Showing papers on "Asymptotic distribution published in 1979"
TL;DR: It is shown that this method of averaging is exact for certain distributions (i.e., the resulting distribution belongs to the same family as the individual distributions) and Monte Carlo studies and application of the method provide evidence that properties derived from the group reaction time distribution are much the same as average propertiesderived from the data of individual subjects.
Abstract: A method of obtaining an average reaction time distribution for a group of subjects is described. The method is particularly useful for cases in which data from many subjects are available but there are only 10-20 reaction time observations per subject cell. Essentially, reaction times for each subject are organized in ascending order, and quantiles are calculated. The quantiles are then averaged over subjects to give group quantiles (cf. Vincent learning curves). From the group quantiles, a group reaction time distribution can be constructed. It is shown that this method of averaging is exact for certain distributions (i.e., the resulting distribution belongs to the same family as the individual distributions). Furthermore, Monte Carlo studies and application of the method to the combined data from three large experiments provide evidence that properties derived from the group reaction time distribution are much the same as average properties derived from the data of individual subjects. This article also examines how to quantitatively describe the shape of reaction time distributions. The use of moments and cumulants as sources of information about distribution shape is evaluated and rejected because of extreme dependence on long, outlier reaction times. As an alternative, the use of explicit distribution functions as approximations to reaction time distributions is considered.
971 citations
TL;DR: In this paper, statistical inference for a system of simultaneous, non-linear, implicit equations is discussed and the authors consider inference as an adjunct to two-and three-stage least squares estimation rather than in a general setting.
353 citations
TL;DR: In this article, it is shown that the distribution of companies by size is approximately log-normal under certain conditions, such as the limiting distribution of the product of positive random variates as the number of terms in the product tends to infinity.
Abstract: IT IS well known that the distribution of companies by size is approximately log-normal. This form arises under certain conditions as the limiting distribution of the product of positive random variates as the number of terms in the product tends to infinity. The current size of a company may be decomposed into the product of past proportionate growth rates and an initial size and sufficiently strong conditionsl on these growth rates ensure that the distribution of the variate 'company size' becomes lognormal as the time elapsed from the start of the growth process tends to infinity.
247 citations
TL;DR: In this paper, the authors considered a modular program in which transfers of control between modules follow a semi-Markov process, where each module is failure-prone, and the different failure processes are assumed to be Poisson.
Abstract: The paper treats a modular program in which transfers of control between modules follow a semi-Markov process. Each module is failure-prone, and the different failure processes are assumed to be Poisson. The transfers of control between modules (interfaces) are themselves subject to failure. The overall failure process of the program is described, and an asymptotic Poisson process approximation is given for the case when the individual modules and interfaces are very reliable. A simple formula gives the failure rate of the overall program (and hence mean time between failures) under this limiting condition. The remainder of the paper treats the consequences of failures. Each failure results in a cost, represented by a random variable with a distribution typical of the type of failure. The quantity of interest is the total cost of running the program for a time t, and a simple approximating distribution is given for large t. The parameters of this limiting distribution are functions only of the means and variances of the underlying distributions, and are thus readily estimable. A calculation of program availability is given as an example of the cost process. There follows a brief discussion of methods of estimating the parameters of the model, with suggestions of areas in which it might be used.
213 citations
TL;DR: The asymptotic properties of Burgers turbulence at extremely large Reynolds numbers and times are investigated by analyzing the exact solution of the Burgers equation, which takes the form of a series of triangular shocks in this situation as discussed by the authors.
Abstract: The asymptotic properties of Burgers turbulence at extremely large Reynolds numbers and times are investigated by analysing the exact solution of the Burgers equation, which takes the form of a series of triangular shocks in this situation. The initial probability distribution for the velocity u is assumed to decrease exponentially as u → ∞. The probability distribution functions for the strength and the advance velocity of shocks and the distance between two shocks are obtained and the velocity correlation and the energy spectrum function are derived from these distribution functions. It is proved that the asymptotic properties of turbulence change qualitatively according as the value of the integral scale of the velocity correlation function J, which is invariant in time, is zero, finite or infinite. The turbulent energy per unit length is shown to decay in time t as t−1 (with possible logarithmic corrections) or according as J = 0 or J ≠ 0.
165 citations
TL;DR: Goodness-of-fit tests for the logistic distribution, based on statistics calculated from the empirical distribution function, are given in this article, where asymptotic percentage points are given, for each of the three cases where one or both of the parameters of the distribution must be estimated from the data.
Abstract: SUMMARY Goodness-of-fit tests are given for the logistic distribution, based on statistics calculated from the empirical distribution function. Emphasis is on the statistics W2, U2 and A2, for which asymptotic percentage points are given, for each of the three cases where one or both of the parameters of the distribution must be estimated from the data. Slight modifications of the calculated statistics are given to enable the points to be used with small samples. Monte Carlo results are included also for statistics D+, D-, D and V.
155 citations
TL;DR: In this article, a general asymptotic theory is developed which includes these adaptive designs and the classical stochastic approximation schemes as special cases, motivated by the cost considerations, some improvements are made in Venter's pairwise sampling scheme of Venter. And the authors make the "cost" of the observations, defined at the nth stage to be Σ 1 n (x i - θ)2, to be of the order of log n instead of n.
Abstract: When y= M(x) + e, where M may be nonlinear, adaptive stochastic approximation schemes for the choice of the levels x 1, x 2, ··· at which y 1, y 2, ··· are observed lead to asymptotically efficient estimates of the value θ of x for which M(θ) is equal to some desired value. More importantly, these schemes make the “cost” of the observations, defined at the nth stage to be Σ 1 n (x i - θ)2, to be of the order of log n instead of n, an obvious advantage in many applications. A general asymptotic theory is developed which includes these adaptive designs and the classical stochastic approximation schemes as special cases. Motivated by the cost considerations, some improvements are made in the pairwise sampling stochastic approximation scheme of Venter.
149 citations
TL;DR: The asymptotic analysis of the stationary distribution in the case of a large number of customers is given and shows that a special nonlinear programming problem must be solved to obtain parameters of the limiting distribution.
Abstract: Two models of a closed queueing network with saturation are proposed. Given that the “reversibility” condition holds in the first model, the stationary distribution is shown to be of product-form for either of them. Queueing networks with the space of admissible states generated by limited capacities of servers is the most important special variant of the general scheme. The asymptotic analysis of the stationary distribution in the case of a large number of customers is given and shows that a special nonlinear programming problem must be solved to obtain parameters of the limiting distribution. In particular, saturation probabilities are asymptotically expressed through the Lagrangian multipliers dual to corresponding linear restrictions of queue-sizes for servers with limited capacities.
136 citations
TL;DR: In this paper, the robustness property of the preliminary test estimator when the assumed restraints may not hold was analyzed for a general multi-sample parametric model and compared with the parallel expressions for the unrestricted and restricted maximum likelihood estimators.
Abstract: Along with the asymptotic distribution, expressions for the asymptotic bias and asymptotic dispersion matrix of the preliminary test maximum likelihood estimator for a general multi-sample parametric model (when the null hypothesis relating to the restraints on the parameters may not hold) are derived and compared with the parallel expressions for the unrestricted and restricted maximum likelihood estimators. This study reveals the robustness property of the preliminary test estimator when the assumed restraints may not hold.
135 citations
01 Jan 1979
117 citations
TL;DR: In this article, the limit distribution of the supremum of the empirical distribution function centered at its expectation and standardized by division by its standard deviation is shown to be degenerate, and the results are also valid for the Studentized version of the above statistic, and corresponding two-sided statistics are treated.
Abstract: It is well known that the limit distribution of the supremum of the empirical distribution function $F_n$ centered at its expectation $F$ and standardized by division by its standard deviation is degenerate, if the supremum is taken on too large regions $\varepsilon_n < F(u) < \delta_n$. So it is natural to look for sequences of linear transformations, so that for given sequences of sup-regions $(\varepsilon_n, \delta_n)$ the limit of the transformed sup-statistics is nondegenerate. In this paper a partial answer is given to this problem, including the case $\varepsilon_n \equiv 0, \delta_n \equiv 1$. The results are also valid for the Studentized version of the above statistic, and the corresponding two-sided statistics are treated, too.
TL;DR: In this article, a sequential procedure is proposed for estimating the difference of the means of two populations when the variances are unknown (and not necessarily equal) and the loss structure for both the problems is the cost of observations plus the squared error loss due to estimating the unknown mean or difference of means.
Abstract: Two problems have been discussed in this paper. First, for independent and identically distributed random variables with unknown mean and unknown variance, a sequential procedure is proposed for point estimation of themean when the distribution is unspecified. Second, a sequential procedure is proposed for estimating the difference of the means of two populations when the variances are unknown (and not necessarily equal). The loss structure for both the problems is the cost of observations plus the squared error loss due to estimating theunknown mean or the difference of means. Without any assumption on the nature of the distribution functions other than the finiteness of the eighth moment, the two procedures are shown to be “asymptotically risk efficient” in the sense of Starr (Ann. Math. Statist. (1966), .37, 1173-1185).
TL;DR: The use of x2 statistics for categorical data problems was initiated by Karl Pearson, but it took several years before the asymptotic distribution of these statistics was well understood.
Abstract: SUMMARY The use of x2 statistics for categorical data problems was initiated by Karl Pearson, but it took several years before the asymptotic distribution of these statistics was well understood. The general structure of asymptotic results for x2 statistics is reviewed and the applicability of the general structure to a variety of problems of practical interest is discussed. These problems include the use of x2 statistics in small-sample situations and in large sparse tables, in cluster sampling, and in cases where they do not have asymptotic x2 distributions.
TL;DR: Asymptotic normality of the posterior distribution of a parameter in a stochastic process is shown to hold under conditions which do little more than ensure consistency of a maximum likelihood estimator as discussed by the authors.
Abstract: Asymptotic normality of the posterior distribution of a parameter in a stochastic process is shown to hold under conditions which do little more than ensure consistency of a maximum likelihood estimator. Much more stringent conditions are required to ensure asymptotic normality of the MLE. This contrast, which has implications of considerable significance, does not emerge in the classical context of independent and identically distributed observations.
TL;DR: In this paper, the studentized form of the above statistic, obtained by division by the estimated standard deviation, is shown to have the same large sample behavior as the uniform distribution.
Abstract: The supremum of the empirical distribution function $F_n$ centered at its expectation $F$ and standardized by division by its standard deviation has recently been shown by Jaeschke to have asymptotically an extreme-value distribution after a second location and scale transformation depending only on the sample size $n$. In this paper the studentized form of the above statistic, obtained by division by the estimated standard deviation, is shown to have the same large sample behavior. This statement is equivalent to the analogous assertion for the standardized sample quantile process for the uniform distribution. The three results imply each other. The present result yields immediately confidence regions that contract to zero width in the tails. The proofs given here rest on a limit theorem by Darling and Erdos on the maxima of standardized partial sums of i.i.d. random variables. In addition, Kolmogorov's theorem is used.
TL;DR: For a general vector linear time series model, this article proved the strong consistency and asymptotic normality of parameter estimates obtained by maximizing a particular time domain approximation to a Gaussian likelihood, although they do not assume that the observations are necessarily normally distributed.
Abstract: For a general vector linear time series model we prove the strong consistency and asymptotic normality of parameter estimates obtained by maximizing a particular time domain approximation to a Gaussian likelihood, although we do not assume that the observations are necessarily normally distributed. To solve the normal equations we set up a constrained Gauss-Newton iteration and obtain the properties of the iterates when the sample size is large. In particular we show that the iterates are efficient when the iteration begins with a VN- consistent estimator. We obtain similar results to the above for a frequency domain approximation to a Gaussian likelihood. We use the asymptotic estimation theory to obtain the asymptotic distribution of several familiar test statistics for testing nonlinear equality constraints.
TL;DR: For quadratic functions with long range memory (not instantaneous) with a normalization nα, 0 <α < 1/2 where n is the sample size.
Abstract: Limit theorems with a non-Gaussian limiting distribution have been obtained, under appropriate conditions for partial sums of instantaneous nonlinear functions of stationary Gaussian sequences with long range dependence by a number of people. The normalization has typically been nα, with 1/2<α<1 where n is the sample size. Here examples of limit theorems are given for quadratic functions with long range memory (not instantaneous) with a normalization nα, 0<α<1/2.
01 Jan 1979
TL;DR: In this paper, the authors present a theory and methodology of robust estimation for time series having two distinctive types of outliers. But the authors do not define a robustness metric for the time series.
Abstract: Publisher Summary This chapter presents some theory and methodology of robust estimation for time series having two distinctive types of outliers. Research on robust estimation in the time series context has lagged behind, and perhaps understandably so in view of the increased difficulties imposed by dependency and the considerable diversity in qualitative features of time series data sets. For time series parameter, estimation problems, efficiency robustness, and min–max robustness are concepts directly applicable. Influence curves for parameter estimates may also be defined without special difficulties. A greater care is needed in defining breakdown points as the detailed nature of the failure mechanism may be quite important. A major problem that remains is that of providing an appropriate and workable definition of qualitative robustness in the time series context. For time series, the desire for a complete probabilistic description of either a nearly-Gaussian process with outliers, or the corresponding asymptotic distribution of parameter estimates, will often dictate that one specify more than a single finite-dimensional distribution of the process. It is only in special circumstances that the asymptotic distribution of the estimate will depend only upon a single univariate distribution or a single multivariate distribution.
TL;DR: In this article, the asymptotic theory of estimates of an unknown parameter in continuous-time Markov processes, which are described by non-linear stochastic differential equations, is investigated.
Abstract: This paper is concerned with the asymptotic theory of estimates of an unknown parameter in continuous-time Markov processes, which are described by non-linear stochastic differential equations. The maximum likelihood estimate and the minimum contrast estimate are investigated. For these estimates strong consistency and asymptotic normality are proved. The unknown parameter is assumed to take its values either in an open or in a compact set of real numbers.
TL;DR: In this paper, the asymptotic normality of permutation statistics derived from weighted sums of bivariate functions of ranks is proven and the result is then combined with work of Jogdeo on statistics that are weighted sums for rank functions.
Abstract: Statistics of the form $H_n = \sum d_{ijn}h_n(X_i, X_j)$ are considered, where $X_1, X_2, \cdots$, are independent and identically distributed random variables, the diagonal terms, $d_{iin}$, are equal to zero, and $h_n(x, y)$ is a symmetric real valued function. The asymptotic normality of such statistics is proven and the result then combined with work of Jogdeo on statistics that are weighted sums of bivariate functions of ranks to find sufficient conditions for asymptotic normality of permutation statistics derived from $H_n$.
TL;DR: In this paper, strong approximation results and methodology are used to obtain inprobability representations of the empirical process when the parameters of the underlying distribution function are estimated under a null hypothesis and a sequence of alternatives converging to the null hypothesis.
Abstract: Strong approximation results and methodology are used to obtain in-probability representations of the empirical process when the parameters of the underlying distribution function are estimated. These representations are obtained under a null hypothesis and a sequence of alternatives converging to the null hypothesis. The fairly general conditions on the estimators are often satisfied by maximum likelihood estimators. The asymptotic distribution of the estimated empirical process depends, in general, on the true value of the unknown parameters. Some useful methods of overcoming this difficulty are discussed.
TL;DR: In this paper, the asymptotic distribution of cross-correlations between univariate autoregressive moving average (ARMA) time series residuals is derived, and its application to the problem of testing for lagged relationships in the presence of instantaneous causality is discussed.
Abstract: Cross-correlations between univariate autoregressive moving average (ARMA) time series residuals are useful in the examination of relationships between time series (Pierce 1977a) and in the identification of dynamic regression models (Haugh and Box 1977). In this article, the asymptotic distribution of these residual cross-correlations is derived, and its application to the problem of testing for lagged relationships in the presence of instantaneous causality is discussed. Some results of a simulation study to investigate the accuracy of the asymptotic variances and covariances of the residual cross-correlations in finite samples are reported.
TL;DR: In this paper, the authors obtain the limiting distribution of the uncovered proportion of the circle, which has a natural interpretation as a noncentral chi-square distribution with zero degrees of freedom by expressing it as a Poisson mixture of mass at zero with central Chi-square deviates having even degree of freedom.
Abstract: Place $n$ arcs, each of length $a_n$, uniformly at random on the circumference of a circle, choosing the arc length sequence $a_n$ so that the probability of completely covering the circle remains constant. We obtain the limiting distribution of the uncovered proportion of the circle. We show that this distribution has a natural interpretation as a noncentral chi-square distribution with zero degrees of freedom by expressing it as a Poisson mixture of mass at zero with central chi-square deviates having even degrees of freedom. We also treat the case of proportionately smaller arcs and obtain a limiting normal distribution. Potential applications include immunology, genetics, and time series analysis.
TL;DR: In this paper, a family of statistics based on mth-order gaps from a uniform sample, obtained by summing a suitably regular function of each gap, is investigated, and it is shown that the sum of squares of gaps is an optimal statistic.
Abstract: SUMMARY The family of statistics based on mth-order gaps from a uniform sample, obtained by summing a suitably regular function of each gap, is investigated. Holst (1979) has established asymptotic normality together with explicit expressions for the mean and variance. This is extended to samples from distributions whose perturbations from uniformity shrink as sample size grows. From these results, Pitman asymptotic relative efficiencies can be calculated, and it is shown that the sum of squares of gaps is an optimal statistic. The properties of this statistic are presented, together with a comparison to the already well treated sum of log gaps. It is shown that the two become indistinguishable as m, the order of the gaps, grows.
TL;DR: In this paper, a general theorem is given which simplifies and extends the techniques of Prakasa Rao (1966) and Brunk (1970), expressing sufficient conditions for a specified limit distribution to obtain are expressed in terms of local and global conditions.
Abstract: : Isotonic estimation involves the estimator of a function which is known to be increasing with respect to a specified partial order. For the case of a linear order, a general theorem is given which simplifies and extends the techniques of Prakasa Rao (1966) and Brunk (1970). Sufficient conditions for a specified limit distribution to obtain are expressed in terms of a local condition and a global condition. The theorem is applied to several examples. The first example is estimation of a monotone function mu on (0,1) based on observations (i/n, X sub ni), where EX sub ni = mu (i/n). In the second example, i/n is replaced by random T sub ni. Robust estimators for this problem are described. Estimation of a monotone density function is also discussed. It is shown that the rate of convergence depends on the order of first non-zero derivative and that this result can obtain even if the function is not monotone over its entire domain. (Author)
TL;DR: In this article, the asymptotic distribution for a certain class of functionals of distribution functions is derived, and the results are applied to the Lorenz-curve and the Gini-measure as special cases of the abovementioned functionals.
Abstract: The asymptotic distribution for a certain class of functionals of distribution functions is derived. This result is used to give distribution free asymptotic confidence intervals for these functionals; for this purpose, a strongly consistent estimate for the asymptotic variance is constructed. These results are applied to the Lorenz-curve and the Gini-measure as special cases of the abovementioned class of functionals.
TL;DR: In this paper, the authors derived the asymptotic distributions of the log-likelihood function and other r.i.d. r.v.'s from a stochastic process whose finite dimensional distributions are of known functional form.
Abstract: Let X
0, X
1,⋯, X
nbe r.v.'s coming from a stochastic process whose finite dimensional distributions are of known functional form except that they involve a k-dimensional parameter. From the viewpoint of statistical inference, it is of interest to obtain the asymptotic distributions of the log-likelihood function and also of certain other r.v.'s closely associated with the likelihood function. The probability measures employed for this purpose depend, in general, on the sample size n. These problems are resolved provided the process satisfies some quite general regularity conditions. The results presented herein generalize previously obtained results for the case of Markovian processes, and also for i.n.n.i.d. r.v.'s. The concept of contiguity plays a key role in the various derivations.
TL;DR: Lancaster's work on the orthogonal expansions of multivariate distributions is used in this paper to show how the asymptotic distribution of degenerate, two-dimensional U-statistics can be extended to higher dimensional U-Statistics, based on a sample of dependent random variables.
Abstract: Summary
Lancaster's work on the orthogonal expansions of multivariate distributions is used to show how recent work on the asymptotic distribution of degenerate, two-dimensional U-statistics can be extended to higher dimensional U-statistics, to the so-called two-sample case and to U-statistics based on a sample of dependent random variables.
31 Oct 1979
TL;DR: The Anderson-Darling statistic A squared as discussed by the authors is a goodness-of-fit statistic, based on the empirical distribution function, which can be found for testing many important distributions when unknown parameters must be estimated from the data.
Abstract: : The Anderson-Darling statistic A squared is a goodness-of-fit statistic, based on the empirical distribution function. Its asymptotic distribution can be found for testing many important distributions when unknown parameters must be estimated from the data. Furthermore, A squared can be easily adapted so that only the asymptotic points are needed for testing purposes. A squared also is easy to calculate, and has overall good power properties. The report gives a review of A squared and tables for testing the following distributions - normal, exponential, gamma, extreme-value and Weibull, and logistic; points are given also for testing any completely specified continuous distribution.