Showing papers on "Asymptotic distribution published in 1972"
TL;DR: In this article, a necessary and sufficient condition for joint asymptotic normality in a new (strong) sense, in the case of independence, is given, in which the dimensionality of the vector random variable under consideration is allowed to increase indefinitely.
Abstract: The concept of asymptotic normality takes on some new aspects when the dimensionality of the vector random variable under consideration is allowed to increase indefinitely. A necessary and sufficient condition for joint asymptotic normality in a new (strong) sense, in the case of independence, is given.
266 citations
TL;DR: The authors present 2 methods for the approximation of a representative schedule recording first marriage frequencies by age, one of which achieves a very close approximation with a simple closed form frequency function, and the other provides a feasible model of nuptiality.
Abstract: The schedule recording first marriage frequencies has been shown to take the same basic form in different populations, with differences only in the origin, area, and horizontal scale. It is shown here that a representative schedule is very closely approximated by a simple closed form frequency function, which is the limiting distribution of the convolution of an infinite number of exponentially distributed components. The schedule is approximated equally well by the convolution of a normal distribution (of age of entry into a marriageable state) and as few as three exponentially distributed delays. The latter convolution provides a plausible model of nuptiality, a model that receives surprising empirical support.
227 citations
TL;DR: Asymptotic expansion and asymptotically optimal tests as discussed by the authors have been used to estimate a family of probability measures by an exponential family of probabilistic functions. But they have not yet been applied to the problem of contiguity.
Abstract: 1. On the concept of contiguity and related theorems 2. Asymptotic expansion and asymptotic distribution of likelihood functions 3. Approximation of a given family of probability measures by an exponential family - asymptotic sufficiency 4. Some statistical applications: AUMP and AUMPU tests for certain testing hypotheses problems 5. Some statistical applications: asymptotic efficiency of estimates 6. Multiparameter asymptotically optimal tests.
211 citations
TL;DR: Two theorems on the asymptotic normality of linear combinations of functions of order statistics are given in this article, one requires a smooth scoring function but the underlying df need not be continuous even and can also depend on the sample size.
Abstract: Two theorems on the asymptotic normality of linear combinations of functions of order statistics are given. Theorem 1 requires a "smooth" scoring function but the underlying df need not be continuous even and can also depend on the sample size. Theorem 2 allows general scoring functions but places additional restrictions on the df. Applications included.
154 citations
TL;DR: In this paper, it was shown that the elegant asymptotic almost-sure representation of a sample quantile for independent and identically distributed random variables, established by Bahadur [1] holds for a stationary sequence of φ-mixing random variables.
138 citations
01 Jan 1972
TL;DR: In this paper, the authors present a statistical theory of spectra based on the self-adjoint linear operator of the Hamiltonian operator, which is used in quantum mechanics to measure the energy of a system.
Abstract: The impetus for this paper comes mainly from work done in recent years by a number of physicists on a statistical theory of spectra. The book by M. L. Mehta [10] and the collection of reprints edited by C. E. Porter [14] are excellent references for this work. The discussion in Section 1.1 is an attempt to present a rationale for such investigations. Our interpretation of linear operators as used in quantum mechanics is based largely on the book by T. F. Jordan [8]. 1.1. Statistical theory of spectra. In quantum mechanics knowledge of the value of measurable quantities of a system is expressed in terms of probabilities. A state of the system specifies these probabilities. Measurable quantities are represented by self-adjoint linear operators on a separable Hilbert space. The only possible values of the measurable quantities are those in the spectrum of the self-adjoint operator which represents the measurable quantity. Experience indicates that energy is represented by the Hamiltonian operator. We are interested in the point spectrum of the Hamiltonian, which is its set of eigenvalues. The eigenvalues E of the Hamiltonian operator H, which are real since H is self-adjoint, are those values of energy for which some state of the system specifies a probability of one that the energy is exactly equal to E [8]. This is expressed in the Schrodinger time independent equation,
117 citations
TL;DR: In this article, a non-linear and non-quadratic estimator of the realizations of a homogeneous random field ζ(t,x1,x2) is proposed.
Abstract: Summary
The estimation of spectra of random stationary processes is an important part of the statistics of random processes. There are several books on spectral analysis, e.g. Blackman & Tukey, Hannan, and Jenkins & Watts. As a rule, spectral estimators are quadratic functions of the realizations. Recently Capon suggested a new method for estimation of spectra of random fields, in which a non-quadratic function of the realization is used: he considered a homogeneous random field ζ(t,x1,x2), i.e. one which is stationary in time and space and a random function of the time and space co-ordinates t, x1, x2. For the sake of expository convenience we shall consider ordinary stationary processes of time only, ζ(t); the generalization of our results to the case of random fields is easy.
Comparison of the conventional spectral estimator and the ‘high-resolution’ estimator for an artificial example showed that the latter has less smoothing effect on the true spectrum (Capon). This was later confirmed by examples using real data (Capon). However, it was not clear whether for a finite realization the high-resolution estimator distorted the true spectrum, i.e. whether it behaved for example like a conventional estimator raised to some power.
In the present paper we introduce and study a new class of spectral estimators which are generally non-linear and non-quadratic functionals of the realizations. These estimators include the conventional and high-resolution ones, for which we shall give the approximate distributions. We derive under rather general conditions the limiting distribution of the new class of estimators, and illustrate them with several examples. As a matter of fact, these new estimators are weighted means of the eigenvalues of the covariance matrix, e.g. the arithmetic mean, geometric mean, and so on.
113 citations
TL;DR: In this paper, the general structure for the distribution functions (reduced density matrices) for systems composed of a number of elements is given by taking the variation with respect to the distribution function in the formalism of the cluster variation method.
Abstract: The general structure for the distribution functions (reduced density matrices) for systems composed of a number of elements is given by taking the variation with respect to the distribution functions in the formalism of the cluster variation method. The parameters or the Lagrange multipliers occurring in the distribution functions must be determined by the reducibility condition of the distribution functions or by the stationariness condition of the free energy.
106 citations
TL;DR: In this article, a Cramer von-Mises type statistic is proposed for testing the symmetry of a continuous distribution function and its asymptotic null distribution is found explicitly.
Abstract: A Cramer von-Mises type statistic is proposed for testing the symmetry of a continuous distribution function Its asymptotic null distribution is found explicitly, and its asymptotic distribution under a sequence of local alternatives is described A Monte Carlo study indicates that the asymptotic formulae are accurate for sample sizes as small as twenty
101 citations
TL;DR: In this article, the authors assume that a random sample of size n has been taken from a multinomial distribution with N cells, and that 4k is the number of observations in the kth cell and set.
Abstract: SUMMARY Assume that a random sample of size n has been taken from a multinomial distribution with N cells. Let 4k be the number of observations in the kth cell and set
100 citations
TL;DR: In this article, the time asymptotic distribution functions corresponding to adiabatic and sudden excitation of an electrostatic wave are calculated and compared and used to calculate the nonlinear response of the plasma, and Poisson's equation is used to find a nonlinear dispersion relation.
Abstract: The time asymptotic distribution functions corresponding to adiabatic and sudden excitation of an electrostatic wave are calculated. These distributions are compared and used to calculate the nonlinear response of the plasma, and Poisson's equation is used to find a nonlinear dispersion relation.
TL;DR: In this paper, conditions for strong consistency and asymptotic normality of the MLE estimator for multiparameter exponential models are given, and the conditions are less restrictive than required by general theorems in this area.
Abstract: Conditions are given for the strong consistency and asymptotic normality of the MLE (maximum likelihood estimator) for multiparameter exponential models. Because of the special structure assumed, the conditions are less restrictive than required by general theorems in this area. The technique involves certain convex functions on Euclidean spaces that arise naturally in the present context. Some examples are considered; among them, the multinomial distribution. Some convexity and continuity properties of multivariate cumulant generating functions are also discussed.
TL;DR: In this article, the authors considered separable mean zero Gaussian processes X(t) with correlation functions rho(t,s) for which 1-rho( t, s) is asymptotic to a regularly varying (at zero) function of /t-s/ with exponent 0=or < alpha =or < 2.
Abstract: : The authors consider two problems for separable mean zero Gaussian processes X(t) with correlation functions rho(t,s) for which 1-rho(t,s) is asymptotic to a regularly varying (at zero) function of /t-s/ with exponent 0=or < alpha =or <2. In showing the existence of such (stationary) processes for 0 = or < alpha < 2, the authors relate the magnitude of the tails of the spectral distributionsto the behavior of the covariance function at the origin. For 0 < alpha = or < 2, the authors obtain the asymptotic distribution of the maximum of X(t). This second result is used to obtain a result for X(t) as t approaches infinity similar to the 'so called' law of the iterated logarithm. (Author)
TL;DR: Asymptotic normality of linear rank statistics for testing the hypothesis of independence is established under fixed alternatives in this article, where a generalization of a result of Bhuchongkul [1] is obtained both with respect to the conditions concerning the orders of magnitude of the score functions and to the smoothness conditions on these functions.
Abstract: Asymptotic normality of linear rank statistics for testing the hypothesis of independence is established under fixed alternatives. A generalization of a result of Bhuchongkul [1] is obtained both with respect to the conditions concerning the orders of magnitude of the score functions and with respect to the smoothness conditions on these functions.
TL;DR: In this article, a simple unbiased estimator, based on a censored sample, is proposed for the scale parameter of the extreme value distribution, and the exact distribution of the estimator is determined for the cases in which only the first two or only three ordered observations are available.
Abstract: A simple, unbiased estimator, based on a censored sample, is proposed for the scale parameter of the extreme-value distribution. The exact distribution of the estimator is determined for the cases in which only the first two or only the first three ordered observations are available. The asymptotic distribution is derived, and an approximate distribution for small sample size is also provided. Interval estimation for the scale parameter is developed and a conservative interval estimate for reliability is also obtained.
TL;DR: In this paper, linearized estimates as functions of the ranks are proposed for the general linear hypothesis, and the asymptotic distribution of the estimates is shown to be the same as the maximum likelihood estimates for fairly general sequences of design matrices.
Abstract: Linearized estimates, as functions of the ranks, are proposed for the general linear hypothesis. These estimates can be computed after a single ranking of the "centered" observations. The asymptotic distribution of the estimates is shown to be the same as the maximum likelihood estimates for fairly general sequences of design matrices.
TL;DR: In this paper, it was shown that a regenerative stochastic process with a non-lattice interarrival-time distribution has a limiting distribution if either the class of inter-arrival time distributions is restricted, or regularity conditions are imposed on the sample paths of the process.
Abstract: Feller (1966) claims that a regenerative stochastic process with a non-lattice interarrival-time distribution has a limiting distribution. This is true if (i) the class of interarrival-time distributions is restricted, or (ii) regularity conditions are imposed on the sample paths of the process.
TL;DR: In this article, the authors considered the discrete-time model of delta modulation for a stationary random input process with a rational spectral density, and an auto-covariance that goes to zero as the lag approaches infinity.
Abstract: The discrete-time model of delta modulation is considered for a stationary random input process with a rational spectral density, and an auto-covariance that goes to zero as the lag approaches infinity. For leaky integration, the joint distribution of input and decoded approximation processes is shown to approach a unique stationary distribution from any initial condition. Under the stationary distribution, the decoded process may take on all values in a bounded interval that is independent of the input process. For the often-studied ideal integration model of delta modulation, it is shown that the successive distributions at even parity time instants converge to a limiting stationary distribution, while at odd parity time instants the distributions converge to a different limiting distribution. Under these limiting distributions, the decoded process is assigned a positive probability for each level of a (discrete) lattice of amplitudes. The mean-absolute approximation error and mean-absolute amplitude of the decoded process are shown to be finite under the limiting distributions. For both ideal and leaky integration cases, an explicit upper bound on mean-absolute approximation error is given, which is independent of the spectral density of the input process.
TL;DR: In this article, the authors consider the problem of finding whether a given set of observations on the circumference of the unit circle indicate any preferred direction or whether the data can be considered to have come from a uniform distribution on the circle.
Abstract: One of the basic problems in the analysis of circularly distributed data is to find whether a given set of observations on the circumference of the unit circle indicate any preferred direction or whether the data can be considered to have come from a uniform distribution on the circumference. We shall assume, throughout this discussion, that the observations are given in terms of angles measured with respect to some suitably chosen origin (or zero direction), taking say, the anticlockwise direction as positive. A "goodness of fit" problem on the circle then is to test whether a random sample (e~, ..., c~,) comes from a population with a completely specified distribution function Fo(e), 0
TL;DR: In this paper, it was shown that every one-sided rank test is asymptotically optimal for a certain nonparametric subclass of contiguous alternatives, provided the test and the associated subclass of alternatives are generated by certain square-integrable functions defined on the unit square.
Abstract: For the one-sample independence problem, the one-sample symmetry problem, and the two-sample problem it is shown that every one-sided rank test is asymptotically optimal for a certain nonparametric subclass of contiguous alternatives, provided the test and the associated subclass of alternatives are generated by certain square-integrable functions defined on the unit square. Then the asymptotic normality of the respective rank statistics under every alternative contiguous to the hypothesis is used in order to give necessary and sufficient conditions for local asymptotic unbiasedness of such tests. Finally, for locally asymptotically unbiased tests there are given necessary and sufficient conditions for having bounds for their asymptotic relative efficiency under contiguous alternatives.
TL;DR: In this article, a continuous estimator for a distribution (based on Pyke's modified sample distribution) is shown to have the property that its expected squared error, for almost all $x$ in the positive sample space of the distribution, is no larger than that of the sample distribution function given $F$ and $n$ sufficiently large.
Abstract: Given a sample of size $n$, a continuous estimator for a distribution $F$ (based on Pyke's modified sample distribution) is shown to have the property that its expected squared error, for almost all $x$ in the positive sample space of $F$, is no larger than that of the sample distribution function given $F$ and $n$ sufficiently large. Letting risk be given by the expected squared error integrated with respect to $F$, it is shown that this estimator dominates both the sample distribution and the other best invariant estimator found by Aggarwal, given $F$ and $n$ sufficiently large. Other common estimators cannot serve in this dominating role. Explicit calculation of risk is made when $F$ is the uniform distribution. In this case the estimator strictly dominates the sample distribution for all $n \geqq 1$.
01 Jan 1972
TL;DR: In this article, an asymptotic distribution of Eigenvalues for wave problems is discussed, and a connecting solution is constructed by using the theory of irregular singular points at x = ∞.
Abstract: Publisher Summary This chapter discusses an asymptotic distribution of Eigen values. In particular, the chapter focuses on the asymptoticdistribution of largepositiveEigenvalues for thisproblem. Problems of this nature arise naturally in investigations of wave phenomena, especially in quantum mechanics. The chapter contain the appropriate connecting solutions by using the theory of irregular singular points at x = ∞. In particular chapter explains the study of the Stokes curves in the whole complex plane and utilizes them to construct such solution. Then subdominant solutions as indicated will exist and appropriate connecting solutions may be constructed under the additional hypothesis.
01 Jan 1972
TL;DR: In this paper, it was shown that for p = 3 or 4, there is no spherically symmetric proper Bayes minimax estimator for multivariate normal distributions.
Abstract: Consider the problem of estimating the mean of a multivariate normal distribution with convariance matrix the identity and sum of squared errors loss. In an earlier paper [5] the author showed that if the dimension p is 5 or greater, then proper Bayes minimax estimators do exist. We review this result briefly in Section 2. The main purpose of the present paper is to show that for p equal to 3 or 4, there do not exist spherically symmetric proper Bayes minimax estimators. The author has been unable, thus far, to disprove the existence of a nonspherical proper Bayes minimax estimator for p equal 3 or 4. Of course, for p = 1, 2, the usual estimator X is unique minimax but not proper Bayes. In Section 3 we derive bounds for the possible bias of a minimax estimator. This result should be of some interest independent of its use in proving the main result of the paper. Section 4 is devoted to the proof of the main result.
TL;DR: It is proved that the asymptotic distribution of L(...) is either Poisson or normal, and under various assumptions about the distribution of the pointsx1, ...,xn and the size ofU is studied.
Abstract: When objects are scattered at random in the plane or in space, some of them overlap to form clumps. It is the object of the present paper to study the asymptotic distribution of the number of clumps of given size and topological structure generated within the following model:
01 Jan 1972
TL;DR: In this paper, the rank tests of symmetry when samples are drawn from purely discrete distributions so that ties of zero and non-zero observations may occur are considered in the same way as nonzero ones.
Abstract: The paper deals with problems of rank tests of symmetry when samples are drawn from purely discrete distributions so that ties of zero and non-zero observations may occur. Zero observations are considered in the same way as nonzero ones. Two ways of treatment of ties are used in the paper, randomization of ties and the method of averaged scores. The asymptotic distributions of the statistics are derived under hypothesis of symmetry and under contiguous alternatives of location. The asymptotic power and efficiency of tests are established.
TL;DR: In this article, the asymptotic normality of the joint distribution of an increasing number of sample quantiles as the sample size increases is investigated in both cases where the basic distributions are equal and are unequal.
Abstract: Uniform (or type (B) d ) asymptotic normality of the joint distribution of an increasing number of sample quantiles as the sample size increases is investigated in both cases where the basic distributions are equal and are unequal. Under fairly general assumptions, sufficient conditions are derived for the asymptotic normality of sample quantiles.
TL;DR: In this paper, it was shown that a dominated family of probability measures with Euclidean parameter space behaves approximately like a family of normal distributions, if each probability measure is the independent product of a great number of identical components.
Abstract: Starting fromLe Cam [1956], it was shown inMichel andPfanzagl [1970] that — under certain regularity conditions — a dominated family of probability measures withEuclidean parameter space behaves approximately like a family of normal distributions, if each probability measure is the independent product of a great number of identical components. It is the purpose of this paper to estimate the accuracy of such a normal approximation.