first stage to find the population with the largest mean out of k ≧ 2 normal populations with unknown means and a common variance is under concern. It was proposed and previousiy studied by Cohen (1959), Alam (1970) and Tamhane and Bechhofer (1977, 1979) using the indifference-zone approach. The conjecture that the least favourable parameter configuration for the probability of a correct selection is of the slippage type remained unproved for k ≧ 3. Miescke and Sehr (1980) proved the conjecture for k=3. The problem was further discussed by Gupta and Miescke (1982). A general proof for rhe conjecture will be given in this paper.     

On a conjecture concerning the least favorable configuration of certain two-stage selection procedures

Given k normal populations with unknown means and a common known variance a two-stage procedure p₁ with screening in the first stage to find the population with the largest mean using the indifference-zone approach is under concern. It was proposed and studied previously by Cohen (1959), Alam (1970) and Tamhane and Bechhofer (1977, 1979). But up to now a conjecture concerning the least favorable parameter configuration of p₁remained unproved for k ≥ 3. In this paper we give a non-standard proof of the conjecture in case of k = 3 for p₁. which (under minor changes) works also for a simplified version of p₁,. Besides, a counterexample is provided to show that another (more intuitive) method of proof fails to work.     

Statistical games against a prophet-proof of a minimax conjecture

“Prophet theory” quantifies the price a statistician has to pay for his lack of information in stochastic sequences. In a recent paper, Schmitz (1991) gave a game-theoretical interpretation of this situation and he formulated in particular a minimax conjecture for the difference case. In this note we prove that conjecture and, moreover, present minimax ran domized stopping times (minimax procedures for the statistician).     

Progress on a conjecture regarding the triangular distribution

Triangular distributions are a well-known class of distributions that are often used as an elementary example of a probability model. Maximum likelihood estimation of the mode parameter of the triangular distribution over the unit interval can be performed via an order statistic based method. It had been conjectured that such a method can be conducted using only a constant number of likelihood function evaluations, on average, as the sample size becomes large. We prove two theorems that validate this conjecture. Graphical and numerical results are presented to supplement our proofs.     

A counterexample to a conjecture on random shadows

We construct a counterexample to a conjecture in stereology. It is well known that two convex solid bodies which cast congruent orthographic shadows on any common plane must necessarily be congruent to each other. This characterization fails if the orientations of the shadows are allowed to differ. Such counterexamples as have been constructed for the latter case fail to have the same distribution of random shadows projected onto planes with random orientation. Therefore P. McCullagh has conjectured that two bodies whose random shadows have the same distribution are necessarily congruent. We construct a counterexample to this conjecture. Some implications are drawn for the development of a general theory of stereological reconstruction based upon random projected shadows or images.     

Proof of a lattice paths conjecture connected to the tennis ball problem

We give a short history of the so-called tennis ball problem, and discuss its relation to lattice path enumeration. We also prove a conjecture related to a solution of the symmetric case, namely when the number of balls removed each turn is exactly half the number inserted.     

C239. The continued fraction for lnr( z): a stieljes conjecture

Estimation in logistic-normal models for correlated and overdispersed binomial data is complicated by the numerical evaluation of often intractable likelihood functions. Penalized quasilikelihood (PQL) estimators of fixed effects and variance components are known to be seriously biased for binary data. A simple correction procedure has been proposed to improve the performance of the PQL estimators. The proposed method is illustrated by analyzing infectious disease data. Its performance is compared, by means of simulations, with that of the Bayes approach using the Gibbs sampler.     

A conjecture on the existence of cyclic Hadamard difference sets

If there exists a cyclic Hadamard difference set of length v, then v = 4n − 1 is conjectured to be either a prime, or a product of “twin primes”, or one less than a power of 2.     

On the role of data,statistics and decisions in a pandemic

AStA Advances in Statistical Analysis - A pandemic poses particular challenges to decision-making because of the need to continuously adapt decisions to rapidly changing evidence and available...     

Discussion on On the role of data,statistics and decisions in a pandemic

AStA Advances in Statistical Analysis - The authors make an important contribution presenting a comprehensive and thoughtful overview about the many different aspects of data, statistics and data...     

On the Feasibility of Basing Horvitz and Thompson's Estimator on a Sample by Rao,Hartley, and Cochran's Scheme

Formulae for the first and second order inclusion probabilities for the Rao et al. (1962 Rao , J. N. K. , Hartley , H. O. , Cochran , W. G. ( 1962 ). On a simple procedure of unequal probability sampling without replacement . J. Roy. Statist. Soc. B 24 : 482 – 491 . [CSA]  [Google Scholar]) (RHC) scheme of sampling are derived. They enable one to evaluate, for a sample drawn according to the RHC scheme, the Horvitz and Thompson's (1952 Horvitz , D. G. , Thompson , D. J. ( 1952 ). A generalization of sampling without replacement from a finite universe . J. Amer. Statist. Assoc. 47 : 663 – 685 . [CSA] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) estimator (HTE) along with its unbiased variance estimator given by Yates and Grundy (1953 Yates , F. , Grundy , P. M. ( 1953 ). Selection without replacement from within strata with probability proportional to size . J. Amer. Statist. Assoc. 75 : 206 – 211 . [CSA]  [Google Scholar]). So, for a sample at hand thus drawn one may choose between the RHCE and the HTE for use on finding which one has the smaller coefficient of variation.     

On randomness, determinism and computability

This paper discusses the meaning and relationship of randomness and determinism. A fundamental development of chaotic dynamical systems is given with examples. Such systems are seen to exhibit randomness in the usual sense of unpredictability. The formal definition of randomness in terms of algorithmic incompressibility is also discussed. The role of recursion in computability and randomness is also discussed.     

On the asymptotic distributions of mean,autocovariance, autocorrelation,crossgovariancb and impulse response estimators of a stationary multidimensional random field

Asymptotic properties of mean, autocovariance, autocorrelation, crosscovariance and impulse response estimators of a stationary M-dimensionai (M-D) random field are studied. It is shown that only unbiased-type estimators of autocovariances, autocorrelations, crosscovariances and impulse responses have the asymptotic distributions when M≧ 2. Moreover, the asymptotic distributions of mean, autocovariance, autocorrelation, crosscovariance and impulse response estimators are presented.     

On equalities between BLUES,WLSEs, and SLSEs

Necessary and sufficient conditions for equalities between the best linear unbiased estimator, the weighted least-squares estimator, and the simple least-squares estimator of the expectation vector in a general Gauss-Markoff model are given in some alternative formulations. The main result states, somewhat surprisingly, that the weighted least-squares estimator cannot be identical with the simple least-squares estimator unless they both coincide with the best linear unbiased estimator.     

On Geometry and Scale of a Stochastic Chemostat

A chemostat is a fixed volume bioreactor in which micro–organisms are grown in a continuously renewed liquid medium. We propose a stochastic model for the evolution of the concentrations in the single species and single substrate case. It is obtained as a diffusion approximation of a pure jump Markov process, whose increments are comparable in mean with the deterministic model. A specific time scale, related to the noise intensity, is considered for each source of variation. The geometric structure of the problem, usable by identification procedures, is preserved both in the drift and diffusion term. We study the properties of this model by numerical experiments.