Density estimation for randomly distributed circular objects

Summary Nearest neighbour methods traditionally used to estimate density of a sessile biological population treat individuals as points. The present paper suggests distance-based density estimators which treat individuals as circles with variable areas. Distribution of distance between a sample point and thek-th (k = 1, 2, 3, …) nearest circle is derived. Maximum likelihood estimator of density is obtained from a random sample of point tok-th order distances. Assuming a skewed distribution for the circle radius, moment estimators of density and mean circle area are derived.     

Two-sample rank tests with truncated populations

LetX ₁,…,X _m andY ₁,…,Y _n be two independent samples from continuous distributionsF andG respectively. Using a Hoeffding (1951) type theorem, we obtain the distributions of the vector S=(S ₍₁₎,…,S _(n)), whereS _(j)=# (X _i ’s≤Y _(j)) andY _(j) is thej-th order statistic ofY sample, under three truncation models: (a)G is a left truncation ofF orG is a right truncation ofF, (b)F is a right truncation ofH andG is a left truncation ofH, whereH is some continuous distribution function, (c)G is a two tail truncation ofF. Exploiting the relation between S and the vectorR of the ranks of the order statistics of theY-sample in the pooled sample, we can obtain exact distributions of many rank tests. We use these to compare powers of the Hajek test (Hajek 1967), the Sidak Vondracek test (1957) and the Mann-Whitney-Wilcoxon test. We derive some order relations between the values of the probagility-functions under each model. Hence find that the tests based onS ₍₁₎ andS _(n) are the UMP rank tests for the alternative (a). We also find LMP rank tests under the alternatives (b) and (c).     

A characterization of the exponential distribution by higher order gap

Summary SupposeX is a non-negative random variable with an absolutely continuous (with respect to Lebesgue measure) distribution functionF (x) and the corresponding probability density functionf(x). LetX ₁,X ₂,...,X _n be a random sample of sizen fromF andX _i,n is thei-th smallest order statistics. We define thej-th order gapg _i,j(n) asg _i,j(n)=X _i+j,n–X_i,n 1i<n, 1nn–i. In this paper a characterization of the exponential distribution is given by considering a distribution property ofg _i,j(n).     

Hypothesis testing and parameter estimation based onM-statistics ink samples with unequal variances

Summary Ink samples with unequal variances,M-tests for homogeneity ofk location parameters are proposed. The asymptoticχ ²-distributions of the test statistics and the robustness of the tests are investigated. NextM-estimators (ME’s) of parameters are discussed. Furthermore positive-part shrinkage versions (PSME’s) of theM-estimators for the location parameters are considered along with modified James-Stein estimation rule. In asymptotic distributional risks based on a special feasible loss, it is shown that the PSME’s dominate the ME’s, and preliminary test and shrinkageM-versions fork≧4.     

On the distribution of a distance function on the sphere

Summary The distribution of the square of the distance between a random point and a fixed point on ap-dimensional unit sphere when (i) the two points lie on the whole sphere and (ii) the two points lie in the positive quadrant, has been derived, assuming that the random point is distributed proportionally to exp (ky ₁), wherek is a concentration parameter. Then-th order moment in both cases is also obtained.     

Combining information for prediction in linear regression

This article deals with the prediction problem in linear regression where the measurements are obtained using k different devices or collected from k different independent sources. For the case of k=2, a Graybill-Deal type combined estimtor for the regression parameters is shown to dominate the individual least squares estimators under the covariance criterion. Two predictors ŷ _c and ŷ _p are proposed. ŷ _c is based on a combined estimator of the regression coefficient vector, and ŷ _p is obtained by combining the individual predictors from different models. Prediction mean square errors of both predictors are derived. It is shown that the predictor ŷ _p is better than the individual predictors for k≥2 and the predictor ŷ _c is better than the individual predictors for k=2. Numerical comparison between ŷ _c and ŷ _p shows that the former is superior to the latter for the case k=2.     

Moment (in-)equalities for differences of order statistics with different sample sizes

Summary Leto _j:n be thej-th order statistic andq _:n the -quantile of sample sizen. Ther-th moment of |o _j1:n1-o _j2:n2| is calculated in terms of hypergeometric distributions. This equality is applied to obtain moment (in-)equalities for |q _:n1-q _:n2|.     

Multivariate multistage methodologies for simultaneous all pairwise comparisons

We consider the problem of constructing simultaneous fixed-width confidence intervals for all pairwise treatment differences μ₁−μ _J , in the presence ofk(≥2) independent populationsN _p (μ₁,Σ), 1≤i≠j≤k. Appropriate purely sequential, accelerated sequential and three-stage sampling strategies have been developed and variousfirst-order asymptotic properties are then derived when Σ _pxp is completely unknown, but positive definite (p.d.). In the two special cases when the largest component variance in Σ is a known multiple of one of the variances or Σ=σ² H where σ(>0) is unknown, butH _pxp is known and p.d., the original multistage sampling strategies are specialized. Under such special circumstances, associatedsecond-order characteristics are then developed. It is to be noted that our present formulation and the methodologies fill important voids in the context of multivariate multiple comparisons which is a challenging area that has not yet been fully explored. Moderate sample performances of the proposed techniques were very encouraging and detailed remarks on these were included in Mukhopadhyay and Aoshima (1997).     

The relative effectiveness of gain-framed and loss-framed messages in charity advertising: Meta-analytic evidence and implications

Meta-analysis is a primary avenue for the development of cumulative knowledge and an essential step in the process theory testing. It provides a more specific quantitative understanding of a subject while correcting for sampling and measurement errors. This project is a modest first attempt to extend existing meta-analytic work on the effect of gain- versus loss-framed messages to charity advertising research. A meta-analysis of 27 studies (k = 40, N = 9,298) finds that gain-framed and loss-framed appeals do not differ significantly on persuasiveness in charity advertising. The random-effects model reveals a mean r of .002 (95% CI −0.061–0.064, p = .96) after correcting for sampling and measurement errors. A re-analysis (k = 38) excluding two outliers yielded a mean r of .03 (95% CI −0.022–.073, p = .29). Among published studies (k = 32), there was tentative indication pointing to the slight advantage of gain-framed appeals. The framing effect does not differ by which type of donation the message calls for (monetary vs. organ), nor is it affected by moderators such as study setting and population. The little difference finding echoes previous conceptual and empirical work and calls for more work identifying moderating factors and boundary conditions on framing research. Practical implications and future directions are discussed.     

Local comparison of linear rank tests,in the Bahadur sense

Summary As continuation ofKremer [1979a] a theory of asymptotic comparison based on local Bahadur efficiency is derived for general linear rank tests of the one-sample symmetry problem and thek-sample problem (k2). The results are similar to former considerations based on Pitman efficiency but hold under weaker conditions on the scores-generating functions or local alternatives.     

Choice of optimal initial designs in sequential experiments

Combined-optimal designs (Li and Lin, 2003) are obviously the best choices for the initial designs if we partition the experiment into two parts with equal size to obtain some information about the process, especially for the case not considering the blocking factor. In this paper, the definition of combined-optimal design is extended to the case when blocking factor is significant, and this new class of designs is called blocked combined-optimal designs. Some general results are obtained which relate 2^k−p_III initial designs with their complementary designs when , where n=2^k−p. By applying these results, we are able to characterize 2^k−p_III combined-optimal designs or blocked combined-optimal designs in terms of their complementary designs. It is also proved that both 2^k−p_III combined-optimal and blocked combined-optimal designs are not minimum aberration designs when and n−1−k > 2. And some combined-optimal and blocked combined-optimal designs with 16 and 32 runs are constructed for illustration. 2000 Mathematics Subject Classifications: 62K15, 62K05     

Het verdelen van steekproeven over subpopulaties bij accountantscontroles

Summary “Stratificationprocedures for a typical auditing problem”. During the past ten years, much experience was gained in The Netherlands in using random sampling methods for typical auditing problems. Especially, a method suggested by VAN. HEERDEN [2] turned out to be very fruitful. In this method a register of entries is considered to be a population of T guilders, if all entries total up to T guilders. The sample size n₀ is determined in such a way that the probability β not to find any mistake in the sample, if a fraction p₀ or more of T is incorrect, is smaller than a preassigned value β₀. So n₀ should satisfy (l-p)ⁿ⁰≤β₀ for p≥p₀. A complication arises if it is not possible to postpone sampling until the whole population T is available. One then wants to take samples from a population which is growing up to T. Suppose one is going to take samples n_i from e.g. r subpopulations Using the minimax procedure, it is shown, that in this case one should choose the sizes n_i equal to (T_i/T)n₀. The minimax-value of the probability not to find any incorrect guilder in the r samples, taken together is equal to β₀.     

One optional observation inflates α by $${100/\sqrt{n}}$$ per cent

For one-sample level α tests ψ _m based on independent observations X ₁, . . . , X _m , we prove an asymptotic formula for the actual level of the test rejecting if at least one of the tests ψ _n , . . . , ψ _n+k would reject. For k = 1 and usual tests at usual levels α, the result is approximately summarized by the title of this paper. Our method of proof, relying on some second order asymptotic statistics as developed by Pfanzagl and Wefelmeyer, might also be useful for proper sequential analysis. A simple and elementary alternative proof is given for k = 1 in the special case of the Gauss test.     

On approximating the distributions ofFriedman's 75-175-175-1and related statistics

Error bounds depending explicitly on parameters of the problem are given for large-sample approximations to the central and noncentral distributions ofFriedman's [1937]x _r ² statistic,Steel's [1959] statistics for comparingr treatments with a control, andNemenyi's [1963] statistics for pair-wise comparisons among all treatments. These bounds are of ordero(N ^–1/2); an improved bound of ordero (N ^–r/(r+1)) is given forFriedman's statistic in the central case. Applications yield bounds on the actual type 1 and type 2 error probabilities in terms of their normal-theory approximations.Supported in part by ONR Contract N00014-72A-0136-003 from the U.S. Office of Naval Research.     

Subjective Bayesian multivariate stratified sampling from finite populations

Summary In this paper we obtain optimum allocation formula in stratified sampling from a finite population withk characteristics under study using the superpopulation approach put forth byEricson [1969a]. Allocation at the second phase is also considered using information obtained from the first phase. Two different approaches are employed: a Bayesian posterior analysis and a Bayesian preposterior analysis. It is also shown that our allocation formulae for the second phase observations include the results ofMohd. Zubair Khan [1976] andDraper/Guttman [1968a] as special cases when the unknown characteristicX _ij possessed by the (i, j)-th element is scalar valued and the stratum sizes are large.     

Minimax estimators for a bounded location parameter

Summary Recently, Bischoff and Fieger (1992) considered the classical problem of estimating a bounded normal mean when the loss is thep-th power of the error. They proved that forp>-2 a two point prior is least favourable in case the parameter interval is small enough. In the present paper it is shown that this result remains valid forp>1. Moreover, the normal family is generalized to location parameter families. Finally, it is proved that no two point prior is least favourable for absolute error loss, i.e., forp=1.     

Characterizations of distributions via linearity of regression of generalized order statistics

Let X (r, n, m, k), 1 r n, denote generalized order statistics based on an absolutely continuous distribution function F. We characterize all distribution functions F for which the following linearity of regression holds E(X(r+l,n,m,k) | X(r,n,m,k))=aX(r,n,m,k)+b.We show that only exponential, Pareto and power distributions satisfy this equation. Using this result one can obtain characterizations of exponential, Pareto and power distributions in terms of sequential order statistics, Pfeifers records and progressive type II censored order statistics. Received July 2001/Revised August 2002     

On maximum likelihood prediction based on Type II doubly censored exponential data

In this paper, the maximum likelihood predictor (MLP) of the kth ordered observation, t _k, in a sample of size n from a two-parameter exponential distribution as well as the predictive maximum likelihood estimators (PMLE's) of the location and scale parameters, θ and β, based on the observed values t _r, …, t _s (1≤r≤s<k≤n), are obtained in closed forms, contrary to the belief they cannot be so expressed. When θ is known, however, the PMLE of β and MLP of t _k do not admit explicit expressions. It is shown here that they exist and are unique; sharp lower and upper bounds are also provided. The derived predictors and estimators are reasonable and also have good asymptotic properties. As applications, the total duration time in a life test and the failure time of a k-out-of-n system may be predicted. Finally, an illustrative example is included. Received: August 1999     

On selecting the best ofk lognormal distributions

Summary Fork lognormal populations, which differ only in one certain parameter Ϙ, the problem of finding the population with the largest value ofϑ is considered. For two-parameter lognormal families, several natural choices ofϑ are treated, where the problem can be solved, through logarithmic transformation of the observations, within the framework of estimating parameters ink, possibly restricted, normal populations. For three-parameter lognormal families, this standard approach of selecting in terms of natural estimators fails to work ifϑ is the “guaranteed lifetime”. For this case, a selection procedure is derived which is based on anL-statistic which has the smallest asymptotic variance. Of importance here is that it is location equivariant, whereas it does not matter what it actually estimates. Comparisons are made with other suitable selection rules, through the asymptotic relative efficiencies, as well as in an example of intermediate sample sizes. It is shown that only in the latter, the selection rule, which is based on the sample minima, compares favorably. The research of this author was supported by the Office of Naval Research Contract N00014-88-K-0170 and NSF Grant Number DMS-8606964 at Purdue University. Reproduction in whole or in part is permitted for any purpose of the United States Government. The research of this author was supported by the Air Force Office of Scientific Research Grant 85-0347 at the University of Illinois at Chicago.     

Distribution of product statistics from aPareto population

Summary Distributions are derived of the product of sample values, the sample geometric mean, the product of two minimum values from sample of unequal size and product ofk minimum values from sample of equal size from aPareto population. The distributions can be conveniently transformed tox ². Paper presented at the Eastern Regional meeting of the Institute of Mathematical Statistics, Upton, Long Island, New York, April 27–29, 1966. Work done when the author was on the faculty of Western Reserve University, Cleveland, Ohio, U.S.A.