Large Sample Results Research Articles

Measurement of noise (cross-) power spectra for frequency-domain system identification purposes: large-sample results

Several frequency-domain estimators of parametric transfer function models assume that the (cross-) power spectra of the disturbing noise sources are known. This paper presents a time-efficient method to measure these noise (cross-) power spectra and studies its influence on the asymptotic properties of the estimated model parameters.

A NEW DIAGNOSTIC TEST OF MODEL INADEQUACY WHICH USES THE MARTINGALE DIFFERENCE CRITERION

Abstract. Let {x(t)} denote a discrete‐time random process. Given a sample of increments e(t) = x(t) ‐ x(t ‐ 1) from the time series, we wish to test formally whether the sample is consistent with the assumption that {e(t)} is a martingale difference. It is shown that the martingale criterion is more general than the white noise criterion in analyzing fitted residuals for signs of model inadequacy. In this paper we present such a test which approximately achieves a given type 1 error probability for samples. We assume that (1) the process is strictly stationary, (2) all its kth‐order cumulant functions exist and (3) the kth‐order cumulants are absolutely summable and satisfy a mixing condition. The martingale assumption implies that most third‐order cumulants of the increment process are zero, and thus the third‐order cumulant sequence is sparse. This result is used to derive test statistics based on a modified sample bispectrum. The test can be regarded as a two‐dimensional portmanteau test of serial dependence. The large‐sample results are demonstrated through the use of artificial data. Finally, the test is applied to a daily financial series.

Large-sample results for optimization-based clustering methods

Many common (nonhierarchical) clustering and classification methods are optimization-based methods, in the sense described by Windham (1987) in this Journal. This paper gives some large sample properties for estimates derived by such methods. Under appropriate conditions, such estimates converge with probability one to a limit, and are asymptotically normally distributed around that limiting value. The conditions are satisfied by most of the common examples of optimization-based methods.

A Survey of Product-Integration with a View Toward Application in Survival Analysis

The correspondence between a survival function and its hazard or failure-rate is a central idea in survival analysis and in the theory of counting processes. This correspondence is shown to be a special case of a more general correspondence between multiplicative and additive matrix-valued measures on the real line. Additive integration of the survival function produces the hazard, while the multiplicative integral, or so-called product-integral, of the hazard yields the survival function. The easy generalization to the matrix case (noncommutative multiplication) allows an elegant and completely parallel treatment of intensity measures of Markov processes, with many possible applications in multistate survival models. However, the difficulties and multiplicity of theories of product-integration in multivariate time explain why so many different multivariate product-limit estimators exist. We give a complete and elementary treatment of the basic theory of the product-integral $\pi(1 + dX)$ together with a discussion of some of its applications. New results are given on the compact differentiability of the product-integral, to be used along with the functional $\delta$-method for getting large-sample results for product-limit estimators.

Income inequality measures based on sample surveys

By relating the Gini coefficient to a general family of inequality measures, the Gini family, methods of estimating inequality parameters from samples are reviewed. Three approaches to making inferences about unknown inequality parameters are discussed and some large-sample results are presented. Alternative variance estimates are presented and compared in the case of the Gini coefficient.

Locally optimal window widths for kernel density estimation with large samples

The smoothing parameter or window width for a kernel estimator of a probability density at a point has been previously specified to minimize either asymptotic mean square error or asymptotic mean absolute error. In this note the ratio of these two widths is shown to be a constant for all kernels and density functions that satisfy the usual smoothness conditions. The fact that this ratio equals 0.985 supports recent comment that in this context these two error criteria do not yield large-sample results that differ by any meaningful amount. Isolated points at which the dominant term of the conventional bias expansion vanishes are examined. Consideration of additional terms and continuity leads to the conclusion that bias is adequately modeled by a multiple of a single rate in all large but finite sample sizes. In practice, for instance, at inflection points with a second-order kernel the abrupt change in exponent from 1/5 to 1/9 is not necessarily a good representation.

Testing for the Constancy of Parameters over Time

Abstract Tests are proposed for detecting possible changes in parameters when the observations are obtained sequentially in time. While deriving the tests the alternative one has in mind specifies the parameter process as a martingale. The distribution theory of these tests relies on the large-sample results; that is, only the limiting null distributions are known (except in very special cases). The main tool in establishing these limiting distributions is weak convergence of stochastic processes. Suppose that we have vector-valued observations x 1, …, x n obtained sequentially in time (or ordered in some other linear fashion). Their joint distribution is described by determining the initial distribution for x 1 and the conditional distribution for each x k given the past up to x k–1. Suppose further that these distributions depend on a p-dimensional parameter vector θ. At least locally (i.e., in a short time period) this may be more or less legitimate. In the long run, however, the possibility of some ch...

Prison crowding: A psychological perspective : by Paul B. Paulus. With the collaboration of Verne C. Cox and Garvin McCain Springer-Verlag of New York, Inc. (P.O. Box 2485, Secaucus, New Jersey 07094), 1988, 103 pp., hardcover—$43.00

The main enzyme for serotonin degradation, monoamine oxidase (MAO) A, has recently emerged as a key biological factor in the predisposition to impulsive aggression. Male carriers of low-activity variants of the main functional polymorphism of the MAOA gene (MAOA-uVNTR) have been shown to exhibit a greater proclivity to engage in violent acts. Thus, we hypothesized that low-activity MAOA-uVNTR alleles may be associated with a higher risk for criminal violence among male offenders. To test this possibility, we analyzed the MAOA-uVNTR variants of violent (n = 49) and non-violent (n = 40) male Caucasian and African-American convicts in a correctional facility. All participants were also tested with the Childhood Trauma Questionnaire (CTQ), Barratt Impulsivity Scale (BIS-11) and Buss-Perry Aggression Questionnaire (BPAQ) to assess their levels of childhood trauma exposure, impulsivity and aggression, respectively. Our results revealed a robust (P < 0.0001) association between low-activity MAOA-uVNTR alleles and violent crime. This association was replicated in the group of Caucasian violent offenders (P < 0.01), but reached only a marginal trend (P = 0.08) in their African American counterparts. While violent crime charges were not associated with CTQ, BIS-11 and BPAQ scores, carriers of low-activity alleles exhibited a mild, yet significant (P < 0.05) increase in BIS-11 total and attentional-impulsiveness scores. In summary, these findings support the role of MAOA gene as a prominent genetic determinant for criminal violence. Further studies are required to confirm these results in larger samples of inmates and evaluate potential interactions between MAOA alleles and environmental vulnerability factors.

Estimation of p(y&lt;x) in the gamma case

We consider estimation of P(Y<X) when X˜Γr(M,λ) and Y˜Γ(N,μ) are independent with M and N known. A concise representation of the UMVUE and several representations for the MLE are derived. Closed-form exact expressions of both MSE's and the bias of the MLE are obtained. Large-sample results are given and numerical comparison of the two point estimators is made. Confidence intervals are given.

Least absolute deviations estimation for the censored regression model

This paper proposes an alternative to maximum likelihood estimation of the parameters of the censored regression (or censored ‘Tobit’) model. The proposed estimator is a generalization of least absolute deviations estimation for the standard linear model, and, unlike estimation methods based on the assumption of normally distributed error terms, the estimator is consistent and asymptotically normal for a wide class of error distributions, and is also robust to heteroscedasticity. The paper gives the regularity conditions and proofs of these large-sample results, and proposes classes of consistent estimators of the asymptotic covariance matrix for both homoscedastic and heteroscedastic disturbances.

Nonparametric Inference for a Class of Semi-Markov Processes with Censored Observations

A class of semi-Markov models, those which have proportional hazards and which are forward-going (if state $j$ can be reached from $i$, then $i$ cannot be reached from $j$), are shown to fit into the multiplicative intensity model of counting processes after suitable random time changes. Standard large-sample results for counting processes following this multiplicative model can therefore be used to make inferences on the above class of semi-Markov models, including the case where observations may be censored. Large-sample results for a four-state model used in clinical trials are presented.

Maximally Selected Chi Square Statistics for Small Samples

Two samples can be compared by selecting a cut point and forming a 2 x 2 table of the numbers of observations above and below the cut point in each sample. Different cut points correspond to the maximum chi square statistic, the maximum squared log odds ratio, and the generalized Kolmogorov-Smirnov statistic. We simulated the small-sample null distribution of these statistics. The distributions of the maximally selected chi square statistic and the squared log odds ratio statistic are compared to each other and to the large-sample results of Miller and Siegmund (1982, Biometrics 38, 1011-1016. The distributions of and agreement among the three cut points are also considered.

JackknifingL-Statistics with Smooth Weight Functions

Abstract The behavior of linear combinations of order statistics (L-statistics) under jackknifing is discussed. The asymptotic behavior of a jackknifed L-statistic is described and the pseudovalue-based variance estimate is shown to be consistent under moderate smoothness and a trimming condition on the weight function. The results extend easily to functions of L-statistics and thus answer a question posed in Miller (1974). Monte Carlo results support small-sample applicability of the large-sample results for the construction of approximate confidence intervals.

On the relationship between two tests for homogeneity of the marginal distributions in a two-way classification

Several tests have been proposed for comparing the marginal distributions in a k x k contingency table with obvious application to studies involving matched pairs and a polytomous response variable. Stuart (1955) proposed the test statistic QO = d'Vo-' d, where d denotes any (k -1) differences of the respective marginal frequencies and VO is the estimated covariance matrix of d under the null hypothesis. Bhapkar (1966) proposed the alternative test statistic Q = d'V' d, where V is the maximum likelihood estimator of the covariance matrix of d. Using standard largesample results, it can be shown that both QO and Q are asymptotically distributed x2 with (k-1) degrees of freedom. For Stuart's example, Bhapkar found the numerical difference between his and Stuart's tests to be negligible but noted that the difference may be quite appreciable if N is small and/or the d's are at least moderately large. The difference betweeen the two test statistics can, in fact, be expressed exactly in an algebraic form, from which the effect of sample size is seen explicitly. The algebraic relationship between Q and QO follows from the matrix relationship between V and VO, namely VO = V + n'dd', from which it can be shown that

Large-Sample Results in the Estimation of a Linear Transformation

Large-Sample Results in the Estimation of a Linear Transformation

Large-sample results in the estimation of a linear transformation

We obtain the asymptotic covariance matrix of the maximum likelihood estimators of the coefficients in a linear functional relationship between two vector variables observed with error. The matrix cannot be found by the usual method of inverting the information matrix due to the presence of incidental parameters. The joint asymptotic distribution of the estimators is shown to be normal under some mild conditions on the incidental parameters. A consistent estimator of the asymptotic co variance matrix is also suggested.

The Large-Sample Behavior of Transformations to Normality

Abstract We investigate the large-sample behavior of the Box-Cox procedures for selecting a transformation to normality. The study of the large-sample behavior clearly reveals the role played by the assumptions. Based on our large-sample results, we introduce an information number approach for transforming a known distribution to near normality. This latter procedure provides bench marks for the maximum amount of improvement achievable through power transformations. We illustrate our procedure with three examples. Finally, we generalize our conclusions to random vector and linear model situations.

The Large-Sample Behavior of Transformations to Normality

Abstract We investigate the large-sample behavior of the Box-Cox procedures for selecting a transformation to normality. The study of the large-sample behavior clearly reveals the role played by the assumptions. Based on our large-sample results, we introduce an information number approach for transforming a known distribution to near normality. This latter procedure provides bench marks for the maximum amount of improvement achievable through power transformations. We illustrate our procedure with three examples. Finally, we generalize our conclusions to random vector and linear model situations.

Treatment Contrasts in Paired Comparisons: Large-Sample Results, Applications, and Some Optimal Designs

Abstract Treatment contrasts in paired comparison experiments are considered. Given that specified orthonormal treatment contrasts are null, remaining treatment parameters, orthonormal contrasts among them, and their logarithms are estimated. The estimators are shown to be consistent and, when suitably scaled and centered, to have distribution functions that are normal in the limit for large sample sizes. Hypothesis tests on treatment contrasts are obtained. The limiting distribution functions of related likelihood ratio tests are derived for defined local alternatives and their noncentrality parameters are shown. The theoretical results are applied to experiments with factorial treatment combinations and some optimal design results are obtained.

Treatment Contrasts in Paired Comparisons: Large-Sample Results, Applications, and Some Optimal Designs

Treatment Contrasts in Paired Comparisons: Large-Sample Results, Applications, and Some Optimal Designs