The main result of the current research describes a monotonicity property of certain invariant tests for the multivariate analysis of variance problem. Suppose $X: r \times p$ has a normal distribution, $EX = \Theta$ and the rows of $X$ are independent, each with unknown covariance matrix $\Sigma: p \times p$. Let $S = p \times p$ have a Wishart distribution $W(\Sigma, p, n), S$ independent of $X$. If $K$ is the acceptance region of an invariant test for the null hypothesis $\Theta = 0$…
Consider the problem of estimating a common mean of two independent normal distributions, each with unknown variances. Note that the problem of recovery of interblock information in balanced incomplete blocks designs is such a problem. Suppose a random sample of size $m$ is drawn from the first population and a random sample of size $n$ is drawn from the second population. We first show that the sample mean of the first population can be improved on (with an unbiased…
Statistics are derived for testing a sequence of observations from an exponential-type distribution for no change in parameter against possible two-sided alternatives involving parameter changes at unknown points. The test statistic can be chosen to have high power against certain of a variety of alternatives. Conditions on functionals on $C\lbrack 0,1\rbrack$ are given under which one can assert that the large sample distribution of the test statistic under the…
A method is derived to place an approximate bound on the mean-square error incurred by using an incorrect covariance matrix in the Gauss-Markov estimator of the coefficient vector in the full-rank general linear model. The bound thus obtained is a function of the incorrect covariance matrix $\tilde{S}$ actually used, the Frobenius norm of $S - \tilde{S}$, where $S$ is the correct covariance matrix, and the basis matrix $\phi$. This estimate can therefore be computed from…
Earlier results on asymptotically optimal sequences of regression designs for autoregressive stochastic processes are extended (nearly) to the equivalence classes of such processes.
Frequency tables are examined in which some cells are not distinguishable. Log-linear models are proposed for these tables which lead to likelihood equations closely related to those associated with log-linear models for conventional frequency tables. Just as in conventional tables, the maximum likelihood equations are shown to be the same under Poisson or multinomial sampling. Applications are made to the problem of estimation of gene frequencies from observed phenotype…
Asymptotic normality of linear rank statistics for testing the hypothesis of independence is established both under fixed alternatives (or the null hypothesis) and under converging alternatives. The results of Ruymgaart, Shorack and van Zwet [14] are used to obtain a further weakening of the smoothness conditions on the score functions. In the present case the score functions are allowed to have a finite number of discontinuities of the first kind. The results of the present…
It is proved that the jackknife estimate $\tilde{\theta} = n\hat{\theta} - (n - 1)(\sum \hat{\theta}_{-i}/n)$ of a function $\theta = f(\beta)$ of the regression parameters in a general linear model $\mathbf{Y} = \mathbf{X\beta} + \mathbf{e}$ is asymptotically normally distributed under conditions that do not require $\mathbf{e}$ to be normally distributed. The jackknife is applied by deleting in succession each row of the $\mathbf{X}$ matrix and $\mathbf{Y}$ vector in order…
For general optimality criteria $\Phi$, criteria equivalent to $\Phi$-optimality are obtained under various conditions on $\Phi$. Such equivalent criteria are useful for analytic or machine computation of $\Phi$-optimum designs. The theory includes that previously developed in the case of $D$-optimality (Kiefer-Wolfowitz) and $L$-optimality (Karlin-Studden-Fedorov), as well as $E$-optimality and criteria arising in response surface fitting and minimax extrapolation.…
Dwyer's (1937) $h$-statistic is extended to the generalized $h$-statistic $h_{p_1\cdots p_u}$ such that $E(h_{p_1\cdots p_u}) = \mu_{p_1} \cdots \mu_{p_u}$, similar to the extension of Fisher's $k$-statistic to the generalized $k$-statistic $k_{p_1\cdots p_u}$ requiring $E(k_{p_1\cdots p_u}) = \kappa_{p_1} \cdots \kappa_{p_u}$. The $h$-statistics follow simpler multiplication rules than for $k$-statistics and involve smaller coefficients. Generalized $h$-statistics are…
This paper provides proof of conjectures about the expected values of the $j$th elementary symmetric function of the roots (i.e. $E(\operatorname{tr}_j A)$) of a non-central Wishart matrix. Utilizing this result we have obtained computationally convenient formula for the expected value of the $j$th elementary symmetric function of the roots of central positive definite quadratic form in matrix argument.
Brunk studied integral regression functions and has obtained strong laws and limiting distributions for estimators of these functions. In this note we will study additional conditions that ensure a rate of convergence of the distribution function of the maximum absolute difference of an integral regression function and its estimator, suitably normalized, to the distribution function of a normalized maximum absolute value of partial sums of random variables. These results are…
The vague statement that "sampling with and without replacement from a finite population are approximately equivalent when the sampling fraction is small" is given a precise meaning in terms of limit theorems for distributions in $R^\infty$ and $D\lbrack 0, \infty)$.
A law of the iterated logarithm is proved for sample $p$-quantiles when the probability distribution function varies regularly at $\\xi$ with $F(\\xi) = p$.
For each $n, X_n(1), \cdots, X_n(n)$ are independent and identically distributed continuous random variables over $(0, 1)$, with common density function equal to $1 + r(x)/n^{\frac{1}{2}}, r(x)$ unknown but satisfying certain regularity conditions. The problem is to test the hypothesis that $r(x) = 0$ for all $x$ in (0, 1). $Y_n(1) < \cdots < Y_n(n)$ are the ordered values of $X_n(1), \cdots, X_n(n). \delta$ is a fixed value in the open interval $(\frac{3}{4}, 1)$. It is…
We feel that a criterion for selecting optimal decision rules in a statistical decision problem should be selected rationally. Specifically, we would like a criterion to satisfy eight properties which we have given. It is known that none of the commonly used criteria satisfy these eight properties. We give sufficient conditions on the decision problem so that the iterated minimax regret criterion does satisfy all eight properties, and give examples to show that these…
We have a population composed of two subpopulations whose probability properties are described by known univariate distribution functions, $G(x)$ and $H(x)$, respectively. The probability of observing an individual from the first population is $\theta$, from the second is $1 - \theta$. We assume $\theta$ is a random variable with a prior distribution on (0, 1) and find the Bayes rule for classifying $n$ observations as from $G$ or from $H$ when the loss function is equal to…
Berk (1970), LeCam (1953) and others have given conditions for the consistency of posterior distributions from a sequence of random variables. They have required that the sequence be i.i.d. We show that their results, Berk's in particular, may be extended to the general linear hypothesis with normal errors model (where the sequence of observations of the dependent variable need not be i.i.d.). We do not assume that the distribution governing the sequence of dependent…
This paper considers linear functions of order statistics of the form $S_n = n^{-1} \sum J(i/(n + 1))X_{(i)}$. The main results are that $S_n$ is asymptotically normal if the second moment of the population is finite and $J$ is bounded and continuous a.e. $F^{-1}$, and that this first result continues to hold even if the unordered observations are not identically distributed. The moment condition can be discarded if $J$ trims the extremes. In addition, asymptotic formulas…
General theorems on asymptotic normality of randomly trimmed and Winsorized means are obtained. A new small sample studentization is proposed. Many examples are presented. The two sample problem is also considered.
This paper proposes and develops a method for selecting a design to estimate a set of linear parametric functions in cases wherein the adequacy of the preliminary linear model is in doubt. The proposed method relies on the norm of the aliasing matrix. If the expected value of the estimator $\hat{\psi}$ of a set of linear functions $\psi = L_1\theta_1$ using a design $\Gamma$, under the true model is $E\lbrack\hat{\psi}\rbrack = \psi + A_\Gamma \theta_2$, then the norm $\|A_\Gamma\| =$…
The paper gives convergence theorems for several sequential Monte-Carlo or stochastic approximation algorithms for finding a local minimum of a function $f(\bullet)$ on a set $C$ defined by $C = \{x: q^i(x) \leqq 0, i = 1, 2, \cdots, s\}. f(\bullet)$ is unknown, but "noise perturbed" values can be observed at any desired parameter $x \in C$. The algorithms generate a sequence of random variables $\{X_n\}$ such that (for a.a. $\omega$) any convergent subsequence of $\{X_n(\omega)\}$…
Let $Y_1, \cdots, Y_n$ be independent random variables with mean zero such that $|Y_i| \leqq i, i = 1, \cdots, n$, and let $\theta_1,\cdots, \theta_n$ be real numbers satisfying $\sum^n_1 \theta_i^2 = 1$. Set $S_n(\theta) = \sum^n_1 \theta_i Y_i$ and let $\varphi(x) = (2\pi)^{-\frac{1}{2}} \exp \lbrack - \frac{1}{2} x^2\rbrack$. THEOREM. For $\alpha > 0$, and for all $\theta_1,\cdots, \theta_n$, \begin{\align*}P\{|S_n(\theta)| \geqq \alpha\} &\leqq 2\inf_{0\leqq u \leqq…
The main result of the current research describes a monotonicity property of certain invariant tests for the multivariate analysis of variance problem. Suppose X:r×p has a normal distribution, EX=Θ and the rows of X are independent, each with unknown covariance matrix Σ:p×p. Let S=p×p have a Wishart distribution W(Σ,p,n),S independent of X. If K is the acceptance region of an invariant test for the null hypothesis Θ=0…
Consider the problem of estimating a common mean of two independent normal distributions, each with unknown variances. Note that the problem of recovery of interblock information in balanced incomplete blocks designs is such a problem. Suppose a random sample of size m is drawn from the first population and a random sample of size n is drawn from the second population. We first show that the sample mean of the first population can be improved on (with an unbiased…
Statistics are derived for testing a sequence of observations from an exponential-type distribution for no change in parameter against possible two-sided alternatives involving parameter changes at unknown points. The test statistic can be chosen to have high power against certain of a variety of alternatives. Conditions on functionals on C[0,1] are given under which one can assert that the large sample distribution of the test statistic under the…
A method is derived to place an approximate bound on the mean-square error incurred by using an incorrect covariance matrix in the Gauss-Markov estimator of the coefficient vector in the full-rank general linear model. The bound thus obtained is a function of the incorrect covariance matrix S~ actually used, the Frobenius norm of S−S~, where S is the correct covariance matrix, and the basis matrix ϕ. This estimate can therefore be computed from…
It is proved that the jackknife estimate θ~=nθ^−(n−1)(∑θ^−i/n) of a function θ=f(β) of the regression parameters in a general linear model Y=Xβ+e is asymptotically normally distributed under conditions that do not require e to be normally distributed. The jackknife is applied by deleting in succession each row of the X matrix and Y vector in order…
For general optimality criteria Φ, criteria equivalent to Φ-optimality are obtained under various conditions on Φ. Such equivalent criteria are useful for analytic or machine computation of Φ-optimum designs. The theory includes that previously developed in the case of D-optimality (Kiefer-Wolfowitz) and L-optimality (Karlin-Studden-Fedorov), as well as E-optimality and criteria arising in response surface fitting and minimax extrapolation.…
Dwyer's (1937) h-statistic is extended to the generalized h-statistic hp1⋯pu such that E(hp1⋯pu)=μp1⋯μpu, similar to the extension of Fisher's k-statistic to the generalized k-statistic kp1⋯pu requiring E(kp1⋯pu)=κp1⋯κpu. The h-statistics follow simpler multiplication rules than for k-statistics and involve smaller coefficients. Generalized h-statistics are…
This paper provides proof of conjectures about the expected values of the jth elementary symmetric function of the roots (i.e. E(trjA)) of a non-central Wishart matrix. Utilizing this result we have obtained computationally convenient formula for the expected value of the jth elementary symmetric function of the roots of central positive definite quadratic form in matrix argument.
The vague statement that "sampling with and without replacement from a finite population are approximately equivalent when the sampling fraction is small" is given a precise meaning in terms of limit theorems for distributions in R∞ and D[0,∞).
A law of the iterated logarithm is proved for sample p-quantiles when the probability distribution function varies regularly at xi with F(xi)=p.
For each n,Xn(1),⋯,Xn(n) are independent and identically distributed continuous random variables over (0,1), with common density function equal to 1+r(x)/n21,r(x) unknown but satisfying certain regularity conditions. The problem is to test the hypothesis that r(x)=0 for all x in (0, 1). Yn(1)<⋯<Yn(n) are the ordered values of Xn(1),⋯,Xn(n).δ is a fixed value in the open interval (43,1). It is…
We have a population composed of two subpopulations whose probability properties are described by known univariate distribution functions, G(x) and H(x), respectively. The probability of observing an individual from the first population is θ, from the second is 1−θ. We assume θ is a random variable with a prior distribution on (0, 1) and find the Bayes rule for classifying n observations as from G or from H when the loss function is equal to…
This paper considers linear functions of order statistics of the form Sn=n−1∑J(i/(n+1))X(i). The main results are that Sn is asymptotically normal if the second moment of the population is finite and J is bounded and continuous a.e. F−1, and that this first result continues to hold even if the unordered observations are not identically distributed. The moment condition can be discarded if J trims the extremes. In addition, asymptotic formulas…
This paper proposes and develops a method for selecting a design to estimate a set of linear parametric functions in cases wherein the adequacy of the preliminary linear model is in doubt. The proposed method relies on the norm of the aliasing matrix. If the expected value of the estimator ψ^ of a set of linear functions ψ=L1θ1 using a design Γ, under the true model is E[ψ^]=ψ+AΓθ2, then the norm ∥AΓ∥=…
The paper gives convergence theorems for several sequential Monte-Carlo or stochastic approximation algorithms for finding a local minimum of a function f(∙) on a set C defined by C={x:qi(x)≦0,i=1,2,⋯,s}.f(∙) is unknown, but "noise perturbed" values can be observed at any desired parameter x∈C. The algorithms generate a sequence of random variables {Xn} such that (for a.a. ω) any convergent subsequence of {Xn(ω)}…
Let Y1,⋯,Yn be independent random variables with mean zero such that ∣Yi∣≦i,i=1,⋯,n, and let θ1,⋯,θn be real numbers satisfying ∑1nθi2=1. Set Sn(θ)=∑1nθiYi and let φ(x)=(2π)−21exp[−21x2]. THEOREM. For α>0, and for all θ1,⋯,θn, \begin{\align*}P\{|S_n(\theta)| \geqq \alpha\} &\leqq 2\inf_{0\leqq u \leqq…