An analog of the spectral analysis of time series is developed for data in general spaces. This is applied to data from an election in which 5738 people rank ordered five candidates. Group theoretic considerations offer an analysis of variance like decomposition which seems natural and fruitful. A variety of inferential tools are suggested. The spectral ideas are then extended to general homogeneous spaces such as the sphere.
We extend the jackknife and the bootstrap method of estimating standard errors to the case where the observations form a general stationary sequence. We do not attempt a reduction to i.i.d. values. The jackknife calculates the sample variance of replicates of the statistic obtained by omitting each block of $l$ consecutive data once. In the case of the arithmetic mean this is shown to be equivalent to a weighted covariance estimate of the spectral density of the observations…
The delete-1 jackknife is known to give inconsistent variance estimators for nonsmooth estimators such as the sample quantiles. This well-known deficiency can be rectified by using a more general jackknife with $d$, the number of observations deleted, depending on a smoothness measure of the point estimator. Our general theory explains why jackknife works or fails. It also shows that (i) for "sufficiently smooth" estimators, the jackknife variance estimators with bounded $d$…
This note concerns the problem of order determination for autoregressive models. Rissanen's "Predictive least squares principle" prescribes that one should choose as order estimate $\hat{k}(n)$ at time $n$ the order of the model which has given the least mean square prediction error up to that time. We show that this procedure is strongly consistent, that is, that $\hat{k}(n) \rightarrow p$ a.s. as $n \rightarrow \infty$ when the data are generated by an AR process of order $p$…
Consider the problem of estimating the ordered means $\mu_1 \leq \mu_2 \leq \cdots \leq \mu_k$ of independent normal random variables, $Y_1, Y_2, \cdots, Y_k$. It is shown that the absolute error of each component $\hat{\mu}_i$ of the isotonic regression estimator is stochastically smaller than that of the usual estimator $Y_i$. Thus $\hat{\mu}_i$ is superior to $Y_i$ under any nonconstant loss which is a nondecreasing function of absolute error.
Certain polynomials of a skew-symmetric matrix are considered. These polynomials can be expressed in terms of the zonal polynomials on the Hermitian matrices, and they are used to obtain a series expansion for the density of the non-null distribution of the maximal invariant corresponding to the problem of testing for reality of the covariance matrix of a complex multivariate normal distribution.
This paper reformulates a result of Hora and Buehler on best equivariant estimators to treat a model admitting an ancillary statistic. The approach itself was established by Pitman, Girshick and Savage and Kiefer, and expanded by Zidek. The model considered in this paper is assumed to be generated as an orbit under a group acting on the parameter space. The general result obtained here is applied to a model in the Nile problem, a model with a known variation coefficient, a…
Conditions are given for the formal posterior of an improper prior to be coherent and applied to translation models. An example is given of a proper countably additive statistical model and a finitely additive prior for which there is no posterior.
In estimation of a $p$-variate normal mean with identity covariance matrix, Stein-type estimators can offer significant gains over the $\operatorname{mle}$ in terms of risk with respect to sum of squares error loss. Their maximum risk is still equal to $p$, however, which will typically be their "reported loss." In this paper we consider use of data-dependent "loss estimators." Two conditions that are attractive for such a loss estimator are that it be an improved loss…
We consider the problem of robustness or sensitivity of given Bayesian posterior criteria to specification of the prior distribution. Criteria considered include the posterior mean, variance and probability of a set (for credible regions and hypothesis testing). Uncertainty in an elicited prior, $\pi_0$, is modelled by an $\varepsilon$-contamination class $\Gamma = \{\pi = (1 - \varepsilon)\pi_0 + \varepsilon q, q \in Q\}$, where $\varepsilon$ reflects the amount of…
Arbitrary, possibly randomized estimators of a one-dimensional parameter are considered. It is shown that suitable averages of their distribution functions are more spread out than particular distribution functions, which are defined in terms of the weight functions by which the averages are taken over the parameter space and in terms of the family of distributions for the random quantity on which the estimators are based. In this way bounds are provided for the performance…
Suppose $\{X_n\}$ is a $p$th order autoregressive process with innovations in the domain of attraction of a stable law and the true order $p$ unknown. The estimate $\hat{p}$ of $p$ is chosen to minimize Akaike's information criterion over the integers $0, 1, \cdots, K$. It is shown that $\hat{p}$ is weakly consistent and the consistency is retained if $K \rightarrow \infty$ as $N \rightarrow \infty$ at a certain rate depending on the index of the stable law.
Motivated by a scheduling problem arising from serial sacrifice experiments, the asymptotic efficiency of irreversible adaptive allocation rules is studied. The asymptotic lower bound for the regret of an adaptive allocation rule is characterized by the minimum of a linear program. Based on a class of one-sided sequential tests, asymptotically efficient rules which achieve the lower bound are constructed. The conditions necessary for this construction are verified in the…
We consider dynamic procedures for sampling from a process (a Brownian motion) and stopping it after a change is detected. The basic idea is to conduct a sequence of similar SPRT's, each one of them done in negligible time, while not sampling at all between them. The procedures detect the change point much faster than the standard procedures with the same sampling rate and time to false alarm, but hold the sampling rate constant.
Methods are described which permit one to work with continuous-time optimal stopping problems, using the heat equation, even when the prior placed on the drift parameter of a Wiener process is not normal. The details of the method are worked out for Chernoff's problem of testing the sign of the drift parameter when the prior is "smooth."
Hani Doss · Steven Freitag · Frank Proschan·刊期 1989-06-01 · 卷 17（2）·The Annals of Statistics已编入正刊
利用相互删失方法联合估计系统与部件可靠性AI 解读
Let $F$ denote the life distribution of a coherent structure of independent components. Suppose that we have a sample of independent systems, all having the same structure. Each system is continuously observed until it fails. For every component in each system, either a failure time or a censoring time is recorded. A failure time is recorded if the component fails before or at the time of system failure; otherwise a censoring time is recorded. We introduce a method for…
To assess the dependence structure in a stationary bivariate point process the second-order distribution can be very useful. We prove that the natural estimates of this distribution, based on a realization $A_1 < A_2 < \cdots < A_{n_A}, B_1 < B_2 < \cdots < B_{n_B}$ are asymptotically normal and we present a method for constructing approximate confidence intervals for this distribution.
The use of moment matrices and their determinants are shown to elucidate the structure of mixture estimation as carried out using the method of moments. The setting is the estimation of a discrete finite support point mixing distribution. In the important class of quadratic variance exponential families it is shown for any sample there is an integer $\hat{\nu}$ depending on the data which represents the maximal number of support points that one can put in the estimated…
Peter Hall · Thomas J. DiCiccio · Joseph P. Romano·刊期 1989-06-01 · 卷 17（2）·The Annals of Statistics已编入正刊
平滑与自助法AI 解读
Recent attention has focussed on possible improvements in performance of estimators which might flow from using the smoothed bootstrap. We point out that in a great many problems, such as those involving functions of vector means, any such improvements will be only second-order effects. However, we argue that substantial and significant improvements can occur in problems where local properties of underlying distributions play a decisive role. This situation often occurs in…
It is proved that $EX^2 < \infty$ is necessary for a very mild form of the bootstrap of the mean to work a.s. and that $X$ must be in the domain of attraction of the normal law if a.s. is weakened to "in probability."
Randomized experiments and observational studies both attempt to estimate the effects produced by a treatment, but in observational studies, subjects are not randomly assigned to treatment or control. A theory of observational studies would closely resemble the theory for randomized experiments in all but one critical respect: In observational studies, the distribution of treatment assignments is not known. The problems that are special to observational studies revolve…
When one uses the unbalanced, mixed linear model $\mathbf{y}_i = \mathbf{X}_i\mathbf{\alpha} + \mathbf{Z}_i\mathbf{\beta}_i + \varepsilon_i, i = 1, \cdots, n$ to analyze data from longitudinal experiments with continuous outcomes, it is customary to assume $\varepsilon_i \sim_{\operatorname{ind}} \mathscr{N}(\mathbf{0}, \sigma^2\mathbf{I}_i)$ independent of $\mathbf{\beta}_i \sim_{\operatorname{iid}} \mathscr{N}(\mathbf{0,\Delta})$, where $\sigma^2$ and the elements of an…
In a recent paper of Gaffke an example was given regarding the $\phi_p$-optimal designs for the highest two coefficients in a one dimensional polynomial regression. The purpose of this paper is to supply a direct proof of this result using the theory of canonical moments and orthogonal polynomials.
We develop asymptotic theory for two polynomial-based methods of estimating orientation in projection pursuit density approximation. One of the techniques uses Legendre polynomials and has been proposed and implemented by Friedman [1]. The other employs Hermite functions. Issues of smoothing parameter choice and robustness are addressed. It is shown that each method can be used to construct $\sqrt n$-consistent estimates of the projection which maximizes distance from…
We extend the jackknife and the bootstrap method of estimating standard errors to the case where the observations form a general stationary sequence. We do not attempt a reduction to i.i.d. values. The jackknife calculates the sample variance of replicates of the statistic obtained by omitting each block of l consecutive data once. In the case of the arithmetic mean this is shown to be equivalent to a weighted covariance estimate of the spectral density of the observations…
The delete-1 jackknife is known to give inconsistent variance estimators for nonsmooth estimators such as the sample quantiles. This well-known deficiency can be rectified by using a more general jackknife with d, the number of observations deleted, depending on a smoothness measure of the point estimator. Our general theory explains why jackknife works or fails. It also shows that (i) for "sufficiently smooth" estimators, the jackknife variance estimators with bounded d…
This note concerns the problem of order determination for autoregressive models. Rissanen's "Predictive least squares principle" prescribes that one should choose as order estimate k^(n) at time n the order of the model which has given the least mean square prediction error up to that time. We show that this procedure is strongly consistent, that is, that k^(n)→p a.s. as n→∞ when the data are generated by an AR process of order p…
Consider the problem of estimating the ordered means μ1≤μ2≤⋯≤μk of independent normal random variables, Y1,Y2,⋯,Yk. It is shown that the absolute error of each component μ^i of the isotonic regression estimator is stochastically smaller than that of the usual estimator Yi. Thus μ^i is superior to Yi under any nonconstant loss which is a nondecreasing function of absolute error.
In estimation of a p-variate normal mean with identity covariance matrix, Stein-type estimators can offer significant gains over the mle in terms of risk with respect to sum of squares error loss. Their maximum risk is still equal to p, however, which will typically be their "reported loss." In this paper we consider use of data-dependent "loss estimators." Two conditions that are attractive for such a loss estimator are that it be an improved loss…
We consider the problem of robustness or sensitivity of given Bayesian posterior criteria to specification of the prior distribution. Criteria considered include the posterior mean, variance and probability of a set (for credible regions and hypothesis testing). Uncertainty in an elicited prior, π0, is modelled by an ε-contamination class Γ={π=(1−ε)π0+εq,q∈Q}, where ε reflects the amount of…
Suppose {Xn} is a pth order autoregressive process with innovations in the domain of attraction of a stable law and the true order p unknown. The estimate p^ of p is chosen to minimize Akaike's information criterion over the integers 0,1,⋯,K. It is shown that p^ is weakly consistent and the consistency is retained if K→∞ as N→∞ at a certain rate depending on the index of the stable law.
Let F denote the life distribution of a coherent structure of independent components. Suppose that we have a sample of independent systems, all having the same structure. Each system is continuously observed until it fails. For every component in each system, either a failure time or a censoring time is recorded. A failure time is recorded if the component fails before or at the time of system failure; otherwise a censoring time is recorded. We introduce a method for…
To assess the dependence structure in a stationary bivariate point process the second-order distribution can be very useful. We prove that the natural estimates of this distribution, based on a realization A1<A2<⋯<AnA,B1<B2<⋯<BnB are asymptotically normal and we present a method for constructing approximate confidence intervals for this distribution.
The use of moment matrices and their determinants are shown to elucidate the structure of mixture estimation as carried out using the method of moments. The setting is the estimation of a discrete finite support point mixing distribution. In the important class of quadratic variance exponential families it is shown for any sample there is an integer ν^ depending on the data which represents the maximal number of support points that one can put in the estimated…
It is proved that EX2<∞ is necessary for a very mild form of the bootstrap of the mean to work a.s. and that X must be in the domain of attraction of the normal law if a.s. is weakened to "in probability."
When one uses the unbalanced, mixed linear model yi=Xiα+Ziβi+εi,i=1,⋯,n to analyze data from longitudinal experiments with continuous outcomes, it is customary to assume εi∼indN(0,σ2Ii) independent of βi∼iidN(0,Δ), where σ2 and the elements of an…
Note on some ϕp-Optimal Designs for Polynomial Regression
In a recent paper of Gaffke an example was given regarding the ϕp-optimal designs for the highest two coefficients in a one dimensional polynomial regression. The purpose of this paper is to supply a direct proof of this result using the theory of canonical moments and orthogonal polynomials.
We develop asymptotic theory for two polynomial-based methods of estimating orientation in projection pursuit density approximation. One of the techniques uses Legendre polynomials and has been proposed and implemented by Friedman [1]. The other employs Hermite functions. Issues of smoothing parameter choice and robustness are addressed. It is shown that each method can be used to construct n-consistent estimates of the projection which maximizes distance from…