Estimating a model through the conditional MLE

The estimation problem of a model through the conditional maximum likelihood estimator (MLE) is explored. The estimated model is compared using the two dual Kullback-Leibler losses with that through the unconditional MLE. The former is found to be superior to the latter under familiar models. This result is applicable to the model selection problem. These suggest a novel extensive use of the conditional likelihood, since the traditional use of the conditional likelihood was restricted only on inference for the structural parameter.     

Computation of the maximum likelihood estimate of a noncentrality parameter

An algorithm and error analysis are presented for finding the maximum likelihood estimator of the noncentrality parameter of the χ² and F distributions.     

STATISTICAL INFERENCE PROCEDURE FOR A BIVARIATE EXPONENTIAL DISTRIBUTION

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Analysis of marginal and conditional density functions for separate inference

Summary This paper discusses, with measure-theoretical rigor, some basic aspects of the theory of separate inference. To analyze densities of marginal and conditional submodels, certain operators are introduced. First a general concept of decomposition of a model is proposed, and the corresponding factorization of densities of the model is established. Next it is shown that the property of smoothness of a family of densities is retained in the operation of conditioning, and therefore it yields the differentiability of the conditional expectation of a real-valued statistic in a certain sense. On the basis of this result, two measures of the effectiveness of a submodel in separate inference are investigated. The Institute of Statistical Mathematics     

Relative homology and the structure of relative approximations

The structure of rightF-^⊥ T-approximations of any finitely generated module over an artin algebra Λ is given, relative to an additive subbifunctorF of Ext _Λ ¹ (-, -) and anF-cotilting moduleT.     

Testing for a multivariate generalized Pareto distribution

It has recently been shown by Rootzén and Tajvidi (Bernoulli, 12:917–930, 2006) that modelling exceedances of a random variable over a high threshold (peaks-over-threshold approach [POT]) can also in the multivariate setup be done rationally only by a multivariate generalized Pareto distribution (GPD). The selection of a proper threshold is, however, a crucial problem. The contribution of this paper is twofold: We develop first a non asymptotic and exact level-α test based on the single-sample t-test, which checks whether multivariate data are actually generated by a multivariate GPD. Secondly, this procedure is utilized for the derivation of a t-test based threshold selection rule in multivariate peaks-over-threshold models. The application to a hydrological data set illustrates this approach.      

A remark on complete positivity of elementary operators

We show that many of the features of the theory of hypercyclic and supercyclic operators extend to that of finitely hypercyclic/supercyclic operators. In particular, subnormal operators, Banach space isometries, and thereforeC ₁ contractions are not finitely supercyclic.     

Minimaxity of a preliminary test estimator for the mean of normal distribution

Summary The problem is to estimate the mean of the normal distribution under the situation where there is vague information that the mean might be equal to zero. A minimax property of the preliminary test estimator obtained by the use of AIC (Akaike information Criterion) procedure is proved under a loss function based on the Kullback-Leibler information measure.     

A note on the Bayesian inference for generalized multivariate gamma distribution

In this paper, we show that the derivation of Lemma 3 of Das and Dey (2010) needs to be corrected by using a logical transformation, instead of the ad-hoc transformation which is partially motivated by its univariate equivalent transformation. The correct derivation is presented by two approaches.     

Connected components of the category of elementary abelian p-subgroups

We determine the maximal number of conjugacy classes of maximal elementary abelian subgroups of rank 2 in a finite p-group G, for an odd prime p. Namely, it is p if G has rank at least 3 and it is p+1 if G has rank 2. More precisely, if G has rank 2, there are exactly 1,2,p+1, or possibly 3 classes for some 3-groups of maximal nilpotency class.     

A compound of the generalized negative binomial distribution with the generalized beta distribution

This paper presents a compound of the generalized negative binomial distribution with the generalized beta distribution. In the introductory part of the paper, we provide a chronological overview of recent developments in the compounding of distributions, including the Polish results. Then, in addition to presenting the probability function of the compound generalized negative binomial-generalized beta distribution, we present special cases as well as factorial and crude moments of some compound distributions.     

Statistical inference for a certain class of bivariate distributions

The paper deals with statistical inference for a certain class of bivariate distributions. The class of marginal distributions is given and is shown to include distributions with only location and scale parameters. A normalizing transformation is applied to the marginal distributions and the parameters are estimated by maximum likelihood. For this class there is a great deal of simplification in the calculations for the asymptotic covariance matrix of the vector of parameter estimators. Statistics for tests of zero correlation are discussed. Also, the analysis is carried out for exponential marginal distributions.     

Model Selection for Variable Length Markov Chains and Tuning the Context Algorithm

We consider the model selection problem in the class of stationary variable length Markov chains (VLMC) on a finite space. The processes in this class are still Markovian of high order, but with memory of variable length. Various aims in selecting a VLMC can be formalized with different non-equivalent risks, such as final prediction error or expected Kullback-Leibler information. We consider the asymptotic behavior of different risk functions and show how they can be generally estimated with the same resampling strategy. Such estimated risks then yield new model selection criteria. In particular, we obtain a data-driven tuning of Rissanen's tree structured context algorithm which is a computationally feasible procedure for selection and estimation of a VLMC.     

A necessary and sufficient condition for the existence of an exponential attractor

We give a necessary and sufficient condition for the existence of an exponential attractor. The condition is formulated in the context of metric spaces. It also captures the quantitative properties of the attractor, i.e., the dimension and the rate of attraction. As an application, we show that the evolution operator for the wave equation with nonlinear damping has an exponential attractor.     

Geometric Generalization of the Exponential Law

For the multivariate ℓ₁-norm symmetric distributions, which are generalizations of the n-dimensional exponential distribution with independent marginals, a geometric representation formula is given, together with some of its basic properties. This formula can especially be applied to a new developed and statistically well motivated system of sets. From that the distribution of a t-statistic adapted for the two-parameter exponential distribution and its generalizations is determined. Asymptotic normality of this adapted t-statistic is shown under certain conditions.     

A note on the conditional probability of <Emphasis Type="Italic">IF</Emphasis>-events

We formulate a version of the conditional probability of an IF-event considering the Łukasiewicz operations with IF-sets. Also some properties of conditional probability are proved. The results are strengthenings of preceding ones published in [LENDELOVá, K.: Conditional IF-probability. In: Soft Methods for Integrated Uncertainty Modelling. Proceedings of the 2006 InternationalWorkshop on Soft Methods in Probability and Statistics (SMPS 2006), Bristol, UK, September 5–7, 2006. Adv. Soft Comput., Springer-Verlag, Berlin, 2006, pp. 275–283]. The paper was supported by Grant VEGA 1/0539/08.     

On the weight distribution of random binary linear codes

We investigate the weight distribution of random binary linear codes. For 0 < λ < 1 and n→∞ pick uniformly at random λn vectors in and let be the orthogonal complement of their span. Given 0 < γ < 1/2 with 0 < λ < h(γ) let X be the random variable that counts the number of words in C of Hamming weight γn. In this paper we determine the asymptotics of the moments of X of all orders .     

On inference concerning time-dependent queue performance: The M/G/1 example

This paper proposes easily-computed approximations to the finite-time expected waiting time for anM/G/1 system starting from an empty state. Both unsaturated (ρ<1) and saturated (ρ>1) conditions are considered. Numerical evidence is presented to indicate that the quality of the approximations is usefully good, especially when ease of computation is an issue. Further, the methodology is adapted to assess expected waiting time when inference must be made from a random sample of service times, and the decision is made to do so nonparametrically, i.e., without fitting a specific function. The results appear reasonable and potentially useful, and are not burdensome to obtain. The methodology investigated can also be applied to the variety of queueing models that are close siblings ofM/G/1: priority and breakdowns and “vacations” being examples. Of course other approximating and inferential options remain to be investigated.     

New Estimates for the Bottom of the Spectrum of Schrödinger Operators

We obtain a bound for the bottom of the negative spectrum of Schrödinger operators on weighted Riemannian manifolds in terms of F-Sobolev inequalities. Also, we show how an estimate for the first negative eigenvalue implies the F-Sobolev inequality.     

On Lower Bounds for the L 1-Norm of Exponential Sums

In the paper a method for finding lower bounds of the L ₁-norm of some exponential sums is described.