Rate of strong uniform convergence of k-NN density estimates

Let f_n(x) be the univariate k-nearest neighbor (k-NN) density estimate proposed by Loftsgaarden and Quesenberry (1965). By using similar techniques as in Bahadur's representation of sample quantiles (1966), and by the recent results on the oscillation of empirical processes by Stute (1982), we derive the rate of strong uniform convergence of f_n(x) on some suitably chosen interval J_δ. Some comparison with the kernel estimates is given, as well as the choice of the bandwidth sequence relative to the sample size.     

Ranked set sampling theory with selective probability vector

In this paper an estimator of the population mean is introduced by using the idea of selective probability vector and the optimization algorithm of linear programming to find the optimal solution of the selective probability vector under the condition of unbiasedness.     

An Estimator of the Exponent of Regular Variation Based on K-Record Values

Let {X _n:n ≥ 1} be an i.i.d. sequence of random variables with a continuous distribution function F. Under the assumption that the upper tail of Fis regularly varying with exponent 1/α, α > 0, we study the asymptotic properties of an estimator of α based on k-record values.     

Numerical estimation of a probability measure

An iterative solution to the problem of maximizing a concave functional ø defined on the set of all probability measures on a topological space is considered. Convergence of this procedure and a rapidly converging algorithm are studied. Computational aspects of this algorithm along with the ones developed earlier by Wynn, Fedorov, Atwood, Wu and others are provided. Examples discussed are taken from the area of mixture likehoods and optimal experimental design.     

Comparing relative importance of the same component in a series system in two environments

The relative performance of a component of a series system in two different environments is considered. The conditional probability of the failure of the system due to the failure of the specified component given that the system failed before time t is regarded as a measure of relative importance of the component to the system. A U-statistic test for checking the equality of the relative importance of the component to the system in two different environments against the alternative that the relative importance is smaller in one of the environments, is proposed. Some simulation results for estimating the power of the test are reported. The proposed test is applied to one real data set and it is seen that a different aspect of the data is brought out by this comparison than that by the comparisons of the absolute importance functions such as the subsurvival functions, considered in earlier studies.     

Analysis of multivariate categorical data with misclassification errors by triple sampling schemes

Previous work has been carried out on the use of double-sampling schemes for inference from categorical data subject to misclassification. The double-sampling schemes utilize a sample of n units classified by both a fallible and true device and another sample of n₂ units classified only by a fallible device. In actual applications, one often hasavailable a third sample of n₁ units, which is classified only by the true device. In this article we develop techniques of fitting log-linear models under various misclassification structures for a general triple-sampling scheme. The estimation is by maximum likelihood and the fitted models are hierarchical. The methodology is illustrated by applying it to data in traffic safety research from a study on the effectiveness of belts in reducing injuries.     

Weak convergence of bounded influence regression estimates with applications to repeated significance testing

We obtained weak convergence results for bounded influence regression M-estimates and apply the results to sequential clinical trials, with special reference to repeated significance tests in the two-sample problem with covariates.     

Stopping times of one-sample rank order sequential probability ratio tests

Weed, Bradley and Grovindarajulu (1974) propose one-sample probability ratio tests based on Lehmann alternatives. They also study the finite sure termination of the stopping times. Motivated by Stein's proof of (1946) of the termination of a sequential probability ratio test (SPRT) in the case of independent and identically distributed (i.i.d.) random variables and the work of Sethuraman (1970) for the two- sample rank order SPRT, we obtain a very mild condition (namely, that a certain random variable U(Z) is not identically zero) for the finite sure termination of the existence of the moment generating function (m.g.f.) for the stopping time of one- sample rank order SPRT's.     

Edgeworth expansion for signed linear rank statistics with regression constants

An Edgeworth expansion with remainder o(N^?1) is obtained for signed linear rank statistics under suitable assumptions. The theorem is proved for a wide class of score generating functions including the Chi-quantile function by adapting van Zwet's methodand Does's conditioning arguments.     

A Jackknife Method for Estimation of Variance Components

This paper concerns a method of estimation of variance components in a random effect linear model. It is mainly a resampling method and relies on the Jackknife principle. The derived estimators are presented as least squares estimators in an appropriate linear model, and one of them appears as a MINQUE (Minimum Norm Quadratic Unbiased Estimation) estimator. Our resampling method is illustrated by an example given by C. R. Rao [7] and some optimal properties of our estimator are derived for this example. In the last part, this method is used to derive an estimation of variance components in a random effect linear model when one of the components is assumed to be known.     

Limit theory of generalized order statistics

Generalized order statistics (gos) were introduced by Kamps [1995. A Concept of Generalized Order Statistics. Teubner, Stuttgart] to unify several models of ordered random variables (rv's), e.g., (ordinary) order statistics (oos), records, sequential order statistics (sos). In a wide subclass of gos that includes oos and sos, the possible limit distribution functions (df's) of the maximum gos are obtained in Nasri-Roudsari [1996. Extreme value theory of generalized order statistics. J. Statist. Plann. Inference 55, 281–297]. In this paper, for this subclass, necessary and sufficient conditions of weak convergence, as well as the form of the possible limit df's of extreme, intermediate and central gos are derived. These results are extended to a wider subclass.     

A class of modified stein estimators with easily computable risk functions

Consider the problem of estimating the mean of a p (≥3)-variate multi-normal distribution with identity variance-covariance matrix and with unweighted sum of squared error loss. A class of minimax, noncomparable (i.e. no estimate in the class dominates any other estimate in the class) estimates is proposed; the class contains rules dominating the simple James-Stein estimates. The estimates are essentially smoothed versions of the scaled, truncated James-Stein estimates studied by Efron and Morris. Explicit and analytically tractable expressions for their risks are obtained and are used to give guidelines for selecting estimates within the class.     

The uniform convergence of the nadaraya-watson regression function estimate

If (X₁,Y₁), …, (X_n,Y_n) is a sequence of independent identically distributed R_d × R-valued random vectors then Nadaraya (1964) and Watson (1964) proposed to estimate the regression function m(x) = ? {Y₁|X₁ = x{ by where K is a known density and {h_n} is a sequence of positive numbers satisfying certain properties. In this paper a variety of conditions are given for the strong convergence to 0 of ess_Xsup|m_n (X)-m(X)| (here X is independent of the data and distributed as X₁). The theorems are valid for all distributions of X₁ and for all sequences {h_n} satisfying h_n → 0 and nh/log n→0.     

The Likelihood Ratio Test for the Presence of Immunes in a Censored Sample

Models for populations with immune or cured individuals but with others subject to failure are important in many areas, such as medical statistics and criminology. One method of analysis of data from such populations involves estimating an immune proportion 1 ? p and the parameter(s) of a failure distribution for those individuals subject to failure. We use the exponential distribution with parameter λ for the latter and a mixture of this distribution with a mass 1 ? p at infinity to model the complete data. This paper develops the asymptotic theory of a test for whether an immune proportion is indeed present in the population, i.e., for H ₀:p = 1. This involves testing at the boundary of the parameter space for p. We use a likelihood ratio test for H ₀. and prove that minus twice the logarithm of the likelihood ratio has as an asymptotic distribution, not the chi-square distribution, but a 50–50 mixture of a chi-square distribution with 1 degree of freedom, and a point mass at 0. The result is proved under an independent censoring assumption with very mild restrictions.     

Unified treatment of Cramér-Rao bound for the nonregular density functions

This paper extends the idea of Vincze (1978) and unifies the approach for the uniparameter and multiparameter situations for obtaining the Cramér-Rao inequality.     

On the normal approximation of an estimate of the mean residual life of a multicomponent system

An estimate of the mean residual life function of a complex system of k independent identically distributed components is proposed and studied with emphasis being on the order of normal approximation.     

A comparison of the independent-samples t-test and the paired-samples t-test when the observations are nonnegatively correlated pairs

An account of the behavior of the independent-samples t-test when applied to homoschedastic bivariate normal data is presented, and a comparison is made with the paired-samples t-test. Since the significance level is not violated when applying the independent-samples t-test to data which consist of positively correlated pairs and since the estimate of the variance is based on a larger number of ‘degrees of freedom’, the results suggest that when the sample size is small, one should not worry much about the possible existence of weak positive correlation. One may do better, powerwise, to ignore such correlation and use the independent-samples t-test, as though the samples were independent.     

Gaussian approximation of signed rank statistics process

We consider the signed linear rank statistics of the form

$\text{S}{}_{\mathrm{\Delta N}}\text{=}\text{∑}\text{i=1}\text{N}\text{c}{}_{\mathrm{Ni}}\text{ø(}\text{R}{}^{\mathrm{\Delta }}{}_{\mathrm{Ni}}\text{(N+1)}\text{)}\text{sgn}\text{Y}{}^{\mathrm{\Delta }}{}_{\mathrm{Ni}}$

where the c_Ni's are known real numbers, Δ∈[0,1] is an unknown real parameter,R^Δ_Ni is the rank of |Y^Δ_Ni| among |Y^Δ_Nj|, 1≤j≤N, ø is a score generating function, sgn y=1 or -1 according as y≥0 or <0, and Y^Δ_Nj, 1≤j ≤N, are independent random variables with continuous cumulative distribution functions F(y?Δd_Nj), 1≤ j≤N, respectively where the d_fNi's are known real numbers. Under suitable assumptions on the c's, d's, φ and F, it is proved that the random process {S_ΔN?S_0N?ES_ΔN, 0≤Δ≤1}, properly normalized, converges weakly to a Gaussian process, and this result is also true if ES_ΔN is replaced by Δb_N, where

$\text{b}{}_{\mathrm{N}}\text{=4}\text{∑}\text{i=1}\text{N}\text{c}{}_{\mathrm{Ni}}\text{d}{}_{\mathrm{Ni}}\text{∫}\text{0}\text{∞}\text{ø′(2F(x)?1)?}{}^2\text{(x)}\text{d}\text{x}\text{and}\text{?=F′}$

. As an application, we derive the asymptotic distribution of the properly normalized length of a confidence interval for Δ.