Showing papers on "Asymptotic distribution published in 1981"

Statistical Properties of the Generalized Inverse Gaussian Distribution

Abstract: 1 Introduction.- 2 Basic properties.- 2.1 Moments and cumulants.- 3 Related distributions.- 3.1 Normal approximations.- 3.2 Powers and logarithms of generalized inverse Gaussian variates.- 3.3 Products and quotients of generalized inverse Gaussian variates.- 3.4 A generalized inverse Gaussian Markov process.- 3.5 The generalized hyperbolic distribution.- 4 Maximum likelihood estimation.- 4.1 Estimation for fixed ?.- 4.2 On the asymptotic distribution of the maximum likelihood estimate for fixed ?.- 4.3 The partially maximized log-likelihood for ?, estimation of ?.- 4.4 Estimation of ? when ? and ? are fixed.- 4.5 Estimation of ? when ? and ?>0 are fixed.- 5 Inference.- 5.1 Distribution results.- 5.2 Inference about ?.- 5.3 Inference about ?.- 5.4 One-way analysis of variance.- 5.5 A regression model.- 6 The hazard function. Lifetime models..- 6.1 Description of the hazard function.- 7 Examples.- 7.1 Failures of airconditioning equipment.- 7.2 Pulses along a nerve fibre.- 7.3 Traffic data.- 7.4 Repair time data.- 7.5 Fracture toughness of MIG welds.- References.- List of symbols.

A Large Sample Study of Cox's Regression Model

Abstract: Strong consistency and asymptotic normality are established for the maximum partial likelihood estimate of the regression parameter in Cox's regression model. Estimates are also derived for the underlying cumulative hazard function and survival distribution. We establish the asymptotic normality of these estimates and calculate the limiting variances.

The asymptotic properties of nonparametric tests for comparing survival distributions

Abstract: The asymptotic distribution under alternative hypotheses is derived for a class of statistics used to test the equality of two survival distributions in the presence of arbitrary, and possibly unequal, right censoring. The test statistics include equivalents to the log rank statistic, the modified Wilcoxon statistic and the class of rank invariant test procedures introduced by Peto & Peto. When there are equal censoring distributions and the hazard functions are proportional the sample size formula for the F test used to compare exponential samples is shown to be valid for the log rank test. In certain situations the power of the log rank test falls as the amount of censoring decreases.

Pseudo Maximum Likelihood Estimation: Theory and Applications

Abstract: : Pseudo maximum likelihood estimation easily extends to k parameter models, and is of interest in problems in which the likelihood surface is ill-behaved in higher dimensions but well-behaved in lower dimensions. Several signal plus noise or convolution models are examined which exhibit such behavior and satisfy the regularity conditions of the asymptotic theory. For specific models, a numerical comparison of asymptotic variances suggests that a psuedo maximum likelihood estimate of the signal parameter is uniformly more efficient than estimators that have been advanced by previous authors. A number of other potential applications are noted.

Inference From Stratified Samples: Properties of the Linearization, Jackknife and Balanced Repeated Replication Methods

Abstract: The asymptotic normality of both linear and nonlinear statistics and the consistency of the variance estimators obtained using the linearization, jackknife and balanced repeated replication (BRR) methods in stratified samples are established The results are obtained as $L \rightarrow \infty$ within the context of a sequence of finite populations $\{\Pi_L\}$ with $L$ strata in $\Pi_L$ and are valid for any stratified multistage design in which the primary sampling units (psu's) are selected with replacement and in which independent subsamples are taken within those psu's selected more than once In addition, some exact analytical results on the bias and stability of these alternative variance estimators in the case of ratio estimation are obtained for small $L$ under a general linear regression model

Robust Estimation in Heteroscedastic Linear Models.

Abstract: We consider a heteroscedastic linear model in which the variances are given by a parametric function of the mean responses and a parameter $\theta$. We propose robust estimates for the regression parameter $\beta$ and show that, as long as a reasonable starting estimate of $\theta$ is available, our estimates of $\beta$ are asymptotically equivalent to the natural estimate obtained with known variances. A particular method for estimating $\theta$ is proposed and shown by Monte-Carlo to work quite well, especially in power and exponential models for the variances. We also briefly discuss a "feedback" estimate of $\beta$.

Convergence to a Stable Distribution Via Order Statistics

Abstract: Let $X_1, X_2, \cdots$ be i.i.d. random variables whose common distribution function $F$ is in the domain of attraction of a nonnormal stable distribution. A simple, probabilistic proof of the convergence of the normalized partial sums to the stable distribution is given. The proof makes use of an elementary property of order statistics and clarifies the manner in which the largest few summands determine the limiting distribution. The method is applied to determine the limiting distribution of self-norming sums and deduce a representation for the limiting distribution. The representation affords an explanation of the infinite discontinuities of the limiting densities which occur in some cases. Application of the technique to prove weak convergence in a separable Hilbert space is explored.

On the Efficiency of Empirical Characteristic Function Procedures

Abstract: SUMMARY The asymptotic normality and arbitrarily high efficiency of some new statistical procedures based on the empirical characteristic function is established under general conditions.

Unbiased Estimators of Ability Parameters, of Their Variance, and of Their Parallel-Forms Reliability

Abstract: Given known item parameters, unbiased estimators are derived i) for an examinee's ability parameterθ and for his proportion-correct true scoreς, ii) for the variances ofθ andς across examinees in the group tested, and iii) for the parallel-forms reliability of the maximum likelihood estimator\(\hat \theta\).

A class of hypothesis tests for one and two sample censored survival data

Abstract: This paper proposes a class of new non-parametric test statistics useful for goodness-of-fit or two-sample hypothesis testing problems when dealing with randomly right censored survival data. The procedures are especially useful when one desires sensitivity to differences in survival distributions that are particularly evident at at least one point in time. This class is also sufficiently rich to allow certain statistics to be chosen which are yery sensitive to survival differences occurring over a specified period of interest. The asymptotic distribution of each test statistic is obtained and then employed in the formulation of the corresponding test procedure. Size and power of the new procedures are evaluated for small and moderate sample sizes using Monte Carlo simulations. The simulations, generated in the two sample situation, also allow comparisons to be made with the behavior of the Gehan-Wilcoxon and log-rank test procedures.

Asymptotic Inference for Eigenvectors

Abstract: : Asymptotic procedures are given for testing certain hypotheses concerning eigenvectors and for constructing confidence regions for eigenvectors. These asymptotic procedures are derived under fairly general conditions on the estimates of the matrix whose eigenvectors are of interest. Applications of the general results to principal components analysis and canonical variate analysis are given. (Author)

The Asymptotic Behavior of Monotone Regression Estimates

Abstract: An estimator for a monotone regression function was proposed by Brunk. He has shown that if the underlying regression function has positive slope at a point, then, based on $r$ observations, the difference of the regression function and its estimate at that point has a nondegenerate limiting distribution if this difference is multiplied by $r^{1/3}$. To understand how the behavior of the regression function at a point influences the asymptotic properties of the estimator at that point, we have generalized Brunk's result to points at which the regression function does not have positive slope. If the first $\alpha - 1$ derivatives of the regression function are zero at a point and the $\alpha$th derivative is positive there, then the norming constants are of order $r^{\alpha/(2\alpha + 1)}$.

Strong Approximations of Some Biometric Estimates under Random Censorship

Abstract: The empirical distribution function has been widely used as an estimator for the distribution function of the elements of a random sample. It is not, however, appropriate when the observations are incomplete. This is the case, for example, when the observations have been randomly censored on the right. In the latter case the so-called product-limit estimate of Kaplan and Meier [16] has been generally accepted as a substitute for the empirical distribution function, and it possesses many of the same properties (cf. Meier [19]). Developing the corresponding statement of Efron [8], weak convergence, over a finite interval, of the Kaplan-Meier product-limit estimate to a Gaussian process when there is censorship on the right, has been obtained by Breslow and Crowley [3] (cf. also Meier [19]). Breslow and Crowley obtained their result via the limiting distribution of the closely related cumulative hazard process under random censorship. A generalisation of the Efron-Breslow-Crowley theorem was formulated by Yang [24] for the case when censorship on the right is performed by more competing risks. These results have numerous statistical applications in areas such as medical follow-up studies, life testing, actuarial sciences and demography. Koziol and Green [18] have computed tables for the limiting distribution of the Cram6r-von Mises statistic based on the Kaplan-Meier product-limit estimate under a specific random censorship model. A point will be stressed in this paper, however, that by a certain transformation, (which seems to have escaped many researchers' attention and is due to Efron [8]), it is always possible to obtain the standard Wiener process in the limit, the distributions of many functionals of which are well known. This transformation was only recently used in a stimulating note by Gillespie and Fisher [14] to construct asymptotic confidence bands for the unknown distribution function in the Kaplan-Meier model. A general scheme of random censorship can be defined in the following way: Let X be a real random variable with distribution function F(t)

Linear Functions of Concomitants of Order Statistics with Application to Nonparametric Estimation of a Regression Function

Abstract: Let (Xi , Yi )(i = 1, 2, …, n) be independent identically distributed as (X, Y). Then the rth ordered X variate is denoted by Xr:n and the associated Y variate, the concomitant of the rth order statistic, by Y [r:n]. This paper considers statistics of the form and more generally of the form , where J is a bounded smooth function and may depend on n. Under certain regularity conditions, the asymptotic normality of these statistics is established. These statistics are used to construct consistent estimators of various conditional quantities, for example E(Y | X = x), P(Y ∈ A | X = x) and var(Y | X = x).

Minimum Distance Estimators for Location and Goodness of Fit

Abstract: A weighted Cramer-von Mises distance between the empirical distribution function and the assumed model F0(x - θ) is minimized to produce estimators θ n that are asymptotically normal. If the weight function is taken proportional to (- ln f 0)″/f 0 , then θ n is asymptotically efficient and the minimized distance has the appropriate loss of one degree of freedom. Special attention is focused on the limiting distribution of this latter goodness-of-fit statistic in both null and alternative situations.

A Representation for Multinomial Cumulative Distribution Functions

Abstract: A re-expression of the usual representation of the multinomial distribution as the conditional distribution of independent Poisson random variables given fixed sum provides a convenient new way to compute multinomial cumulative distribution functions.

On the Asymptotic Distribution of Ratio and Regression Estimators

Abstract: We give a formal derivation of the asymptotic normality of ratio and regression estimators of a finite population total for simple random sampling and for sampling with probability proportional to aggregate size. The results are all well known and widely used, of course, but a rigorous development, spelling out relatively simple sufficient conditions under which the standard results are valid, does not seem to be available at present.

Robust Methods and Asymptotic Theory in Nonlinear Econometrics

Abstract: 1 Introduction.- 1.1 Specification and misspecification of the econometric model.- 1.2 The purpose and scope of this study.- 2 Preliminary Mathematics.- 2.1 Random variables, independence, Borel measurable functions and mathematical expectation.- 2.1.1 Measure theoretical foundation of probability theory.- 2.1.2 Independence.- 2.1.3 Borel measurable functions.- 2.1.4 Mathematical expectation.- 2.2 Convergence of random variables and distributions.- 2.2.1 Weak and strong convergence of random variables.- 2.2.2 Convergence of mathematical expectations.- 2.2.3 Convergence of distributions.- 2.2.4 Convergence of distributions and mathematical expectations.- 2.3 Uniform convergence of random functions.- 2.3.1 Random functions. Uniform strong and weak convergence.- 2.3.2 Uniform strong and weak laws of large numbers.- 2.4 Characteristic functions, stable distributions and a central limit theorem.- 2.5 Unimodal distributions.- 3 Nonlinear Regression Models.- 3.1 Nonlinear least-squares estimation.- 3.1.1 Model and estimator.- 3.1.2 Strong consistency.- 3.1.3 Asymptotic normality.- 3.1.4 Weak consistency and asymptotic normality under weaker conditions.- 3.1.5 Asymptotic properties if the error distribution has infinite variance. Symmetric stable error distributions.- 3.2 A class of nonlinear robust M-estimators.- 3.2.1 Introduction.- 3.2.2 Strong consistency.- 3.2.3 Asymptotic normality.- 3.2.4 Properties of the function h(?). Asymptotic efficiency and robustness.- 3.2.5 A uniformly consistent estimator of the function h(?).- 3.2.6 A two-stage robust M-estimator.- 3.2.7 Some weaker results.- 3.3 Weighted nonlinear robust M-estimation.- 3.3.1 Introduction.- 3.3.2 Strong consistency and asymptotic normality.- 3.3.3 A two-stage weighted robust M-estimator.- 3.4 Miscellaneous notes on robust M-estimation.- 3.4.1 Uniform consistency.- 3.4.2 The symmetric unimodality assumption.- 3.4.3 The function ?.- 3.4.4 How to decide to apply robust M-estimation.- 4 Nonlinear Structural Equations.- 4.1 Nonlinear two-stage least squares.- 4.1.1 Introduction.- 4.1.2 Strong consistency.- 4.1.3 Asymptotic normality.- 4.1.4 Weak consistency.- 4.2 Minimum information estimators: introduction.- 4.2.1 Lack of instruments.- 4.2.2 Identification without using instrumental variables.- 4.2.3 Consistent estimation without using instrumental variables.- 4.2.4 Asymptotic normality.- 4.2.5 A problem concerning the nonsingularity assumption.- 4.3 Minimum information estimators: instrumental variable and scaling parameter.- 4.3.1 An instrumental variable.- 4.3.2 An example.- 4.3.3 A scaling parameter and its impact on the asymptotic properties.- 4.3.4 Estimation of the asymptotic variance matrix.- 4.3.5 A two-stage estimator.- 4.3.6 Weak consistency.- 4.4 Miscellaneous notes on minimum information estimation.- 4.4.1 Remarks on the function $$S_{ - n}^* (\theta |\gamma )$$.- 4.4.2 A consistent initial value.- 4.4.3 An upperbound of the variance matrix.- 4.4.4 A note on the symmetry assumption.- 5 Nonlinear Models with Lagged Dependent Variables.- 5.1 Stochastic stability.- 5.1.1 Stochastically stable linear autoregressive processes.- 5.1.2 Multivariate stochastically stable processes.- 5.1.3 Other examples of stochastically stable processes.- 5.2 Limit theorem for stochastically stable processes.- 5.2.1 A uniform weak law of large numbers.- 5.2.2 Martingales.- 5.2.3 Central limit theorem for stochastically stable martingale differences.- 5.3 Dynamic nonlinear regression models and implicit structural equations.- 5.3.1 Dynamic nonlinear regression models.- 5.3.2 Dynamic nonlinear implicit structural equations.- 5.4 Remarks on the stochastic stability concept.- 6 Some Applications.- 6.1 Applications of robust M-estimation.- 6.1.1 Municipal expenditure.- 6.1.2 An autoregressive model of money demand.- 6.2 An application of minimum information estimation.- References.

Chi-squared tests for contingency tables with proportions estimated from survey data

Abstract: We investigate the effect of clustering and stratification on the standard chi-squared tests for homogeneity and independence. Although the form of the asymptotic distribution (a weighted sum of independent xt random variables with weights related to particular design effects) is the same for both test statistics, the test for homogeneity is more seriously affected in general than that for independence. We suggest some simple corrections to the usual chi-squared statistics and investigate the performance of the modified tests empirically and through the use of some simple models for clustering.

Asymptotic Normality of Linear Combinations of Order Statistics with a Smooth Score Function

Abstract: Asymptotic normality of linear combinations of order statistics of the form $T_n = n^{-1} \sum J(i/(n + 1))X_{in}$ is investigated along with a slightly trimmed version of $T_n$. Theorem 5 of Stigler (1974) is extended to show asymptotic normality of $T_n$ for a wide class of score functions. In addition, a proof of Theorem 4 of Stigler (1974) is given.

On the distribution of the maximum of the longrank statistic and the modified Wilcoxon statistic

Abstract: The asymptotic distribution of the maximum of the logrank statistic and the modified Wilcoxon statistic is examined [in order to test] the equality of two survival distributions. A table is provided to facilitate use of the maximum statistics and Monte Carlo simulation experiments are used to investigate the small-sample properties of the maximum statistic. An example is given to illustrate the use of the methods developed in this paper. (SUMMARY IN FRE) (EXCERPT)

Passage-time statistics for the decay of unstable equilibrium states

Abstract: The decay of unstable equilibrium states is accompanied by large-scale fluctuations. The statistical properties of such processes can be characterized by using the time at which a representative observable first passes through a fixed threshold value. We present an asymptotic probability distribution for that passage time which is valid when the threshold is set sufficiently far from the initial state. For the simplest example of linear isotropic amplification of an $n$-component vector we calculate both the exact first-passage time distribution and our asymptotic distribution. We verify that the asymptotic distribution coincides with the exact one in the appropriate limit. We then evaluate our asymptotic distribution for a number of more complicated systems including one in which an $n$-component vector field in $d$ spatial dimensions departs from an unstable equilibrium state. The resulting expression has a considerable degree of universality. Its form is independent of $d$ and of details of the field dynamics. It is insensitive, in particular, to whether the underlying field considered is conserved or not. Our procedure is applicable to a wide variety of problems in which an order parameter departs spontaneously from an unstable initial value.

Optimal Sequential Selection of a Monotone Sequence From a Random Sample

Abstract: : The length of the longest monotone increasing subsequence of a random sample of size n is known to have expected value asymptotic to (2n)(to the 1/2 power). We prove that it is possible to make sequential choices which give an increasing subsequence of expected length asymptotic to (2n)(to the 1/2 power). Moreover, this rate of increase is proved to be asymptotically best possible. (Author)

The Calculation of the Limiting Distribution of the Least Squares Estimator of the Parameter in a Random Walk Model

Abstract: The limit distribution of the least squares estimator $\hat{\alpha}$ of the parameter $\alpha$ of the first order stochastic difference equation, in the boundary case $\alpha = 1$, is calculated.

Simple Approximate Distributional Results for Confidence and Tolerance limits for the Weibull Distribution Based on Maximum likelihood Estimators

Abstract: One form of the generalized gamma distribution brings out an association between the normal distribution and the extreme-value distribution. This relationship suggests chi-squared, Student's t and noncentral t approximations for certain functions of the maximum likelihood estimators of the parameters of the extreme-value distribution. The accuracy of these approximations is studied by comparing them to existing Monte Carlo simulation results. These approximations provide simple procedures for obtaining approximate confidence intervals for the parameters, tolerance limits, and confidence limits on reliability for the Weibull or extremevalue distribution.