Let $\mu$ be a continuous mean regression function defined on $U$, the unit cube in $N$-dimensional Euclidean space. Let $F$ be a distribution function with support in $U$, and let $M$ denote the indefinite integral of $\mu$ with respect to $F$. This paper provides consistency results, including rates of convergence, for a certain estimator of $M$ in the case that the $n$th estimate is based on observations at points $\mathbf{t}_{n1},\cdots, \mathbf{t}_{nn}$ of $U$. The…
Arnold (1973) studied testing problems where the covariance matrix is assumed to have the generalized correlation structure under both hypotheses. That paper showed how to transform such problems to "products" of unpatterned problems. This paper extends those results to testing problems where the covariance matrix is assumed to have Geisser's (1963) generalization of the pattern of compound symmetry. We prove theorems indicating how to transform such problems to products of…
The problem of estimating the mean of a $p$-variate $(p \geqq 3)$ normal distribution is considered. It is assumed that the covariance matrix $\not\sum$ is known and that the loss function is quadratic. A class of minimax estimators is given, out of which admissible minimax estimators are developed.
Kendall's rank correlation statistic $T_n = \sum i > j \operatorname{sgn} (X_i - X_j)\cdot$ sgn $(Y_i - Y_j)$ is well known to be asymptotically normally distributed under the null hypothesis of independence as the sample size $n\rightarrow\infty$. In the note it is shown that this assertion can be obtained easily from the recurrence formula $p_n(t) = (1/n) \sum^n_{k =1}p_{n - 1}(t - 2k + n + 1)$ for the probability distribution $p_n$ of $T_n$ (see Kendall (1970), e.g.).…
As a "robust" alternative to the least squares estimates for a regression parameter, Koul (1969) proposed new estimates based on signed-rank statistics. To find out their asymptotic distribution Koul proved that under quite general assumptions, the signed-rank statistics of Wilcoxon type are asymptotically linear in the sense that they are uniformly approximable by linear forms in the regression parameter. More general results have been obtained by Van Eeden (1972) in a…
Without assumptions on the underlying distributions we prove the asymptotic normality of averaged scores rank statistics under all product distributions which are contiguous to the null hypothesis, and find a very simple form of the centering constants. In the one-sided two-sample and trend situations this enables us to show that monotonicity of the scores generating function is equivalent to the asymptotic unbiasedness of the corresponding averaged scores rank test. For…
Asymptotic expansions are established for the power of distribution free tests in the one-sample problem. These expansions are then used to obtain deficiencies in the sense of Hodges and Lehmann (1970) for distribution free tests with respect to their parametric competitors and for the estimators of location associated with these tests.
Huber's theory of robust estimation of a location parameter is adapted to obtain estimators that are robust against a class of asymmetric departures from normality. Let $F$ be a distribution function that is governed by the standard normal density on the set $\lbrack - d, d\rbrack$ and is otherwise arbitrary. Let $X_1,\cdots, X_n$ be a random sample from $F(x - \theta)$, where $\theta$ is the unknown location parameter. If $\psi$ is in a class of continuous skew-symmetric…
Let $X = (X_1,\cdots, X_p)^t, p \geqq 3$, have density $f(x - \theta)$ with respect to Lebesgue measure. It is desired to estimate $\theta = (\theta_1,\cdots, \theta_p)^t$ under the loss $L(\delta - \theta)$. Assuming the problem has a minimax risk $R_0$, an estimator is defined to be tail minimax if its risk is no larger than $R_0$ outside some compact set. Under quite general conditions on $f$ and $L$, sufficient conditions for an estimator to be tail minimax are given. A…
Ever since Stein's result, that the sample mean vector $\mathbf{X}$ of a $k \geqq 3$ dimensional normal distribution is an inadmissible estimator of its expectation $\mathbf{\theta}$, statisticians have searched for uniformly better (minimax) estimators. Unbiased estimators are derived here of the risk of arbitrary orthogonally-invariant and scale-invariant estimators of $\mathbf{\theta}$ when the dispersion matrix $\sum$ of $\mathbf{X}$ is unknown and must be estimated.…
Let $\mathbf{x}_1,\cdots, \mathbf{x}_n$ be a sample from an $m$-variate distribution which is spherically symmetric up to an affine transformation. This paper deals with the robust estimation of the location vector $\mathbf{t}$ and scatter matrix $\mathbf{V}$ by means of "$M$-estimators," defined as solutions of the system: $\sum_i u_1(d_i)(\mathbf{x}_i - \mathbf{t}) = \mathbf{0}$ and $n^{-1}\sum_i u_2(d_i^2)(\mathbf{x}_i - \mathbf{t})(\mathbf{x}_i - \mathbf{t})' = \mathbf{V}$…
The problem of estimating several normal mean vectors in an empirical Bayes situation is considered. In this case, it reduces to the problem of estimating the inverse of a covariance matrix in the standard multivariate normal situation using a particular loss function. Estimators which dominate any constant multiple of the inverse sample covariance matrix are presented. These estimators work by shrinking the sample eigenvalues toward a central value, in much the same way as…
In the statistical literature Le Cam's third lemma (cf. Hajek and Sidak (1967), page 208) is extensively used in order to get asymptotic normality of a statistic $S_n$ under contiguous alternatives from asymptotic normality of $S_n$ under the nullhypothesis. Since Le Cam's lemma utilizes the joint asymptotic normality of $S_n$ and $\log$-likelihood-ratio $\log L_n$, which is a sufficient but in general not a necessary condition for contiguity, it is not possible to get…
G. F. de Montricher · R. A. Tapia · J. R. Thompson·刊期 1975-11-01 · 卷 3（6）·The Annals of Statistics已编入正刊
惩罚函数法下概率密度的非参数最大似然估计AI 解读
Tne maximum likelihood estimate of a probability density function based on a random sample does not exist in the nonparametric case. For this reason and others based on heuristic Bayesian considerations Good and Gaskins suggested adding a penalty term to the likelihood. They proposed two penalty terms; however they did not establish existence or uniqueness of their maximum penalized likelihood estimates. Good and Gaskins also suggested an alternate approach for calculating…
Assume $X = (X_1,\cdots, X_p)^t$ has a $p$-variate density, with respect to Lebesgue measure, of the form $f((x - \theta)^t\not\sum^{-1}(x - \theta))$. Here $\not\sum$ is a known positive definite $p \times p$ matrix and $p \geqq 3$. Assume either (i) $f$ is completely monotonic, or (ii) there exist $\alpha > 0$ and $K > 0$ for which $h(s) = f(s)e^{\alpha s}$ is nondecreasing and nonzero if $s > K$. Then for estimating $\theta$ under a known quadratic loss, classes of…
Let $\{X_1, \cdots, X_N\}$ be $N$ observations on an $m$-state Markov chain with stationary transition probability matrix $\mathbf{P} = (p_{ij}), p_{ij} > 0, i,j = 1,\cdots, m$, where $N$ is a random variable. For any parametric function of $\mathbf{P}$, the information inequality gives a lower bound on the variance of an unbiased estimator; attaining the lower bound depends on whether the sampling plan or stopping rule $S$, the estimator $f = f(X_1,\cdots, X_N)$, and the…
The autoregressive process with moving average residuals is a stationary process $\{y_t\}$ satisfying $\sum^p_{s = 0} \beta_sy_{t - s} = \sum^q_{j = 0} \alpha_j\nu_{t - j}$, where the sequence $\{\nu_t\}$ consists of independently identically distributed (unobservable) random variables. The distribution of $y_1,\cdots, y_T$ can be approximated by the distribution of the $T$-component vector $\mathbf{y}$ satisfying $\sum^p_{s = 0} \beta_s\mathbf{K}_s\mathbf{y} = \sum^q_{j = 0} \alpha_j\mathbf{J}_j\mathbf{v}$…
Several invariant tests for uniformity of a distribution on the circle, the sphere and the hemisphere have been proposed by Rayleigh, Watson, Bingham, Ajne, Beran and others. In this paper a class of invariant tests for uniformity on compact Riemannian manifolds containing many of the known ones is presented and studied (the asymptotic theory as well as some local optimality properties for this class of tests are given). The examples include two new tests, one for the sphere…
Assume given the $(n + 1)$-first generation sizes of a supercritical branching process. An estimator is proposed for the variance $\sigma^2$ of this process when the mean is known. It is shown to be unbiased, consistent and asymptotically normal. From that one deduces a consistent and asymptotically normal estimator for $\sigma^2$ in the case of an unknown mean. Finally, the maximum likelihood estimator of $\sigma^2$, based on a richer sample, is found and asymptotic…
The following proposition is sometimes used in distribution theory: for each fixed $z$ suppose that $T(X, z)$ has the distribution $Q$ and is independent of $Y$; then $T(X, Z(Y))$ has the distribution $Q$ and is independent of $Y$. An example is presented to show this result is false in general. Additional conditions under which the proposition becomes valid are presented.
Spjotvoll [1967] has obtained a test associated with an unbalanced oneway layout for Model II ANOVA. Under the assumption of normality, his test possesses several optimum properties. Without the normality assumption, the significance level is (in general) highly nonrobust. An attempt to remedy this situation, using a test based on the jackknife technique, appears in Arvesen [1969]. The present paper proposes as an alternative a jackknifed version of Spjotvoll's test. The new…
Estimation of linear functions of a vector parameter $\theta$ when an observed discrete- or continuous-time vector-valued stationary process has mean value $H\theta, H$ a known matrix, is considered. Large-sample comparisons of best linear unbiased estimators and estimators based on the sample mean are made. Limits and rates of convergence of the variances of these estimators are obtained. It is shown that under general conditions there are asymptotically efficient…
Conditions are given for the weak convergence of weighted empirical cumulative processes of three types of mixing random variables (rv's) on $\\lbrack 0, 1\\rbrack$.
This paper investigates the effect of serial dependence in the data on the efficiency of some robust estimators. When the observations are from a stationary process satisfying certain mixing conditions, linear combinations of order statistics and the Hodges-Lehmann estimator are shown to be asymptotically normally distributed. Gaussian processes are studied in detail and it is shown that when all the serial correlations $(\rho_n)$ are $\geqq 0$, the efficiency of the robust…
Let μ be a continuous mean regression function defined on U, the unit cube in N-dimensional Euclidean space. Let F be a distribution function with support in U, and let M denote the indefinite integral of μ with respect to F. This paper provides consistency results, including rates of convergence, for a certain estimator of M in the case that the nth estimate is based on observations at points tn1,⋯,tnn of U. The…
The problem of estimating the mean of a p-variate (p≧3) normal distribution is considered. It is assumed that the covariance matrix ∑ is known and that the loss function is quadratic. A class of minimax estimators is given, out of which admissible minimax estimators are developed.
Kendall's rank correlation statistic Tn=∑i>jsgn(Xi−Xj)⋅ sgn (Yi−Yj) is well known to be asymptotically normally distributed under the null hypothesis of independence as the sample size n→∞. In the note it is shown that this assertion can be obtained easily from the recurrence formula pn(t)=(1/n)∑k=1npn−1(t−2k+n+1) for the probability distribution pn of Tn (see Kendall (1970), e.g.).…
Huber's theory of robust estimation of a location parameter is adapted to obtain estimators that are robust against a class of asymmetric departures from normality. Let F be a distribution function that is governed by the standard normal density on the set [−d,d] and is otherwise arbitrary. Let X1,⋯,Xn be a random sample from F(x−θ), where θ is the unknown location parameter. If ψ is in a class of continuous skew-symmetric…
Let X=(X1,⋯,Xp)t,p≧3, have density f(x−θ) with respect to Lebesgue measure. It is desired to estimate θ=(θ1,⋯,θp)t under the loss L(δ−θ). Assuming the problem has a minimax risk R0, an estimator is defined to be tail minimax if its risk is no larger than R0 outside some compact set. Under quite general conditions on f and L, sufficient conditions for an estimator to be tail minimax are given. A…
Ever since Stein's result, that the sample mean vector X of a k≧3 dimensional normal distribution is an inadmissible estimator of its expectation θ, statisticians have searched for uniformly better (minimax) estimators. Unbiased estimators are derived here of the risk of arbitrary orthogonally-invariant and scale-invariant estimators of θ when the dispersion matrix ∑ of X is unknown and must be estimated.…
Robust M-Estimators of Multivariate Location and Scatter
Let x1,⋯,xn be a sample from an m-variate distribution which is spherically symmetric up to an affine transformation. This paper deals with the robust estimation of the location vector t and scatter matrix V by means of "M-estimators," defined as solutions of the system: ∑iu1(di)(xi−t)=0 and n−1∑iu2(di2)(xi−t)(xi−t)′=V…
In the statistical literature Le Cam's third lemma (cf. Hajek and Sidak (1967), page 208) is extensively used in order to get asymptotic normality of a statistic Sn under contiguous alternatives from asymptotic normality of Sn under the nullhypothesis. Since Le Cam's lemma utilizes the joint asymptotic normality of Sn and log-likelihood-ratio logLn, which is a sufficient but in general not a necessary condition for contiguity, it is not possible to get…
Assume X=(X1,⋯,Xp)t has a p-variate density, with respect to Lebesgue measure, of the form f((x−θ)t∑−1(x−θ)). Here ∑ is a known positive definite p×p matrix and p≧3. Assume either (i) f is completely monotonic, or (ii) there exist α>0 and K>0 for which h(s)=f(s)eαs is nondecreasing and nonzero if s>K. Then for estimating θ under a known quadratic loss, classes of…
Let {X1,⋯,XN} be N observations on an m-state Markov chain with stationary transition probability matrix P=(pij),pij>0,i,j=1,⋯,m, where N is a random variable. For any parametric function of P, the information inequality gives a lower bound on the variance of an unbiased estimator; attaining the lower bound depends on whether the sampling plan or stopping rule S, the estimator f=f(X1,⋯,XN), and the…
The autoregressive process with moving average residuals is a stationary process {yt} satisfying ∑s=0pβsyt−s=∑j=0qαjνt−j, where the sequence {νt} consists of independently identically distributed (unobservable) random variables. The distribution of y1,⋯,yT can be approximated by the distribution of the T-component vector y satisfying ∑s=0pβsKsy=∑j=0qαjJjv…
Assume given the (n+1)-first generation sizes of a supercritical branching process. An estimator is proposed for the variance σ2 of this process when the mean is known. It is shown to be unbiased, consistent and asymptotically normal. From that one deduces a consistent and asymptotically normal estimator for σ2 in the case of an unknown mean. Finally, the maximum likelihood estimator of σ2, based on a richer sample, is found and asymptotic…
The following proposition is sometimes used in distribution theory: for each fixed z suppose that T(X,z) has the distribution Q and is independent of Y; then T(X,Z(Y)) has the distribution Q and is independent of Y. An example is presented to show this result is false in general. Additional conditions under which the proposition becomes valid are presented.
Estimation of linear functions of a vector parameter θ when an observed discrete- or continuous-time vector-valued stationary process has mean value Hθ,H a known matrix, is considered. Large-sample comparisons of best linear unbiased estimators and estimators based on the sample mean are made. Limits and rates of convergence of the variances of these estimators are obtained. It is shown that under general conditions there are asymptotically efficient…
Convergence of Empirical Processes of Mixing rv's on [0,1]
Conditions are given for the weak convergence of weighted empirical cumulative processes of three types of mixing random variables (rv's) on lbrack0,1rbrack.
This paper investigates the effect of serial dependence in the data on the efficiency of some robust estimators. When the observations are from a stationary process satisfying certain mixing conditions, linear combinations of order statistics and the Hodges-Lehmann estimator are shown to be asymptotically normally distributed. Gaussian processes are studied in detail and it is shown that when all the serial correlations (ρn) are ≧0, the efficiency of the robust…