Measurement error in regression analysis

Consider the linear regression model Y = Xθ+ ε where Y denotes a vector of n observations on the dependent variable, X is a known matrix, θ is a vector of parameters to be estimated and e is a random vector of uncorrelated errors. If X'X is nearly singular, that is if the smallest characteristic root of X'X s small then a small perurbation in the elements of X, such as due to measurement errors, induces considerable variation in the least squares estimate of θ. In this paper we examine for the asymptotic case when n is large the effect of perturbation with regard to the bias and mean squared error of the estimate.     

Asymptotic expansions in non-linear renewal theory

We obtain first order asymptotic expansions for the distribution of the excess of a standard normal random walk over a curved boundary and the error probabilities of some repeated significance tests. The key step in the analysis is an asymptotic expansion for the conditional probability that the random walk has not crossed the boundary before the N step, given that it is near the boundary after the n^th step.     

Estimation of the mean of the exponential distribution with an outlying observation

Given a life testing experiment consisting of n items, n-1 of which have the expected life λ while one could have an expected life λ/α with 0 < α < 1 the problem is. to find a mean square error (MSE) minimizing estimation function. The standard estimators for the homogeneous case (α = 1) overestimate the expected life and their MSE tend to infinity when a tends to 0.

Looking at the estimation problem as an insurance (see Anscombe (1960)) two different “testimators” are compared with respect to their MSE, Numerical results show that an estimation function based on the “Epstein-statistic” x_(n)/[xbar] is the best one.     

Nonparametric estimation of the survival distribution in censored data

Two nonparametric estimators o f the survival distributionare discussed. The estimators were proposed by Kaplan and Meier (1958) and Breslow (1972) and are applicable when dealing with censored data. It is known that they are asymptotically unbiased and uniformly strongly consistent, and when properly normalized that they converge weakly to the same Gaussian process. In this paper, the properties of the estimators are carefully inspected in small or moderate samples. The Breslow estimator, a shrinkage version of the Kaplan-Meier, nearly always has the smaller mean square error (MSE) whenever the truesurvival probabilityis at least 0.20, but has considerably larger MSE than the Kaplan-Meier estimator when the survivalprobability is near zero.     

From Ordinary to Shrinkage Square-Root Estimators

Consider a skewed population. Suppose an intelligent guess could be made about an interval that contains the population mean. There may exist biased estimators with smaller mean squared error than the arithmetic mean within such an interval. This article indicates when it is advisable to shrink the arithmetic mean towards a guessed interval using root estimators. The goal is to obtain an estimator that is better near the average of natural origins. An estimator proposed. This estimator contains the Thompson (1968 Thompson , J. R. ( 1968 ). Accuracy borrowing in the estimation of the mean by shrinkage towards an interval . J. Amer. Statist. Assoc. 63 : 953 – 963 . [CSA] [CROSSREF] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) ordinary shrinkage estimator, the Jenkins et al. (1973 Jenkins , O. C. , Ringer , L. J. , Hartley , H. O. ( 1973 ). Root estimators . J Amer. Statist. Assoc. 68 : 414 – 419 . [CSA] [CROSSREF] [Taylor & Francis Online], [Web of Science ®] , [Google Scholar]) square-root estimator, and the arithmetic sample mean as special cases. The bias and the mean squared error of the proposed more general estimator is compared with the three special cases. Shrinkage coefficients that yield minimum mean squared error estimators are obtained. The proposed estimator is considerably more efficient than the three special cases. This remains true for highly skewed populations. The merits of the proposed shrinkage square-root estimator are supported by the results of numerical and simulation studies.     

A Class of Estimators Using Auxiliary Information for Estimating Finite Population Variance in Presence of Measurement Errors

This article addresses the problem of estimating the population variance using auxiliary information in the presence of measurement errors. When the measurement error variance associated with study variable is known, a class of estimators of the population variance using auxiliary information has been proposed. We obtain the bias and mean squared errors of the suggested class of estimators upto the terms of order n ^?1, and also optimum estimators in asymptotic sense of the class with approximate mean squared error formula.     

Variable Bandwidths for Nonparametric Hazard Rate Estimation

A smoothing parameter inversely proportional to the square root of the true density is known to produce kernel estimates of the density having faster bias rate of convergence. We show that in the case of kernel-based nonparametric hazard rate estimation, a smoothing parameter inversely proportional to the square root of the true hazard rate leads to a mean square error rate of order n ^?8/9, an improvement over the standard second order kernel. An adaptive version of such a procedure is considered and analyzed.     

Kernel estimation of a distribution function

A distribution function is estimated by a kernel method with

a poinrwise mean squared error criterion at a point x. Relation- ships between the mean squared error, the point x, the sample size and the required kernel smoothing parazeter are investigated for several distributions treated by Azzaiini (1981). In particular it is noted that at a centre of symmetry or near a mode of the distribution the kernei method breaks down. Point- wise estimation of a distribution function is motivated as a more useful technique than a reference range for preliminary medical diagnosis.     

A prior-valued estimator applied to multinomial classification

A multinomial classification rule is proposed based on a prior-valued smoothing for the state probabilities. Asymptotically, the proposed rule has an error rate that converges uniformly and strongly to that of the Bayes rule. For a fixed sample size the prior-valued smoothing is effective in obtaining reason¬able classifications to the situations such as missing data. Empirically, the proposed rule is compared favorably with other commonly used multinomial classification rules via Monte Carlo sampling experiments