Uncorrelated residuals from linear models

In the usual linear model $\text{y}\text{=}\text{Xβ}\text{+}\text{u}$, the error vector u is not observable and the vector r of least squares residuals has a singular covariance matrix that depends on the design matrix X. We approximate u by a vector$\text{r}{}^1\text{=}\text{G}\text{(}\text{JA'y}\text{+}\text{Kz}\text{)}$ of uncorrelated ‘residuals’, where G and $\text{(}\text{J, K}\text{)}$ are orthogonal matrices, $\text{A'X}\text{=}\text{0}$ and $\text{A'A}\text{=}\text{I}$, while z is either 0 or a random vector uncorrelated with u satisfying $\text{E(}\text{z}\text{) = E(}\text{J'u}\text{) =}\text{0}$, $\text{V(}\text{z}\text{) = V(}\text{J'u}\text{) = σ}{}^2\text{I}$. We prove that $\text{r}{}^1\text{-}\text{r}$ is uncorrelated with $\text{r}\text{-}\text{u}$, for any such r¹, extending the results of Neudecker (1969). Building on results of Hildreth (1971) and Tiao and Guttman (1967), we show that the BAUS residual vector $\text{r}{}_{\mathrm{h}}\text{=}\text{r}\text{+}\text{P}{}_{\mathrm{<mtext>1</mtext>}}\text{z}$, where P₁ is an orthonormal basis for X, minimizes each characteristic root of $\text{V(}\text{r}{}^1\text{-}\text{u}\text{)}$, while the vector r_b of Theil's BLUS residuals minimizes each characteristic root of $\text{V(}\text{Jr}{}_{\mathrm{a}}\text{-}\text{r}\text{)}$, cf. Grossman and Styan (1972). We find that $\text{tr V(}\text{r}{}_{\mathrm{h}}\text{-}\text{u}\text{) < tr V(}\text{Jr}{}_{\mathrm{b}}\text{-}\text{u}\text{)}$ if and only if the average of the singular values of $\text{P}\text{′}{}_1\text{K}$ is less than $\text{1}\text{2}$, and give examples to show that BAUS is often better than BLUS in this sense.     

Continua of stochastic dominance relations for bounded probability distributions

Let P={F,G,…} be the set of probability distribution functions on [0,b]. For each αε[1, ∞), F_αG means that ∫^x_o(x−y^α−1dF(y)∫^x_o(x−y)^α−1dG(y) for all xε[0, b], and F>_αG means that F_αG and F≠G. Each _α is reflexive and transitive and each>_α is asymmetric and transitive. Both _α and>_α increase as α increases but their limits are not complete. A class U_α of utility functions is defined to give F>_αG iff ∫udF>∫udG for all uεU_α. These classes decrease as α increases, and their limit is empty. Similar decreasing classes are defined for each _α, and their limit is essentially the constant functions on (0, b].     

Locally minimax test of independence in elliptically symmetrical distributions with additional observations

Summary LetX=(X _ij )=(X ₁, ...,X _n )’,X’ _i =(X _i1, ...,X _ip )’,i=1,2, ...,n be a matrix having a multivariate elliptical distribution depending on a convex functionq with parameters, 0,σ. Let ϱ²=ϱ ₂ ^-2 be the squared multiple correlation coefficient between the first and the remainingp ₂+p ₃=p−1 components of eachX _i . We have considered here the problem of testingH ₀:ϱ²=0 against the alternativesH ₁:ϱ ₁ ^-2 =0, ϱ ₂ ^-2 >0 on the basis ofX andn ₁ additional observationsY ₁ (n ₁×1) on the first component,n ₂ observationsY ₂(n ₂×p ₂) on the followingp ₂ components andn ₃ additional observationsY ₃(n ₃×p ₃) on the lastp ₃ components and we have derived here the locally minimax test ofH ₀ againstH ₁ when ϱ ₂ ^-2 →0 for a givenq. This test, in general, depends on the choice ofq of the familyQ of elliptically symmetrical distributions and it is not optimality robust forQ.     

Inequalities for the Gini coefficient of composite populations

Unlike other popular measures of income inequality, the Gini coefficient is not decomposable, i.e., the coefficient G($\text{X}$) of a composite population $\text{X}\text{=}\text{X}{}_1\text{∪…∪}\text{X}{}_{\mathrm{r}}$ cannot be computed in terms of the sizes, mean incomes and Gini coefficients of the components $\text{X}$_i. In this paper upper and lower bounds (best possible for r=2) for G($\text{X}$) in terms of these data are given. For example, $\text{G(}\text{X}{}_1\text{∪…∪}\text{X}{}_{\mathrm{r}}\text{)≧Σα}{}_{\mathrm{i}}\text{G(}\text{X}{}_{\mathrm{i}}\text{)}$, where α_i is the proportion of the pop ulation in $\text{X}$_i. One of the tools used, which may be of interest for other applications, is a lower bound for ∫^∞₀f(x)g(x)dx (converse to Cauchy's inequality) for monotone decreasing functions f and g.     

Strong consistency of the MLE under random censoring

LetX ₁, ...,X _n be an i.i.d. sample from some parametric family {_θ :θ (Θ} of densities. In the random censorship model one observesZ _i =min (X _i ,Y _i ) andδ _i =1_{ x _i ≤Y _i}, whereY _i is a censoring variable being independent ofX _i . In this paper we investigate the strong consistency ofθ _n maximizing the modified likelihood function based on (Z _i ,δ _i , 1≤i≤n. The main result constitutes an extension of Wald’s theorem for complete data to censored data. Work partially supported by the “Deutsche Forschungsgemeinschaft”.     

Identification in models with autoregressive errors

Consider the model

$\text{A(L)x}{}_{\mathrm{t}}\text{=B(L)y}{}_{\mathrm{t}}\text{+C(L)z}{}_{\mathrm{t}}\text{=u}{}_{\mathrm{t}}\text{, t=1,…,T}$

, where

$\text{A(L)=(B(L):C(L))}$

is a matrix of polynomials in the lag operator so that $\text{L}{}^{\mathrm{r}}\text{x}{}_{\mathrm{t}}\text{=x}{}_{\mathrm{t?r}}$, and $\text{y}{}_{\mathrm{t}}$ is a vector of n endogenous variables,

$\text{B(L)=}\text{∑}\text{s=0}\text{k}\text{B}{}_{\mathrm{s}}\text{L}{}^{\mathrm{s}}$

$\text{B}{}_0\text{I}{}_{\mathrm{n}}$, and the remaining $\text{B}{}_{\mathrm{s}}$ are $\text{n × n}$ square matrices,

$\text{C(L)=}\text{∑}\text{s=0}\text{k}\text{C}{}_{\mathrm{s}}\text{L}{}^{\mathrm{s}}$

, and $\text{C}{}_{\mathrm{s}}$ is $\text{n × m}$.Suppose that $\text{u}{}_{\mathrm{t}}$ satisfies

$\text{R(L)u}{}_{\mathrm{t}}\text{=e}{}_{\mathrm{t}}$

, where

$\text{R(L)=}\text{∑}\text{s=0}\text{r}\text{R}{}_{\mathrm{s}}\text{L}{}^{\mathrm{s}}$

, $\text{R}{}_0\text{=I}{}_{\mathrm{n}}$, and $\text{R}{}_{\mathrm{s}}$ is a $\text{n × n}$ square matrix. $\text{e}{}_{\mathrm{t}}$ may be white noise, or generated by a vector moving average stochastic process.Now writing

$\text{Ψ(L)=R(L)A(L)}$

, it is assumed that ignoring the implicit restrictions which follow from eq. (1), $\text{Ψ(L)}$ can be consistently estimated, so that if the equation

$\text{Ψ(L)x}{}_{\mathrm{t}}\text{=e}{}_{\mathrm{t}}$

has a moving average error stochastic process, suitable conditions [see E.J. Hannan] for the identification of the unconstrained model are satisfied, and that the appropriate conditions (lack of multicollinearity) on the data second moments matrices discussed by Hannan are also satisfied. Then the essential conditions for identification of the A(L) and R(L) can be considered by requiring that for the true $\text{Ψ(L)}$ eq. (1) has a unique solution for $\text{A(L)}$ and $\text{R(L)}$.There are three types of lack of identification to be distinguished. In the first there are a finite number of alternative factorisations. Apart from a factorisation condition which will be satisfied with probability one a necessary and sufficient condition for lack of identification is that $\text{A(L)}$ has a latent root λ in the sense that for some non-zero vector β,

$\text{β′A(λ)=0}$

.The second concept of lack of identification corresponds to the Fisher conditions for local identifiability on the derivatives of the constraints. It is shown that a necessary and sufficient condition that the model is locally unidentified in this sense is that R(L) and A(L) have a common latent root, i.e., that for some vectors δ and β,

$\text{R(λ)δ=0 and β′A(λ)=0}$

.Firstly it is shown that only if further conditions are satisfied will this lead to local unidentifiability in the sense that there are solutions of the equation

$\text{Ψ(z)=R(z)A(z)}$

in any neighbourhood of the true values.     

Multivariate L1 mean

The center of a univariate data set {x ₁,…,x _n} can be defined as the point μ that minimizes the norm of the vector of distances y′=(|x ₁−μ|,…,|x _n−μ|). As the median and the mean are the minimizers of respectively the L ₁- and the L ₂-norm of y, they are two alternatives to describe the center of a univariate data set. The center μ of a multivariate data set {x ₁,…,x _n} can also be defined as minimizer of the norm of a vector of distances. In multivariate situations however, there are several kinds of distances. In this note, we consider the vector of L ₁-distances y′₁=(∥x ₁- μ∥₁,…,∥x _n- μ∥₁) and the vector of L ₂-distances y′₂=(∥x ₁- μ∥₂,…,∥x _n-μ∥₂). We define the L ₁-median and the L ₁-mean as the minimizers of respectively the L ₁- and the L ₂-norm of y ₁; and then the L ₂-median and the L ₂-mean as the minimizers of respectively the L ₁- and the L ₂-norm of y ₂. In doing so, we obtain four alternatives to describe the center of a multivariate data set. While three of them have been already investigated in the statistical literature, the L ₁-mean appears to be a new concept. Received January 1999     

The Perron-Frobenius theorem without additivity

We prove the following non-linear generalization of the Perron-Frobenius theorem. Let A:R^m₊→R^m₊ be continuous, homogeneous of degree 1 and primitive (i.e., for some integer l, x≥yA^lxA^ly); then A has a positive eigenvector x₀, unique up to multiplication by a positive scalar, and for all x0, Aⁿx/|Aⁿx| converges to x₀/|x₀|.     

Largest excess of boundary crossings for martingales

Summary LetX ₁,X ₂, ... form a sequence of martingale differences and denote byZ(a, α) = sup n (S _n −an ^α)⁺ the largest excess forS _n =X ₁ + ... +X _n crossing the boundaryan ^α. We give a sufficient condition for the finiteness ofEZ(a, α)^β which is formulated in terms of bounds forE(X _i ^{+ p} andE(|X _i |γ|X ₁, ...,X _i-1), whereα, β, γ, p are suitably related. This general result is then applied to the case of independent random variables.     

Correcting for truncation bias caused by a latent truncation variable

We discuss estimation of the model Y_i=X_ib_Y+e_Yi, T_i=X_ib_T+ e_Ti, when data on the continuous dependent variable Y and on the independent variables X are observed iff the ‘truncation variable’ T>0 and when T is latent. This case is distinct from both (i) the‘censored sample’ case, in which Y data are available iff T>0, T is latent and X data are available for all observations, and (ii) the ‘observed truncation variable’ case, in which both Y and X are observed iff T>0 and in which the actual value of T is observed whenever T>0. We derive a maximum-likelihood procedure for estimating this model and discuss identification and estimation.     

Determination of uniformly most powerful tests in discrete sample spaces

Uniformly most powerful (UMP) tests are known to exist in one-parameter exponential families when the hypothesis H ₀ and the alternative hypothesis H ₁ are given by (i) H ₀ : θ≤θ₀, H ₁ : θ>θ₀, and (ii) H ₀ : θ≤θ₁ or θ≥θ₂, H ₁ : θ₁<θ<θ₂, where θ₁<θ₂.  Likewise, uniformly most powerful unbiased (UMPU) tests do exist when the hypotheses H ₀ and H ₁ take the form (iii) H ₀ : θ₁≤θ≤θ₂, H ₁ : θ<θ₁ or θ>θ₂, where θ₁<θ₂, and (iv) H ₀ : θ=θ₀, H ₁:θ≠θ₀.  To determine tests in case (i), only one critical value c and one randomization constant γ have to be computed. In cases (ii) through (iv) tests are determined by two critical values c ₁, c ₂ and two randomization constants γ₁, γ₂. Unlike determination of tests in case (i), computation of critical values and randomization constants in the remaining cases is rather difficult, unless distributions are symmetric. No straightforward method to determine two-sided UMP tests in discrete sample spaces seems to be known. The purpose of this note is to disclose a distribution independent principle for the determination of UMP tests in cases (ii) through (iv). Received: March 1999