Inference concerning the population correlation coefficient from bivariate normal samples based on minimal observations

Summary Let (X, Y) be bivariate normally distributed with means (μ ₁,μ ₂), variances (σ ₁ ² ,σ ₂ ² ) and correlation betweenX andY equal to ρ. Let (X _i ,Y _i ) be independent observations on (X,Y) fori=1,2,...,n. Because of practical considerations onlyZ _i =min (X _i ,Y _i) is observed. In this paper, as in certain routine applications, assuming the means and the variances to be known in advance, an unbiased consistent estimator of the unknown distribution parameter ρ is proposed. A comparison between the traditional maximum likelihood estimator and the unbiased estimator is made. Finally, the problem is extended to multivariate normal populations with common mean, common variance and common non-negative correlation coefficient.     

Maximum Likelihood Estimation of Means and Variances from Normal Populations Under Simultaneous Order Restrictions

For k normal populations with unknown means μ_i and unknown variances σ²_i, i = 1, ..., k, assume that there are some order restrictions among the means and variances, respectively, for example, simple order restrictions: μ₁ ≤ μ₂ ≤ ... ≤ μ_k and σ²₁ ≥ σ²₂ ≥ ... ≥ σ²_k > 0. Some properties of maximum likelihood estimation of μ_is and σ²_i are discussed and an algorithm of obtaining the maximum likelihood estimators under the order restrictions is proposed.     

The finite simple groups with at most two fusion classes of every order

Elements a,b of a group G are said to be fused if a = b^σ and to be inverse-fused if a =(b^-1)^σ for some σ ? Aut(G). The fusion class of a ? G is the set {a^σ | σ ? Aut(G)}, and it is called a fusion class of order i if a has order iThis paper gives a complete classification of the finite nonabelian simple groups G for which either (i) or (ii) holds, where:

(i) G has at most two fusion classes of order i for every i (23 examples); and

(ii) any two elements of G of the same order are fused or inversenfused.

The examples in case (ii) are: A₅, A₆,L₂(7),L₂(8), L₃(4), Sz(8), M₁₁ and M₂₃An application is given concerning isomorphisms of Cay ley graphs.     

PiecewiseL-splines

Leta=x ₀<x ₁<...<x _N =b be a partition of the interval [a, b] and letL be a normalm-th order linear differential operator. The purpose of this note is to point out that spline functions in one variable need not be excluded to piecewise fits of functions belonging to the null spaceN(L ^* L) on each closed subinterval [x _i,x _i+1], 0in-1 but may be extended to piecewise fits of functions belonging toN(L _i ^* L _i) on each subinterval [x _i,x _i+1] provided theL _i's are selected from a uniformly bounded family of normal linear differential operators. Furthermore when theL _i's are so selected one obtains the usual integral relations and error estimates obtained for splines [2, 8 and 9] including the extended error estimates obtained by Swartz and Varga [10].     

Acceptance inspection by variables when the measurements are subject to error

The situation considered is that in which measurement of the characteristic of interest is not exact but subject to appreciable error. The error is assumed to be unbiased and independent of the actual value of the characteristic measured. The population and error variances,σ ² andσ _e ² , are assumed to be such thatσ/σ _e has a known lower limit which is greater than zero. The probability distributions involved are assumed to be normal while the actual values and measurement errors each form a random sample. For suitable specified acceptable and unacceptable fractions defective, and forσ _e assumed known and unknown, this paper presents one-sided acceptance inspection criteria which are optimum in a specified sense, and which have the property that the producer’s and consumer’s risks have specified upper bounds.     

Functional central limit theorems for self-normalized least squares processes in regression with possibly infinite variance data

Based on an R²-valued random sample {(y_i,x_i),1≤i≤n} on the simple linear regression model y_i=x_iβ+α+ε_i with unknown error variables ε_i, least squares processes (LSPs) are introduced in D[0,1] for the unknown slope β and intercept α, as well as for the unknown β when α=0. These LSPs contain, in both cases, the classical least squares estimators (LSEs) for these parameters. It is assumed throughout that {(x,ε),(x_i,ε_i),i≥1} are i.i.d. random vectors with independent components x and ε that both belong to the domain of attraction of the normal law, possibly both with infinite variances. Functional central limit theorems (FCLTs) are established for self-normalized type versions of the vector of the introduced LSPs for (β,α), as well as for their various marginal counterparts for each of the LSPs alone, respectively via uniform Euclidean norm and sup–norm approximations in probability. As consequences of the obtained FCLTs, joint and marginal central limit theorems (CLTs) are also discussed for Studentized and self-normalized type LSEs for the slope and intercept. Our FCLTs and CLTs provide a source for completely data-based asymptotic confidence intervals for β and α.     

Robust M-estimators of location vectors

Let X₁,…,X_n be i.i.d. random vectors in R^m, let θεR^m be an unknown location parameter, and assume that the restriction of the distribution of X₁−θ to a sphere of radius d belongs to a specified neighborhood of distributions spherically symmetric about 0. Under regularity conditions on and d, the parameter θ in this model is identifiable, and consistent M-estimators of θ (i.e., solutions of Σ_i=1ⁿψ(|X_i− |)(X_i− )=0) are obtained by using “re-descenders,” i.e., ψ's wh satisfy ψ(x)=0 for x≥c. An iterative method for solving for is shown to produce consistent and asymptotically normal estimates of θ under all distributions in . The following asymptotic robustness problem is considered: finding the ψ which is best among the re-descenders according to Huber's minimax variance criterion.     

On non-orthogonal main effect plans for asymmetrical factorials

Summary Following the lines of Raktoe and Federer [19] a unified approach for constructing main effect plans in any factorials wherek _i's are the numbers of equispaced levels of each of then _i factors, andk _i's are not necessarily primes or prime powers and need not satisty any relations among themselves, is presented. The method consists of, first, dividing the totality of treatment combinations, omitting, of course, some, if necessary, in to pairs such that the differences within the pairs are clear of ‘even’ effects and the sums are clear of ‘odd’ effects, and then, depending on the number of error d.f. wanted, selecting a suitable sub-set of these pairs which lead to the solution of the estimates of main effects. A general class of non-orthogonal main effect plans for 2 ^m ×2 ⁿ factorials is proposed. Information matrices and their inverses for such plans are worked out. An example followed by discussions and comparison statements is presented.     

Relatively pseudocomplemented semilattices amalgamate strongly

Letk be any finite or infinite cardinal andS _ω the symmetric group of denumerable infinite degree. It is shown that fori<k ifG _i is thei-th row of a matrix whose columns are allk-termed sequences of elements ofS _ω in each of which all but a finite number of terms are equal to the identity ofS _ω thenG _i 's (withG _i ⁻¹ 's defined in an obvious way and with coordinatewise multiplication amongG _i 's andG _i ⁻¹'s) generate the Free Group onk free generatorsG _i . Analogously, Free Abelian and other types of free groups are also constructed. Presented by L. Fuchs.     

Admissibility of the usual confidence interval for the normal mean

Suppose that X₁,…,X_n are independent and identically N(μ,σ²) distributed, where μ and σ are unknown parameters (μ∈R and σ>0). We prove that the usual confidence interval for μ is admissible within a broad class of confidence intervals.     

Fixed Width Confidence Region for the Mean of a Multivariate Normal Distribution

Srivastava gave an asymptotically efficient and consistent sequential procedure to obtain a fixed-width confidence region for the mean vector of any p-dimensional random vector with finite second moments. For normally distributed random vectors, Srivastava and Bhargava showed that the specified coverage probability is attained independent of the width, the mean vector, and the covariance matrix by taking a finite number of observations over and above T prescribed by the sequential rule. However, the problem of showing that E(T−n₀) is bounded, where n₀ is the sample size required if the covariance matrix were known, has not been available. In this paper, we not only show that it is bounded but obtain sharper estimates of E(T) on the lines of Woodroofe. We also give an asymptotic expansion of the coverage probability. Similar results have recently been obtained by Nagao under the assumption that the covariance matrix Σ=∑^k_i=1 σ_iA_i and ∑^k_i=1 A_i=I, where A_i's are known matrices. We obtain the results of this paper under the sole assumption that the largest latent root of Σ is simple.     

An asymptotic minimax risk bound for estimation of a linear functional relationship

We consider estimation of the parameter B in a multivariate linear functional relationship X_i=ξ_i+ξ_1i, Y_i=Bξ_i+ξ_2i, i=1,…,n, where the errors (ζ_1i^′, ζ_2i^′) are independent standard normal and (ξ_i, i ) is a sequence of unknown nonrandom vectors (incidental parameters). If there are no substantial a priori restrictions on the infinite sequence of incidental parameters then asymptotically the model is nonparametric but does not fit into common settings presupposing a parameter from a metric function space. A special result of the local asymptotic minimax type for the m.1.e. of B is proved. The accuracy of the normal approximation for the m.l.e. of order n^−1/2 is also established.     

Eigenvalue Intervals for 2mth Order Sturm—Liouville Boundary Value Problems

Values of λ are determined for which there exist positive solutions of the 2mth order differential equation on a measure chain, (-1)^m x ^?2m (t)=λa(t)f(u(σ(t))), y? [0,1], satisfying α _i+1 u ^?21(0)+0, γ _i+1 u ^?21(σ(1))=0, 0≤i≤m?1 with αi,βi,γi,δi≥0, where a and f are positive valued, and both lim _x-0+ (f(x)/x) and lim _x-0+ (f(x)/x) exist.     

Identification of the Parameters of a Multivariate Normal Vector by the Distribution of the Maximum

This paper continues the work started by Basu and Ghosh (J. Mult. Anal. (1978), 8, 413–429), by Gilliland and Hannan (J. Amer. Stat. Assoc. (1980), 75, No. 371, 651–654), and then continued on by Mukherjea and Stephens (Prob. Theory and Rel. Fields (1990), 84, 289–296), and Elnaggar and Mukherjea (J. Stat. Planning and Inference (1990), 78, 23–37). Let (X₁, X₂,..., X_n) be a multivariate normal vector with zero means, a common correlation and variances ² ₁, ² ₂,..., ² _n such that the parameters , ² ₁, ² ₂,..., s² _n are unknown, but the distribution of the max{X_i: 1in} (or equivalently, the distribution of the min{X_i: 1in}) is known. The problem is whether the parameters are identifiable and then how to determine the (unknown) parameters in terms of the distribution of the maximum (or its density). Here, we solve this problem for general n. Earlier, this problem was considered only for n3. Identifiability problems in related contexts were considered earlier by numerous authors including: T. W. Anderson and S. G. Ghurye, A. A. Tsiatis, H. A. David, S. M. Berman, A. Nadas, and many others. We also consider here the case where the X_i's have a common covariance instead of a common correlation.     

Functions, Measures, and Equipartitioning Convex k-Fans

A k-fan in the plane is a point x∈?² and k halflines starting from x. There are k angular sectors σ ₁,…,σ _k between consecutive halflines. The k-fan is convex if every sector is convex. A (nice) probability measure μ is equipartitioned by the k-fan if μ(σ _i )=1/k for every sector. One of our results: Given a nice probability measure μ and a continuous function f defined on sectors, there is a convex 5-fan equipartitioning μ with f(σ ₁)=f(σ ₂)=f(σ ₃).     

WAVELET ESTIMATION IN NONPARAMETRIC MODEL UNDER MARTINGALE DIFFERENCE ERRORS

§1　IntroductionConsider the following heteroscedastic regression model:Yi =g(xi) +σiei,　1≤i≤n,(1.1)whereσ2i=f(ui) ,(xi,ui) are nonrandom design points,0≤x0 ≤x1 ≤...≤xn=1and0≤u0≤u1 ≤...≤un=1,Yi are the response variables,ei are random errors,and f(·) andg(·) are unknown functions defined on closed interval[0 ,1] .It is well known thatregression model has many applications in practical problems,sothe model (1.1) and its special cases have been studied extensively. For instance,…     

Optimal maps for the multidimensional Monge-Kantorovich problem

Let μ₁,…, μ_N be Borel probability measures on ℝ^d. Denote by Γ(μ₁,…, μ_N) the set of all N-tuples T=(T₁,…, T_N) such that T_i:ℝ^d ↔ ℝ^d (i = 1,…, N) are Borel-measurable and satisfy μ₁ = μ_i[V] for all Borel V ⊂ ℝ^d. The multidimensional Monge-Kantorovich problem investigated in this paper consists of finding S=(S₁,…, S_N) ∈ Γ(μ₁,…, μ_N) minimizing over the set Γ(μ₁, ···, μ_N). We study the case where the μ_i's have finite second moments and vanish on (d - 1)-rectifiable sets. We prove existence and uniqueness of optimal maps S when we impose that S₁( x ) ≡ x and give an explicit form of the maps S_i. The result is obtained by variational methods and to the best of our knowledge is the first available in the literature in this generality. As a consequence, we obtain uniqueness and characterization of an optimal measure for the multidimensional Kantorovich problem. © 1998 John Wiley & Sons, Inc.     

The exact density function of the ratio of two dependent linear combinations of chi-square variables

A computable expression is derived for the raw moments of the random variableZ=N/D whereN= ₁ ⁿ m _iX_i+ _n ^+1s m _iX_i,D= _n ^+1s l _iX_i+ _s ^+1r n _iX_i, and theX _i's are independently distributed central chi-square variables. The first four moments are required for approximating the distribution ofZ by means of Pearson curves. The exact density function ofZ is obtained in terms of sums of generalized hypergeometric functions by taking the inverse Mellin transform of theh-th moment of the ratioN/D whereh is a complex number. The casen=1,s=2 andr=3 is discussed in detail and a general technique which applies to any ratio having the structure ofZ is also described. A theoretical example shows that the inverse Mellin transform technique yields the exact density function of a ratio whose density can be obtained by means of the transformation of variables technique. In the second example, the exact density function of a ratio of dependent quardratic forms is evaluated at various points and then compared with simulated values.     

Almost Optimal Differentiation Using Noisy Data

We study differentiation of functionsfbased on noisy dataf(t_i)+_i. We recoverf^(k)either at a single point or on the interval [0, 1] inL₂-norm. Under stochastic assumptions onfand_i, we determine the order of the errors of the best linear methods which use n noisy function values. Polynomial interpolation for the pointwise problem and smoothing splines for the problem inL₂-norm are shown to be almost optimal. The analysis involves worst case estimates in reproducing kernel Hilbert spaces and a Landau inequality.