Let $P_\eta, \eta \in \Theta \times \Gamma \subset \mathbb{R} \times \mathbb{R}^k$, be an exponential family. It is shown that the sequence of tests $(\varphi^\ast_n)_{n\in\mathbb{N}}$, where $\varphi^\ast_n, n \in \mathbb{N}$, is u.m.p. in the class of all tests similar with respect to the nuisance-parameter $\gamma$ for the hypothesis $\{P^n_{(\theta, \gamma)}: \gamma \in \Gamma\}$ against alternatives $P^n_{(\theta_1, \eta_1)}, \theta_1 > \theta, \eta_1 \in \Gamma$, is…
The notion of Bahadur efficiency is used to compare multivariate linear hypothesis tests based on six criteria: (1) Roy's largest root, (2) the likelihood ratio test, (3) the Lawley-Hotelling trace, (4) Pillai's trace, (5) Wilks' $U$, and (6) Olson's statistic. Bahadur exact slope is computed for each statistic as a function of noncentrality parameters using results for probabilities of large deviations. The likelihood ratio test is shown to be asymptotically optimal in the…
This paper considers Robbins-Monro stochastic approximation when the regression function changes with time. At time $n$, one can select $X_n$ and observe an unbiased estimator of the regression function evaluated at $X_n$. Let $\theta_n$ be the root of the regression function at time $n$. Our goal is to select the sequence $X_n$ so that $X_n - \theta_n$ converges to 0. It is assumed that $\theta_n = f(s_n)$ for $s_n$ known at time $n$ and $f$ an unknown element of a class of…
A rotationally invariant exponential model, which includes the Fisher-von Mises and Bingham distributions as special cases, is proposed for directional data in $R^p(p \geqslant 2)$. A new regression estimator for the model parameters is developed as a competitor to the maximum likelihood estimator. Both the new estimator and the MLE are asymptotically efficient at the postulated model and are robust under small departures from that model. Computationally, the regression…
For linear rank tests in the two-sample case the concept of approximate Bahadur efficiency (BE) is developed, and as the main result of this paper the equality of the approximate and exact local BE is shown. According to a result of Wieand, local approximate BE equals Pitman efficiency under rather general conditions and as a consequence these three approaches to efficiency generally coincide for the class of linear rank tests.
When $y = M(x) + \varepsilon$, where $M$ may be nonlinear, adaptive stochastic approximation schemes for the choice of the levels $x_1, x_2, \cdots$ at which $y_1, y_2, \cdots$ are observed lead to asymptotically efficient estimates of the value $\theta$ of $x$ for which $M(\theta)$ is equal to some desired value. More importantly, these schemes make the "cost" of the observations, defined at the $n$th stage to be $\sum^n_1(x_i - \theta)^2$, to be of the order of $\log n$ ins…
A family of estimators which dominate the maximum likelihood estimators of regression coefficients is given when the dependent variable and (3 or more) independent variables have a joint normal distribution.
Let $X_1, \cdots, X_n$ be a sequence of independent random vectors taking values in $\mathbf{\mathbb{R}}^d$ with a common probability density $f$. If $f_n(x) = (1/h) h^{-d}_n\sum^n_{i = 1}K((x - X_i)/h_n)$ is the kernel estimate of $f$ from $X_1, \cdots, X_n$ then conditions on $K$ and $\{h_n\}$ are given which insure that $\int|f_n(x) - f(x)|dx \rightarrow_n 0$ in probability or with probability one. No continuity conditions are imposed on $f$.
For the two-action two-state compound decision problem, a class of decision rules is found that are both asymptotically optimal and admissible at each stage $n$. These rules are Bayes rules with respect to appropriate exchangeable prior distributions.
The monotone regression of a variable $X$ on another variable $Y$ is of particular interest when $Y$ cannot be directly observed. The correlation of $X$ and $Y$ can be tested if at least high and low values of $Y$ can be recognized. If all the components of a random vector have monotone regression on a variable $Y$, and if they are all uncorrelated given $Y$, then an inequality due to Chebyshev shows that marginal zero covariances imply that all but at most one of the…
Consider the realization of the process $y(t) = \Sigma^n_{k=1}\theta_kf_k(t) + \xi(t)$ on the interval $T = \lbrack 0, 1\rbrack$ for functions $f_1(t), f_2(t), \cdots, f_n(t)$ in $H(R)$, the reproducing kernel Hilbert space with reproducing kernel $R(s, t)$ on $T \times T$, where $R(s, t) = E\xi(s)\xi(t)$ is assumed to be continuous and known. Problems of the selection of functions $\{f_k(t)\}^n_{k=1}$ are discussed for $D$-optimal, $A$-optimal and other criteria of optimal…
S. Rincon-Gallardo · C. P. Quesenberry · Federico J. O'Reilly·刊期 1979-09-01 · 卷 7（5）·The Annals of Statistics已编入正刊
多元正态分布的条件概率积分变换与拟合优度检验AI 解读
Let $X_1, \cdots, X_n$ be a random sample from a full-rank multivariate normal distribution $N(\mu, \Sigma)$. The two cases (i) $\mu$ unknown and $\Sigma = \sigma^2\Sigma_0, \Sigma_0$ known, and (ii) $\mu$ and $\Sigma$ completely unknown are considered here. Transformations are given that transform the observation vectors to a (smaller) set of i.i.d. uniform rv's. These transformations can be used to construct goodness-of-fit tests for these multivariate normal…
The problems of estimating sequentially the intensity parameter of a homogeneous Poisson process and of estimating sequentially the mean of a sequence of i.i.d. Poisson rv's, are considered. The procedures suggested are shown to perform well for large values of the parameter and/or for small sampling cost. Having bounded regret, the procedure for estimating the mean of the Poisson sequence is asymptotically Bayes w.r.t. any sequence of a priori densities, which spread mass…
We give a decision-theoretic justification of a heuristic principle suggested by Robbins. Suppose a sequence $X_1, X_2, \cdots$ of independent random variables can be observed; they are known to be identically distributed with unknown mean and variance $u, \sigma^2(u)$ respectively. We assume further that the common distribution is of exponential type, so that the sample mean $\bar{X}$ is a sufficient statistic. The problem is to estimate $u$ taking into account the cost of…
Along with the asymptotic distribution, expressions for the asymptotic bias and asymptotic dispersion matrix of the preliminary test maximum likelihood estimator for a general multi-sample parametric model (when the null hypothesis relating to the restraints on the parameters may not hold) are derived and compared with the parallel expressions for the unrestricted and restricted maximum likelihood estimators. This study reveals the robustness property of the preliminary test…
A method is presented for the construction of all families of smallest simultaneous confidence sets (SCS) in a given class, for a family $\{\psi_i(\gamma)\}$ of parametric functions of the parameter of interest $\gamma = \gamma(\theta)$. The method is applied to the MANOVA problem (in its canonical form) of inference about $M = EX$, where $X$ is $q \times p$ and has rows that are independently multivariate normal with common covariance matrix $\Sigma$. Let $S$ be the usual…
When samples are taken independently from different populations, the sample means are jointly admissible for the population means with the squared error as loss function. The result supplements a previous result that when there are many variate values associated with the population units, the sample means of the variate values are jointly admissible for the population means for general loss functions.
Questions of admissibility of statistical estimators are reduced to considerations involving differential inequalities. The coefficients of these inequalities involve moments of the underlying distributions; and so are, in principle, not difficult to derive. The methods are "heuristic" because it is necessary to verify on an ad-hoc basis that error terms are small. Some conditions on the structure of the problem are given which we believe will guarantee that these error…
The functional $T(F) = \int F^{-1}(t)J(t) dt$ associated with linear combinations of order statistics is shown to have a Frechet-type differential. As a corollary, the statistic $T(F_n)$ obtained by evaluating $T(\cdot)$ at the sample df $F_n$ is seen to be asymptotically normal and to obey a law of the iterated logarithm.
In this paper we consider adaptive one-sample rank tests of the following type: the score function $J$ of the test is estimated from the sample under the restriction that $J \in \mathscr{J}$, for some given one-parameter family $\mathscr{J} = \{J_r, r \in I \subset R^1\}$. Using deficiencies, we compare the performance of such tests to that of rank tests with fixed scores. Conditions on the estimator $S$ of the parameter $r$ and on $J_r$ are given, under which the deficiency…
Let $X_1, \cdots, X_n$ be an ordered sample from a distribution $A_n$ on [0, 1]. The $k$-spacings $D_1(N, k), \cdots, D_N(N, k)$ are defined and the weak convergence of their empirical distribution function under a sequence of alternatives $A_n$ approaching the uniform distribution is established. This is then applied to find the limiting distribution of $W_n(g, k) = N^{-\frac{1}{2}}\Sigma^N_{i=l}(g(NkD_i(N, k)) - a)$ where $g$ is a smooth function and $k$ is fixed. The…
The weak convergence of the Kaplan-Meier estimate for censored survival data to a Gaussian process is used to construct asymptotic confidence bands for the survival curves. The method involves transforming to obtain Brownian motion and using straight line boundaries and hitting probabilities for Brownian motion.
Asymptotically optimal (a.o.) empirical Bayes (EB) estimators are proposed. Speeds and the best possible speed at which these estimators are a.o. are investigated. The underlying component problem is the squared error loss estimation of $\theta$ based on an observation $X$ whose conditional (on $\theta$) $\operatorname{pdf}$ is of the form $u(x)C(\theta)\exp(\theta x)$. The function $u$ could have infinitely many discontinuities; $\theta$ is distributed according to an…
Let Pη,η∈Θ×Γ⊂R×Rk, be an exponential family. It is shown that the sequence of tests (φn∗)n∈N, where φn∗,n∈N, is u.m.p. in the class of all tests similar with respect to the nuisance-parameter γ for the hypothesis {P(θ,γ)n:γ∈Γ} against alternatives P(θ1,η1)n,θ1>θ,η1∈Γ, is…
The notion of Bahadur efficiency is used to compare multivariate linear hypothesis tests based on six criteria: (1) Roy's largest root, (2) the likelihood ratio test, (3) the Lawley-Hotelling trace, (4) Pillai's trace, (5) Wilks' U, and (6) Olson's statistic. Bahadur exact slope is computed for each statistic as a function of noncentrality parameters using results for probabilities of large deviations. The likelihood ratio test is shown to be asymptotically optimal in the…
This paper considers Robbins-Monro stochastic approximation when the regression function changes with time. At time n, one can select Xn and observe an unbiased estimator of the regression function evaluated at Xn. Let θn be the root of the regression function at time n. Our goal is to select the sequence Xn so that Xn−θn converges to 0. It is assumed that θn=f(sn) for sn known at time n and f an unknown element of a class of…
A rotationally invariant exponential model, which includes the Fisher-von Mises and Bingham distributions as special cases, is proposed for directional data in Rp(p⩾2). A new regression estimator for the model parameters is developed as a competitor to the maximum likelihood estimator. Both the new estimator and the MLE are asymptotically efficient at the postulated model and are robust under small departures from that model. Computationally, the regression…
When y=M(x)+ε, where M may be nonlinear, adaptive stochastic approximation schemes for the choice of the levels x1,x2,⋯ at which y1,y2,⋯ are observed lead to asymptotically efficient estimates of the value θ of x for which M(θ) is equal to some desired value. More importantly, these schemes make the "cost" of the observations, defined at the nth stage to be ∑1n(xi−θ)2, to be of the order of logn ins…
The L1 Convergence of Kernel Density Estimates
Let X1,⋯,Xn be a sequence of independent random vectors taking values in Rd with a common probability density f. If fn(x)=(1/h)hn−d∑i=1nK((x−Xi)/hn) is the kernel estimate of f from X1,⋯,Xn then conditions on K and {hn} are given which insure that ∫∣fn(x)−f(x)∣dx→n0 in probability or with probability one. No continuity conditions are imposed on f.
For the two-action two-state compound decision problem, a class of decision rules is found that are both asymptotically optimal and admissible at each stage n. These rules are Bayes rules with respect to appropriate exchangeable prior distributions.
The monotone regression of a variable X on another variable Y is of particular interest when Y cannot be directly observed. The correlation of X and Y can be tested if at least high and low values of Y can be recognized. If all the components of a random vector have monotone regression on a variable Y, and if they are all uncorrelated given Y, then an inequality due to Chebyshev shows that marginal zero covariances imply that all but at most one of the…
Let Yt satisfy the stochastic difference equation Yt=∑j=1pηjYt−j+et for t=1,2,⋯, where the et are independent identically distributed (0,σ2) random variables and the initial conditions (Y−p+1,Y−p+2,⋯,Y0) are fixed constants. It is assumed the true, but unknown, roots m1,m2,⋯,mp of mp−∑j=1pηjmp−j=0 satisfy m1=m2=1 and ∣mj∣<1 for j=3,4,⋯,p.…
Consider the realization of the process y(t)=Σk=1nθkfk(t)+ξ(t) on the interval T=[0,1] for functions f1(t),f2(t),⋯,fn(t) in H(R), the reproducing kernel Hilbert space with reproducing kernel R(s,t) on T×T, where R(s,t)=Eξ(s)ξ(t) is assumed to be continuous and known. Problems of the selection of functions {fk(t)}k=1n are discussed for D-optimal, A-optimal and other criteria of optimal…
Let X1,⋯,Xn be a random sample from a full-rank multivariate normal distribution N(μ,Σ). The two cases (i) μ unknown and Σ=σ2Σ0,Σ0 known, and (ii) μ and Σ completely unknown are considered here. Transformations are given that transform the observation vectors to a (smaller) set of i.i.d. uniform rv's. These transformations can be used to construct goodness-of-fit tests for these multivariate normal…
We give a decision-theoretic justification of a heuristic principle suggested by Robbins. Suppose a sequence X1,X2,⋯ of independent random variables can be observed; they are known to be identically distributed with unknown mean and variance u,σ2(u) respectively. We assume further that the common distribution is of exponential type, so that the sample mean Xˉ is a sufficient statistic. The problem is to estimate u taking into account the cost of…
A method is presented for the construction of all families of smallest simultaneous confidence sets (SCS) in a given class, for a family {ψi(γ)} of parametric functions of the parameter of interest γ=γ(θ). The method is applied to the MANOVA problem (in its canonical form) of inference about M=EX, where X is q×p and has rows that are independently multivariate normal with common covariance matrix Σ. Let S be the usual…
The functional T(F)=∫F−1(t)J(t)dt associated with linear combinations of order statistics is shown to have a Frechet-type differential. As a corollary, the statistic T(Fn) obtained by evaluating T(⋅) at the sample df Fn is seen to be asymptotically normal and to obey a law of the iterated logarithm.
In this paper we consider adaptive one-sample rank tests of the following type: the score function J of the test is estimated from the sample under the restriction that J∈J, for some given one-parameter family J={Jr,r∈I⊂R1}. Using deficiencies, we compare the performance of such tests to that of rank tests with fixed scores. Conditions on the estimator S of the parameter r and on Jr are given, under which the deficiency…
On the Asymptotic Distribution of k-Spacings with Applications to Goodness-of-Fit Tests
Let X1,⋯,Xn be an ordered sample from a distribution An on [0, 1]. The k-spacings D1(N,k),⋯,DN(N,k) are defined and the weak convergence of their empirical distribution function under a sequence of alternatives An approaching the uniform distribution is established. This is then applied to find the limiting distribution of Wn(g,k)=N−21Σi=lN(g(NkDi(N,k))−a) where g is a smooth function and k is fixed. The…
Asymptotically optimal (a.o.) empirical Bayes (EB) estimators are proposed. Speeds and the best possible speed at which these estimators are a.o. are investigated. The underlying component problem is the squared error loss estimation of θ based on an observation X whose conditional (on θ) pdf is of the form u(x)C(θ)exp(θx). The function u could have infinitely many discontinuities; θ is distributed according to an…