This paper gives a class of minimum $L_2$-distance estimators of the autoregression parameter in the first-order autoregression model when the errors have an unknown symmetric distribution. Within the class an asymptotically efficient estimator is exhibited. The asymptotic efficiency of this estimator relative to the least-squares estimator is the same as that of a certain signed rank estimator relative to the sample mean in the one sample location model. The paper also…
In this paper we construct a class of minimum distance Cramer-von Mises-type estimators for the parameter in the first-order stationary autoregressive time series. The estimator is proved to be asymptotically normal under appropriate assumptions. The proofs involve some results of independent interest.
Consider a regression model $Y_i=x'_i\beta+R_i$, i=1,\cdots,n, where ${R_i}$ are i.i.d. with c.d.f., F; $x_i\inR^p$ and $\beta\inR^p$. Let $\hat{\beta}$ be a M-estimator defined using kernel, $\psi$; let $\hat{F}_n(x)$ denote the empiric distribution of the residuals, $Y_i-x'_i\hat{\beta}$ and let $\hat{F}^\ast_n$ be the empiric c.d.f. of the errors, ${R_i}$. Under suitable smoothness conditions on $\psi$, F, and the density F' = f and conditions requiring essentially that ${x_i}$…
This paper deals with the problem of efficiently estimating (asymptotically minimax) a distribution function when essentially nothing is known about it except that it is unimodal. The sample distribution function $F_n$ is shown to be asymptotically minimax among the family $\mathscr{E}$ of all unimodal distribution functions. Since $F_n$ does not belong to this family, estimators belonging to this family are constructed and are shown to be asymptotically minimax relative to…
We construct nonparametric estimators of the distribution function F and its hazard function in the class of all increasing failure rate (IFR) distribution. Denoting the empirical distribution and empirical hazard function by $F_n$ and $H_n$ , respectively, let $C_n$ be the greatest convex minorant of $H_n$, and $G_n$ the distribution with hazard function $C_n$. The estimator $G_n$ is itself IFR. We prove that under suitable restrictions on F, and for any fixed $\lambda$ with…
The asymptotic optimality of Mallows' $C_L$ and generalized cross-validation is demonstrated in the setting of ridge regression. An application is made to spline smoothing in nonparametric regression. A counterexample is given to help understand why sometimes GCV may not be asymptotically optimal. The coefficient of variation for the eigenvalues of the information matrix must be large in order to guarantee the optimality of GCV. The proff is based on the connection between…
In the present paper we will improve the results concerning the rate of convergence of the error of second kind of the Neyman-Pearson test if the Kullback-Leibler information $K(P_0,P_1)$ is infinite. It is pointed out that in certain cases the sequence $exp(-q_\infty,n)$ is the correct rate of convergence if $-q_\infty,n$ denotes the logarithm of the critical value of the Neyman-Pearson test of level and sample size n. Therefore we generalize the classical results of Stein,…
Let $X_1,X_2,\cdots$ be iid with density $f_y$ with respect to a sigma finite measure $\mu$ where ${f_y}_(y\in\omega$, $\omega\subseteqR$ is an exponential family. Let F be a probability measure on $\omega$ and let $\theta_0\in\omega$. Define $T(B,F)=\min \big\{n \left| \int_omega \frac{f_y(X_1) \1dots f_y(X_n)} {f_{\theta_0}(X_1)\dots f_{\theta_0 (X_1) \1dots f_{\theta_0} (X_n)} dF(y)\geq B \big\}$, $T (B,F) =\infty$ if no such n exists. Previous studies have found that if F has a positive and continuous density with respect to Lebesgue measure on $…
Richard Arratia · Louis Gordon · Michael Waterman·刊期 1986-09-01 · 卷 14（3）·The Annals of Statistics已编入正刊
序列匹配的极值理论AI 解读
Consider finite sequences $X_1, X_2,\cdots, X_m$ and $Y_1, Y_2,\cdots, Y_n$ where the letters ${X_i}$ and ${Y_i}$ are chosen i.i.d. on a countable alphabet with $p=P(X_1=Y_1)\in(0,1)$ We study the distribution of the longest contiguous run of matches between the X's and Y's allowing at most k mismatches. The distribution is closely approximated by that of the maximum of (1 - p)mn i.i.d. negative binomial random variables. The latter distribution is in turn shown to behave…
The risk in a trial to compare two medical treatments is borne by the patients who receive the inferior treatment during the experimental phase and by those remaining after the experiment who will all receive the inferior treatment if the results are misleading. The Bayes rule indicates, for the observed progression of successes and failures, when it is optimal to stop this experimental phase. This stopping rule can be described exactly, or nearly so, for symmetric two-point…
It is desired to estimate a real valued function F on the unit square having observed F with error at N points in the square. F is assumed to be drawn from a particular Gaussian process and measured with independent Gaussian errors. The proposed estimate is the Bayes estimate of F given the data. The roughness penalty corresponding to the prior is derived and it is shown how the Bayesian technique can be regarded as a generalisation of variance components analysis. The…
Consider a supercritical Galton-Watson process $(Z_n)$ with offspring distribution a member of the power series family, and having unknown mean $\theta$. The conditional asymptotic normality of the suitably normalized maximum likelihood estimator of $\theta$ given the conditional information is established. The conditional information here is proportional to the total number of ancestors $V_n$, and it is also seen that this statistic is asymptotically ancillary for $\theta$ i…
Suppose $u_1,\cdots,u_n$ are fixed points on the sphere, $v_1,\cdots,v_n$ are random points such that the distribution of $v_i$ depends only upon $v^t_iAu_i$ for some unknown rotation A. This paper provides asymptotic tests and confidence regions for A and for the axis of rotation of A. Results are given in arbitrary dimension.
This paper gives a method for constructing conservative Scheffe-type simultaneous confidence bands for curvilinear regression functions over finite intervals. The method is based on the use of a geometric inequality giving an upper bound for the uniform measure of the set of points within a given distance from y, an arbitrary piecewise differentiable path with finite length in $S^{k-1}$, the unit sphere in $R^k$. The upper bound is obtained by "straightening" the path so…
In the differential geometric approach to parametric statistics, developed by Chentsov, Efron, Amari, and others, the parameter space is set up as a differentiable manifold with expected information as metric tensor and with a family of affine connections, the a-connections, determined from the expected information and the skewness tensor of the score vector. The usefulness of this approach is particularly notable in connection with Edgeworth expansions of estimators.…
A definition is given for influence functionals of parameter estimates in time-series models. The definition involves the use of a contaminated observations process of the form $y^\gamma_t=(1-z^\gamma_t){x_t+z^\gamma_tw_t}$, $p=1,2,...,0\leq\gamma\leq1$ where $x_t$ is a core process (usually Gaussian), $w_t$ is a contaminating process, and $z^\gamma_t"$ is a zero-one process with $P(z^\gamma_t=1)={\gamma+0(\gamma)}$. This form is sufficiently general to model such diverse…
Let $Z_i:-\infty<i <+\infty}$ be a strictly stationary $\infty$ -mixing sequence. Without specifying the dependence model giving rise to ${Z_i}$ and without specifying the marginal distribution of $Z_i$ , we address the question of variance estimation for a general statistic $t_n=t_n (Z_1,...,Z_n)$. For estimating $Var{t_n}$ from just the available data $(Z_1,...,Z_n)$ we propose computing subseries values: $t_m(Z_{i+1},Z_{i+2}, Z_{i+m})$, $0\leq i
A general method for the construction of asymptotically efficient estimates in semiparametric models is presented. It improves and modifies Bickel's (1982) construction of adaptive estimates and obtains asymptotically efficient estimates under conditions weaker than those in Bickel.
As a modification of the notion of algorithmic complexity, the stochastic complexity of a string of data, relative to a class of probabilistic models, is defined to be the fewest number of binary digits with which the data can be encoded by taking advantage of the selected models. The computation of the stochastic complexity produces a model, which may be taken to incorporate all the statistical information in the data that can be extracted with the chosen model class. This…
A Cramer type large deviation theorem for signed linear rank statistics under the symmetry hypothesis is obtained. The theorem is proved for a wide class of scores covering most of the commonly used ones (including the normal scores). Furthermore, the optimal range $0 < x \leq o(n^{1/4})$ can be obtained for bounded scores, whereas the range $0 < x \leq o(n^\delta), \delta \in (0, \frac{1}{4})$ is obtainable for many unbounded ones. This improves the earlier result under the…
The kernel estimator is a widely used tool for the estimation of a density function. In this paper its adaptation to censored data using the Kaplan-Meier estimator is considered. Asymptotic properties of four estimators, arising naturally as a result of considering various types of bandwidths, are investigated. In particular we show that (i) both proposed estimators stemming from the nearest neighbor estimator have censoring-free variances and (ii) one of them is pointwise…
Two Pearson-type goodness-of-fit test statistics for parametric families are considered for randomly right-censored data. Asymptotic distribution theory for the test statistics is based on the result that the product-limit process with MLE for nuisance parameters converges weakly to a Gaussian process. The Chernoff-Lehmann (1954) result extends to a generalized Pearson statistic. A modified Pearson statistic is shown to have a limiting chi-square null distribution.
We exhibit those distributions with minimum Fisher information for location in various Kolmogorov neighbourhoods $\{F|\sup_x|F(x) - G(x)| \leq \varepsilon\}$ of a fixed, symmetric distribution $G$. The associated $M$-estimators are then most robust (in Huber's minimax sense) for location estimation within these neighbourhoods. The previously obtained solution of Huber (1964) for $G = \Phi$ and "small" $\varepsilon$ is shown to apply to all distributions with strongly…
The univariate weak convergence theorem of Murota and Takeuchi (1981) is extended for the Mahalanobis transform of the $d$-variate empirical characteristic function, $d \geq 1$. Then a maximal deviation statistic is proposed for testing the composite hypothesis of $d$-variate normality. Fernique's inequality is used in conjunction with a combination of analytic, numerical analytic, and computer techniques to derive exact upper bounds for the asymptotic percentage points of…
This paper gives a class of minimum L2-distance estimators of the autoregression parameter in the first-order autoregression model when the errors have an unknown symmetric distribution. Within the class an asymptotically efficient estimator is exhibited. The asymptotic efficiency of this estimator relative to the least-squares estimator is the same as that of a certain signed rank estimator relative to the sample mean in the one sample location model. The paper also…
Consider a regression model Yi=xi′β+Ri, i=1,\cdots,n, where Ri are i.i.d. with c.d.f., F; $x_i\inR^p$ and $\beta\inR^p$. Let β^ be a M-estimator defined using kernel, ψ; let F^n(x) denote the empiric distribution of the residuals, Yi−xi′β^ and let F^n∗ be the empiric c.d.f. of the errors, Ri. Under suitable smoothness conditions on ψ, F, and the density F' = f and conditions requiring essentially that xi…
This paper deals with the problem of efficiently estimating (asymptotically minimax) a distribution function when essentially nothing is known about it except that it is unimodal. The sample distribution function Fn is shown to be asymptotically minimax among the family E of all unimodal distribution functions. Since Fn does not belong to this family, estimators belonging to this family are constructed and are shown to be asymptotically minimax relative to…
We construct nonparametric estimators of the distribution function F and its hazard function in the class of all increasing failure rate (IFR) distribution. Denoting the empirical distribution and empirical hazard function by Fn and Hn , respectively, let Cn be the greatest convex minorant of Hn, and Gn the distribution with hazard function Cn. The estimator Gn is itself IFR. We prove that under suitable restrictions on F, and for any fixed λ with…
Asymptotic Optimality of CL and Generalized Cross-Validation in Ridge Regression with Application to Spline Smoothing
The asymptotic optimality of Mallows' CL and generalized cross-validation is demonstrated in the setting of ridge regression. An application is made to spline smoothing in nonparametric regression. A counterexample is given to help understand why sometimes GCV may not be asymptotically optimal. The coefficient of variation for the eigenvalues of the information matrix must be large in order to guarantee the optimality of GCV. The proff is based on the connection between…
In the present paper we will improve the results concerning the rate of convergence of the error of second kind of the Neyman-Pearson test if the Kullback-Leibler information K(P0,P1) is infinite. It is pointed out that in certain cases the sequence exp(−q∞,n) is the correct rate of convergence if −q∞,n denotes the logarithm of the critical value of the Neyman-Pearson test of level and sample size n. Therefore we generalize the classical results of Stein,…
Let X1,X2,⋯ be iid with density fy with respect to a sigma finite measure μ where fy(y∈ω, $\omega\subseteqR$ is an exponential family. Let F be a probability measure on ω and let θ0∈ω. Define $T(B,F)=\min \big\{n \left| \int_omega \frac{f_y(X_1) \1dots f_y(X_n)} {f_{\theta_0}(X_1)\dots f_{\theta_0 (X_1) \1dots f_{\theta_0} (X_n)} dF(y)\geq B \big\},T (B,F) =\inftyifnosuchnexists.PreviousstudieshavefoundthatifFhasapositiveandcontinuousdensitywithrespecttoLebesguemeasureon…
Consider finite sequences X1,X2,⋯,Xm and Y1,Y2,⋯,Yn where the letters Xi and Yi are chosen i.i.d. on a countable alphabet with p=P(X1=Y1)∈(0,1) We study the distribution of the longest contiguous run of matches between the X's and Y's allowing at most k mismatches. The distribution is closely approximated by that of the maximum of (1 - p)mn i.i.d. negative binomial random variables. The latter distribution is in turn shown to behave…
Consider a supercritical Galton-Watson process (Zn) with offspring distribution a member of the power series family, and having unknown mean θ. The conditional asymptotic normality of the suitably normalized maximum likelihood estimator of θ given the conditional information is established. The conditional information here is proportional to the total number of ancestors Vn, and it is also seen that this statistic is asymptotically ancillary for θ i…
Suppose u1,⋯,un are fixed points on the sphere, v1,⋯,vn are random points such that the distribution of vi depends only upon vitAui for some unknown rotation A. This paper provides asymptotic tests and confidence regions for A and for the axis of rotation of A. Results are given in arbitrary dimension.
This paper gives a method for constructing conservative Scheffe-type simultaneous confidence bands for curvilinear regression functions over finite intervals. The method is based on the use of a geometric inequality giving an upper bound for the uniform measure of the set of points within a given distance from y, an arbitrary piecewise differentiable path with finite length in Sk−1, the unit sphere in Rk. The upper bound is obtained by "straightening" the path so…
A definition is given for influence functionals of parameter estimates in time-series models. The definition involves the use of a contaminated observations process of the form ytγ=(1−ztγ)xt+ztγwt, p=1,2,...,0≤γ≤1 where xt is a core process (usually Gaussian), wt is a contaminating process, and ztγ" is a zero-one process with P(ztγ=1)=γ+0(γ). This form is sufficiently general to model such diverse…
Let $Z_i:-\infty<i <+\infty}$ be a strictly stationary ∞ -mixing sequence. Without specifying the dependence model giving rise to Zi and without specifying the marginal distribution of Zi , we address the question of variance estimation for a general statistic tn=tn(Z1,...,Zn). For estimating Vartn from just the available data (Z1,...,Zn) we propose computing subseries values: tm(Zi+1,Zi+2,Zi+m), $0\leq i
A Cramer type large deviation theorem for signed linear rank statistics under the symmetry hypothesis is obtained. The theorem is proved for a wide class of scores covering most of the commonly used ones (including the normal scores). Furthermore, the optimal range 0<x≤o(n1/4) can be obtained for bounded scores, whereas the range 0<x≤o(nδ),δ∈(0,41) is obtainable for many unbounded ones. This improves the earlier result under the…
Minimax Variance M-Estimators of Location in Kolmogorov Neighbourhoods
We exhibit those distributions with minimum Fisher information for location in various Kolmogorov neighbourhoods {F∣supx∣F(x)−G(x)∣≤ε} of a fixed, symmetric distribution G. The associated M-estimators are then most robust (in Huber's minimax sense) for location estimation within these neighbourhoods. The previously obtained solution of Huber (1964) for G=Φ and "small" ε is shown to apply to all distributions with strongly…
The univariate weak convergence theorem of Murota and Takeuchi (1981) is extended for the Mahalanobis transform of the d-variate empirical characteristic function, d≥1. Then a maximal deviation statistic is proposed for testing the composite hypothesis of d-variate normality. Fernique's inequality is used in conjunction with a combination of analytic, numerical analytic, and computer techniques to derive exact upper bounds for the asymptotic percentage points of…