On a theorem of graybill

Graybill (1976) gives a theorem for testing variance components in balanced random models. Unfortunately, the theorem does not hold true. We pin down the reason that causes the theorem to fail and give a correct version of it.     

On a theorem of Kelley

Kelley (1974) claims to give necessary and sufficient conditions for the myopic strategy to be optimal for the two-armed Bernoulli bandit problem with a two-point prior. In this paper we argue that his conditions are not necessary. We further explore the myopicity of the optimal strategy for this particular bandit problem.     

On a threshold representation for complex load-sharing systems

Complex load-sharing systems are studied to incorporate dependencies among components through a load-sharing rule. As the load on the system increases, a series of cycles of Phase I/II failures occur where Phase I failure is a single component failure, which then causes a cascade of component failures (Phase II) due to the load transfer as these components fail. A threshold representation for the process of system failure is given. This representation is a gamma-type mixture representation when the component strengths are independent exponentials. In this case, for a given breaking pattern the mixture is over the gamma scale parameter and is based on a convolution of uniforms defined by the load-sharing parameters. Such convolutions can be approximated by normal densities which reduces the dimension of the parameter space. This representation can be generalized to independent component strengths with arbitrary distributions by transforming the strength and load-sharing to pseudo-strength and pseudo-load-sharing rules.     

On structural models and a theorem of d. basu

Structural inference as a method of statistical analysis seems to have escaped the attention of many statisticians. This paper focuses on Fraser’s necessary analysis of structural models as a tool to derive classical distribution results.

A structural model analyzed by Zacks (1971) by means of conventional statistical methods and fiducial theory is re-examined by the structural method. It is shown that results obtained by the former methods come as easy consequences of the latter analysis of the structural model. In the process we also simplify Zacks1 methods of obtaining a minimum risk equivariant estimator of a parameter of the model.

A theorem of Basu (1955), often used to prove independence of a complete sufficient statistic and an ancillary statistic, is also reexamined in the light of structural method. It is found that for structural models more can be achieved by necessary analysis without the use of Basu’s theorem. Bain’s (1972) application of Basu’s theorem of constructing confidence intervals for Weibull reliability is given as an example.     

On σ-coherence and a theorem of heath and sudderth

For the important case that the space of the possible states of nature is a compact Hausdorff space it is easy to generalize a result of Heath and Sudderth (1978) to the countably additive setting. No topological assumption is made on the space of observations. We also observe that posteriors need not always exist.     

On a Berry-Esséen theorem for a Studentized jackknife L-estimate

Consider a linear function of order statistics (“L-estimate”) which can be expressed as a statistical function T(F_n) based on the sample cumulative distribution function F_n. Let T*(F_n) be the corresponding jackknifed version of T(F_n), and let V²_n be the jackknife estimate of the asymptotic variance of n ^1/2T(F_n) or n ^1/2T*(F_n). In this paper, we provide a Berry-Esséen rate of the normal approximation for a Studentized jackknife L-estimate n^1/2[T*(F_n) - T(F)]/V_n, where T(F) is the basic functional associated with the L-estimate.     

On Bayes' theorem for fuzzy data

There are some ideas concerning a generalization of Bayes' theorem to the situation of fuzzy data. Some of them are given in the references [1], [5], and [7]. But the proposed methods are not generalizations in the sense of the probability content of Bayes' theorem for precise data. In the present paper a generalization of Bayes' theorem to the case of fuzzy data is described which contains Bayes' theorem for precise data as a special case and allows to use the information in fuzzy data in a coherent way. Moreover a generalization of the concept of HPD-regions is explained which makes it possible to model and analyze the situation of fuzzy data. Also a generalization of the concept of predictive distributions is given in order to calculate predictive densities based on fuzzy sample information.     

On some properties of variable size simple random sampling and a limit theorem

This paper introduces a sampling plan for finite populations herein called “variable size simple random sampling” and compares properties of estimators based on it with results from the usual fixed size simple random sampling without replacement. Necessary and sufficient conditions (in the spirit of Hajek (1960)) for the limiting distribution of the sample total (or sample mean) to be normal are given.     

On a theorem of Stein relating Bayesian and classical inferences in group models

Let a group G act on the sample space. This paper gives another proof of a theorem of Stein relating a group invariant family of posterior Bayesian probability regions to classical confidence regions when an appropriate prior is used. The example of the central multivariate normal distribution is discussed.     

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.     

On bahadur's representation of quantiles in nonregular cases

The paper considers Bahadur’s representation of quantiles (see Bahadur (1966), Ann. Math. Statist., 37, 577-580) in cases where the usual assumptions regarding the existence and boundedness of the derivatives of the distribution function in a neighbourhood of the population quantile(s) of interest are not met. Following Kiefer((1967), Ann. Math. Statist., 38, 1323-1342) we provide an exact order of the remainder term in Bahadur’s representation. A weaker result regarding the order of the remainder term is also provided under weaker regularity assumptions.     

On the real representation of quaternion random variables

For the first time, the matrix-variate quaternion normal and quaternion Wishart distributions are derived from first principles, that is, from their real counterparts, exposing the relations between their respective densities and characteristic functions. Applications of this theory in hypothesis testing are presented, and the density function of Wilks’ statistic is derived for quaternion Wishart matrices.     

On the error term in bahadur's representation of an order statistic

Bahadur (1966) presented a representation of an order statistic, giving its asymptotic distribution and the rate of convergence, under weak assumptions on the density function of the parent distribution. In this paper we consider the mean(squared) deviation of the error term in Bahadur’s approximation of the q th sample quantile (q_n ). We derive a uniform bound on the mean (squared) deviation of q_n , not depending on the value of q. An application of the given result provides the corresponding result for a kernel type estimator of the q th quantile.     

On the mean square convergence of the convolution representation of linear filters

Mean square convergence is the most frequently considered mode of convergence for the infinite series convolution expressions representing filter outputs in stationary time series analysis. There is confusion, however, also in the literature, about which conditions guarantee that this convergence holds. If only general properties of the input series to the filter are known, it is appropriate to consider the class of series with these properties. For each of several classes of full rank, wide sense stationary, zero-mean, vector time series, a weakest possible condition on the frequency response function of a linear filter is given which guarantees that the time-domain convolution representation of the filter converges to the filter output in mean square, whenever the input series belongs to the class under consideration. The classes considered are (i) the purely nondeterministic series with essentially bounded spectral density matrix, (ii) all purely nondeterministic series, (iii) all series. We then show that more unified resttlts can be obtained if Cesiro sums are utilized to define the convergence of the convolution representation. The mean square convergence of infinite autoregressions is also discussed.     

A note on a strong renewal theorem

Let {S_n, n ≥ 1} be a sequence of partial sums of independent and identically distributed non-negative random variables with a common distribution function F. Let F belong to the domain of attraction of a stable law with exponent α, 0 < α < 1. Suppose H(t) = ? N(t), t ? 0, where N(t) = max(n : S_n ≥ t). Under some additional assumptions on F, the difference between H(t) and its asymptotic value as t → ∞ is estimated.     

On the Gaussian representation of the Riesz probability distribution on symmetric matrices

AStA Advances in Statistical Analysis - The Riesz probability distribution on symmetric matrices represents an important extension of the Wishart distribution. It is defined by its Laplace...     

A Lusin theorem for a class of Choquet capacities

We extend the Lusin Theorem to supermodular Dempster capacities. Received: July 2000; revised version: May 2001     

On the Bahadur representation of sample quantiles and order statistics for NA sequences

In this paper, the Bahadur representation of sample quantiles for negatively associated (NA) sequences under mild conditions is established, which improves the results of Xing and Yang (2011). Furthermore, we also obtain the Bahadur representation of order statistics based on the NA sequences.     

On the distributional transform, Sklar's theorem, and the empirical copula process

We review the distributional transform of a random variable, some of its applications, and some related multivariate distributional transformations. The distributional transform is a useful tool, which allows in many respects to deal with general distributions in the same way as with continuous distributions. In particular it allows to give a simple proof of Sklar's theorem in the general case. It has been used in the literature for stochastic ordering results. It is also useful for an adequate definition of the conditional value at risk measure and for many further purposes. We also discuss the multivariate quantile transform as well as the multivariate extension of the distributional transform and some of their applications. In the final section we consider an application to an extension of a limit theorem for the empirical copula process, also called empirical dependence function, to general not necessarily continuous distributions. This is useful for constructing and analyzing tests of dependence properties for general distributions.