Uniform asymptotic joint normality of sample quantities in censored cases

Summary The asymptotic joint distribution of an increasing number of sample quantiles as the sample size increases, when the underlying sample is censored, is shown to be asymptotically uniformly (or type (B) _d ) normally distributed under fairly general conditions. The discussions for uncensored cases have been given by [4].     

Null distribution of the sum of squared z-transforms in testing complete independence

Brien et al. (1984, Biometrika, 71, 545–554; 1988, Biometrika, 75, 469–476) have proposed, illustrated and discussed advantages of using Fisher's z-transforms for analyzing correlation structures of multinormal data. Chen and Mudholkar (1988, Austral. J. Statist., 31, 105–110) have studied the sum of squared z-transforms of sample correlations as a test statistic for complete independence. In this paper Brown's (1987, Ann. Probab., 15, 416–422) graph-theoretic characterization of the dependence structure of sample correlations is used to evaluate moments of the test statistic. These moments are then used to approximate its null distribution accurately over a broad range of parameters, including the case where the population dimension exceeds the sample size.     

On the uniform asymptotic joint normality of sample quantiles

Summary 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. Type (B) _d asymptotic normality is a strictly stronger notion than the usual one which is based on the convergence in law, and the results obtained in this article will be helpful to widen the applicability of results on asymptotic normality of sample quantiles to related statistical inferences.     

Quenouille-type theorem on autocorrelations

The central result is a limit theorem for not necessarily stationary processes resembling AR (p). Assumption of a vector limit distribution for standardized sample autocorrelations leads to the convergence of a vector limit distribution for ordinary sample partial autocorrelations, and to a clear relationship between the two limit distributions. The motivation is the study of the case p=1 by Mills and Seneta (1989, Stochastic Process Appl., 33, 151–161). The central result is used to explain the nature of the relationship between the two results of Quenouille in the classical stationary AR (p) setting.     

Closer estimators of a common mean in the sense of Pitman

Consider the problem of estimating the common mean of two normal populations with different unknown variances. Suppose a random sample of sizem is drawn from the first population and a random sample of sizen is drawn from the second population. The paper gives a family of estimators closer than the sample mean of the first population in the sense of Pitman (1937,Proc. Cambridge Phil. Soc.,33, 212–222). In particular, the Graybill-Deal estimator (1959,Biometrics,15, 543–550) is shown to be closer than each of the sample means ifm5 andn5.     

On Sample Size and Quick Simultaneous Confidence Interval Estimations for Multinomial Proportions

Abstract

Asymptotically, sample proportions from a multinomial distribution converge in distribution to a multivariate normal distribution with a singular negative product correlation structure. Based on this result, we propose a new approach to estimate the sample size requirement for constructing quick simultaneous confidence intervals (QSCI) for multinomial proportions. In addition, this new approach can be used to construct QSCI and provides a statistical justification to the reports of the opinion polling.     

Maximum variance of order statistics

Yang (1982,Bull. Inst. Math. Acad. Sinica,10(2), 197–204) proved that the variance of the sample median cannot exceed the population variance. In this paper, the upper bound for the variance of order statistics is derived, and it is shown that this is attained by Bernoulli variates only. The proof is based on Hoeffding's identity for the covariance.     

A DISCRETE-TIME ITÔ'S FORMULA

Abstract

The numerical methods on stochastic differential equations (SDEs) have been well established. There are several papers that study the numerical stability of SDEs with respect to sample paths or moments. In this paper, we study the stability in distribution of numerical solution of SDEs.     

Confident Search

Abstract

The task of searching for the best element or a good element in a large set P is central to many problems in artificial intelligence and related fields. Often, heuristic information is used to reduce the scope of the search; however, in many instances, this information carries no guarantee of good performance. This article begins with an arbitrary heuristic search procedure and supplies it with a confidence statement of the following form: With specified high probability β, the output of the confidence procedure will be among the best 100α% of the elements of P. The confidence procedure will report either the outcome of the heuristic search or a better alternative with the required properties; that is, it will either certify that the heuristic answer has the desired confidence property or it will produce a better answer having the property. The approach involves combining heuristic search with a form of heuristic sampling that tends to sample the better elements of P. The sample is designed in such a way that the best element in the sample has the desired confidence property—if the answer produced by the heuristic search is better still, it inherits the confidence property. Various devices permit the sampling procedure to retain its confidence property while (1) moving the sample in the direction suggested by the heuristic, (2) adjusting the heuristic preference in response to what is learned during sampling, and (3) reorganizing the sampling whenever promising discoveries are made by chance.     

A class of multivariate skew-normal models

The existing model for multivariate skew normal data does not cohere with the joint distribution of a random sample from a univariate skew normal distribution. This incoherence causes awkward interpretation for data analysis in practice, especially in the development of the sampling distribution theory. In this paper, we propose a refined model that is coherent with the joint distribution of the univariate skew normal random sample, for multivariate skew normal data. The proposed model extends and strengthens the multivariate skew model described in Azzalini (1985,Scandinavian Journal of Statistics,12, 171–178). We present a stochastic representation for the newly proposed model, and discuss a bivariate setting, which confirms that the newly proposed model is more plausible than the one given by Azzalini and Dalla Valle (1996,Biometrika,83, 715–726).     

Almost sure exponential stability for stochastic neutral partial functional differential equations

In this paper, we initiate a study on stochastic neutral partial functional differential equations in a real separable Hilbert space. Our goal here is to study the existence and uniqueness of a mild solution of this class of equations and also the exponential stability of the moments of a mild solution as well as its sample paths. The results obtained here generalize the main results from [Taniguchi, Stochastics and Stochastics Reports, 53, (1995) 41–52], [Taniguchi, Stochastic Analysis and Applications, 16, (1998) 965–975] and [Liu and Truman, Statistics Probability Letters, 50, (2000) 273–278]. An example is given to illustrate the theory.     

Clinical decision support system for dental treatment

BackgroundIn this research, a decision making system, based on fuzzy inference mechanism as proposed by Mamdani, is presented. Literature suggests that there is a lack of consistency among dentists in choosing treatment plan(s). So, this research work aims to facilitate the dentist decide the treatment plan(s) of the broken tooth.MethodsAn expert system based on fuzzy logic has been designed to accept inaccurate and vague values of dental signs and symptoms associated with the broken tooth. We designed a knowledge base with 60 rules and used Mamdani inference algorithm to decide the possible one or more treatment(s) and suggest the same to the dentist.ResultsThe results proposed by the system are compared with the dentists’ suggestions. The Chi-square test of homogeneity is conducted on 100 randomly generated sample cases with the help of three professional dentists. It is found that the results produced by the system are consistent with the treatment plan(s) proposed by the dentists. Chi-square value of the test is 3.843565 which is less than the critical value which is 12.592. Hence, we are unable to reject the null hypothesis that assumes the two populations are homogeneous with respect to treatments.ConclusionsThe accuracy of the proposed decision support system for the treatment of broken tooth enhances the confidence level of the dentists while making decision regarding the treatment plan(s). Simple and interactive GUI makes it easy to use.     

ARIES: A Visual Path in the Investigation of Statistical Data

Abstract

The massive flood of numbers in ongoing large-scale periodic economic and social surveys commonly leaves little time for anything but a cursory examination of the quality of the data, and few techniques exist for giving an overview of data activity. At the U.S. Bureau of Labor Statistics, a graphical and query-based solution to these problems has recently been adopted for data review in the Current Employment Statistics survey. Chief among the motivations for creating the new system were: (1) Reduce or eliminate the arduous paper review of thousands of sample reports by review analysts; (2) allow the review analysts a more global view of sample activity and at the same time make outlier detection less of a strain; and (3) present global views of estimates over time and among groups of subestimates. The specific graphics approaches used in the new system were designed to quickly portray both time series and cross-sectional aspects of the data, as these are both critical elements in the review process. The described system allows the data analysts to track down suspicious sample members by first graphically pinpointing questionable estimates, and then pinpointing questionable sample data used to produce those estimates. Query methods are used for cross-checking relationships among different sample data elements. Although designed for outlier detection and estimation, the data-representation methods employed in the system have opened up new possibilities for further statistical and economic uses of the data. The authors were torn between the desire for a completely automatic system of data review and the practical demands of an actual survey operating under imperfect conditions, and thus viewed the new system as an evolutionary advance, not as an ideal final solution. Possibilities opened up by the new system prompted some further thinking on finding an ideal state.     

Estimation of two normal means which may be common

Consider the problem of estimating the mean of a normal population when independent samples from this as well as a second normal population are available. Pre-test estimators which combine the two sample means if a test of the hypothesis of equal population means accepts but otherwise use only the first sample mean, are compared to limited translation estimators which are derived in the spirit of Bickel (1984, Ann. Statist., 12, 864–879) (we also cover the cases of unknown variances). Our conclusion is that if the accuracy with which the second population mean can be estimated is of the same or better order of magnitude as teh accuracy with which the first can be estimated, then the limited translation estimators largely dominate the pre-test estimators in terms of mean square error loss.This research was supported by grants from the FRD of the CSIR of South Africa.     

Trimmed minimax estimator of a covariance matrix

Summary In the problem of estimating the covariance matrix of a multivariate normal population, James and Stein (Proc. Fourth Berkeley Symp. Math. Statist. Prob.,1, 361–380, Univ. of California Press) obtained a minimax estimator under a scale invariant loss. In this paper we propose an orthogonally invariant trimmed estimator by solving certain differential inequality involving the eigenvalues of the sample covariance matrix. The estimator obtained, truncates the extreme eigenvalues first and then shrinks the larger and expands the smaller sample eigenvalues. Adaptive version of the trimmed estimator is also discussed. Finally some numerical studies are performed using Monte Carlo simulation method and it is observed that the trimmed estimate shows a substantial improvement over the minimax estimator. The second author's research was supported by NSF Grant Number MCS 82-12968.     

Asymptotic Behavior of a Set-Statistic

The existence theorem of Minkowski for a polytope with given facet normals and areas is adapted to a data-analytic context. More precisely, we show that a centered, random point sample arising from an absolutely continuous distribution in R ^d can be uniquely mapped into such a polytope almost surely. With increasing sample size, the sequence of (scaled) polytopes converges almost surely to a limiting convex body that is associated with the underlying distribution. An accompanying central limit theorem is proved using methods from the theory of empirical processes. Received January 28, 1999, and in revised form April 12, 1999.     

On the deviation between the distributions of sums and maximum sums of IID random vectors

Summary For a sequence of independent and identically distributed random vectors, with finite moment of order less than or equal to the second, the rate at which the deviation between the distribution functions of the vectors of partial sums and maximums of partial sums is obtained both when the sample size is fixed and when it is random, satisfying certain regularity conditions. When the second moments exist the rate is of ordern ^−1/4 (in the fixed sample size case). Two applications are given, first, we compliment some recent work of Ahmad (1979,J. Multivariate Anal.,9, 214–222) on rates of convergence for the vector of maximum sums and second, we obtain rates of convergence of the concentration functions of maximum sums for both the fixed and random sample size cases.     

Nonparametric test of restricted interchangeability

Exact and large sample distributions of the rank order test under the null hypothesis of restricted interchangeability are obtained. Under given regularity conditions and under Pitman's shift in location alternative, the asymptotic relative efficiency of this nonparametric test in comparison with Votaw's (1948, Ann. Math. Statist., 19, 447–473) likelihood ratio test is given.     

Gibbs Sampling with Diffuse Proper Priors: A Valid Approach to Data-Driven Inference?

Abstract

This article demonstrates by example that the use of the Gibbs sampler with diffuse proper priors can lead to inaccurate posterior estimates. Our results show that such inaccuracies are not merely limited to small sample settings.     

The Limit Distribution of the Largest Interpoint Distance from a Symmetric Kotz Sample

Generalizing recent work of P. C. Matthews and A. L. Rukhin (Ann. Appl. Probab.3(1993), 454–466), we obtain the limit law of the largest interpoint Euclidean distance for a spherically symmetric multivariate sample of the Kotz distribution. While going through the proof, some errors in the reasoning given by Matthews and Rukhin are pointed out and corrected.