SIZE AND SHAPE AS LINEAR COMBINATIONS OF UNTRANSFORMED MEASUREMENTS

A numerical specification of ‘size’ and ‘shape’ is of interest for making interpretations in morphometrics. Starting from a, possibly large, set m ₁,…, m_r of size measurements, e.g. m ₁= height, m ₂= sitting height, etc., a preliminary analysis provides the set x ₁,…,x_p of size measurements to be used, e.g. x ₁= m ₁? m ₂= subischial leg length, x ₂= m ₂= sitting height, and x ₃= head circumference. In general these x_j are constructed as appropriately scaled linear combinations of the original measurements. A constant term should not be included because size measurements have to be 0 if all x_j are 0. Our theory requires a (compromise) vector μof means and a matrix Σof (co)variances. Size being specified as an optimalsize characteristic of the form c ′ x , the remaining morphological information is expressed by, at most, p? 1 components of shapeof the form d ′ x. Relations with Darroch-Mosimann [9] Darroch, J. N. and Mosimann, J. E. 1985. Canonical and Principal Components of Shape. Biometrika, 72: 241–252. [Crossref], [Web of Science ®] , [Google Scholar]are indicated. An application to human growth is made and other applications are suggested.

Don't read my book, think for yourself.

C. R. Rao, personal communications, 1981     

Some subset selection problems

Consider that we have a collection of k populations π₁, π₂…,π_k. The quality of the ith population is characterized by a real parameter θ_i and the population is to be designated as superior or inferior depending on how much the θ_i differs from θ_max = max{θ₁, θ₂,…,θ_k}. From the set {π₁, π₂,…,π_k}, we wish to select the subset of superior populations. In this paper we devise rules of selection which have the property that their selected set excludes all the inferior populations with probability at least 1?α, where a is a specified number.     

ESTIMATING A PARAMETER WHEN IT IS KNOWN THAT THE PARAMETER EXCEEDS A GIVEN VALUE

In some statistical problems a degree of explicit, prior information is available about the value taken by the parameter of interest, θ say, although the information is much less than would be needed to place a prior density on the parameter's distribution. Often the prior information takes the form of a simple bound, ‘θ > θ₁ ’ or ‘θ < θ₁ ’, where θ₁ is determined by physical considerations or mathematical theory, such as positivity of a variance. A conventional approach to accommodating the requirement that θ > θ₁ is to replace an estimator, , of θ by the maximum of and θ₁. However, this technique is generally inadequate. For one thing, it does not respect the strictness of the inequality θ > θ₁ , which can be critical in interpreting results. For another, it produces an estimator that does not respond in a natural way to perturbations of the data. In this paper we suggest an alternative approach, in which bootstrap aggregation, or bagging, is used to overcome these difficulties. Bagging gives estimators that, when subjected to the constraint θ > θ₁ , strictly exceed θ₁ except in extreme settings in which the empirical evidence strongly contradicts the constraint. Bagging also reduces estimator variability in the important case for which is close to θ₁, and more generally produces estimators that respect the constraint in a smooth, realistic fashion.     

Non-parametric estimation of an acceleration parameter

Let X₁,… X_m be a random sample of m failure times under normal conditions with the underlying distribution F(x) and Y₁,…,Y_n a random sample of n failure times under accelerated condititons with underlying distribution G(x);G(x)=1?[1?F(x)]^θ with θ being the unknown parameter under study.Define:U_ij=1 otherwise.The joint distribution of _ijdoes not involve the distribution F and thus can be used to estimate the acceleration parameter θ.The second approach for estimating θ is to use the ranks of the Y-observations in the combined X- and Y-samples.In this paper we establish that the rank of the Y-observations in the pooled sample form a sufficient statistic for the information contained in the U_ii 's about the parameter θ and that there does not exist an unbiassed estimator for the parameter θ.We also construct several estimators and confidence interavals for the parameter θ.     

Nonparametric estimation of a multivariate multiple regression function

Let (X₁, X₂, Y₁, Y₂) be a four dimensional random variable having the joint probability density function f(x₁, x₂, y₁, y₂). In this paper we consider the problem of estimating the regression function \({{E[(_{Y_2 }^{Y_1 } )} \mathord{\left/ {\vphantom {{E[(_{Y_2 }^{Y_1 } )} {_{X_2 = X_2 }^{X_1 = X_1 } }}} \right. \kern-0em} {_{X_2 = X_2 }^{X_1 = X_1 } }}]\) on the basis of a random sample of size n. We have proved that under certain regularity conditions the kernel estimate of this regression function is uniformly strongly consistent. We have also shown that under certain conditions the estimate is asymptotically normally distributed.     

Estimation bayesienne des effets dans le modele a effets aleatoires de classification double avec interaction

The problem of estimating the effects in a balanced two-way classification with interaction \documentclass{article}\pagestyle{empty}\begin{document}$i = 1, \ldots ,I;j = 1, \ldots ,J;k = 1, \ldots ,K$\end{document} using a random effect model is considered from a Bayesian view point. Posterior distributions of r_i, c_j and t_ij are obtained under the assumptions that r_i, c_j, t_ij and e_ijk are all independently drawn from normal distributions with zero meansand variances \documentclass{article}\pagestyle{empty}\begin{document}$\sigma _r^2 ,\sigma _c^2 ,\sigma _t^2 ,\sigma _e^2$\end{document} respectively. A non informative reference prior is adopted for \documentclass{article}\pagestyle{empty}\begin{document}$\mu ,\sigma _r^2 ,\sigma _c^2 ,\sigma _t^2 ,\sigma _e^2$\end{document}. Various features of thisposterior distribution are obtained. The same features of the psoterior distribution for a fixed effect model are also obtained. A numerical example is given.     

Distribution of the correlation coefficient for the class of bivariate elliptical models

We consider n pairs of random variables (X₁₁,X₂₁),(X₁₂,X₂₂),… (X_1n,X_2n) having a bivariate elliptically contoured density of the form where θ₁ θ₂ are location parameters and Δ = ((λ_ik)) is a 2 × 2 symmetric positive definite matrix of scale parameters. The exact distribution of the Pearson product-moment correlation coefficient between X₁ and X₂ is obtained. The usual case when a sample of size n is drawn from a bivariate normal population is a special case of the abovementioned model.     

Uniform asymptotic linearity in a regression parameter of a process based on a rank statistic

For X₁, …, X_N a random sample from a distribution F, let the process S^δ_N(t) be defined as where K²_N = σ^N_i=1(c_i ? c?)² and R x_i, + Δd, is the rank of X_i + Δd_i, among X₁ + Δd₁, …, X_N + Δd_N. The purpose of this note is to prove that, under certain regularity conditions on F and on the constants c_i and d_i, S^Δ_N (t) is asymptotically approximately a linear function of Δ, uniformly in t and in Δ, |Δ| ≤ C. The special case of two samples is considered.     

Simultaneous selection of extreme populations:a bayesian approach

The problem of simultaneously selecting two non-empty subsets, S_Land S_U, of k populations which contain the lower extreme population (LEP) and the upper extreme population (UEP), respectively, is considered. Unknown parameters θ₁,…,θ_kcharacterize the populations π₁,…,π_kand the populations associated with θ_[1]=min θ_i. and θ_[k]= max θ_i. are called the LEP and the UEP, respectively. It is assumed that the underlying distributions possess the monotone likelihood ratio property and that the prior distribution of θ= (θ₁,…,θ_k) is exchangeable. The Bayes rule with respect to a general loss function is obtained. Bayes rule with respect to a semi-additive and non-negative loss function is also determined and it is shown that it is minimax and admissible. When the selected subsets are required to be disjoint, it shown that the Bayes rule with respect to a specific loss function can be obtained by comparing certain computable integrals, Application to normal distributions with unknown means θ₁,…,θ_kand a common known variance is also considered.