Statistical properties of a variant of the histospline density estimate introduced by Boneva-Kendall-Stefanov are obtained. The estimate we study is formed for $x$ in a finite interval, $x\in\lbrack a, b\rbrack = \lbrack 0, 1\rbrack$ say, by letting $\hat{F}_n(x), x \epsilon\lbrack 0, 1\rbrack$ be the unique cubic spline of interpolation to the sample cumulative distribution function $F_n(x)$ at equi-spaced points $x = jh, j = 0, 1,\cdots, l + 1, (l + 1)h = 1$, which…
Let $W_p^{(m)}(M) = \{f: f^{(\nu)} \operatorname{abs}. \operatorname{cont}., \nu = 0, 1,\cdots, m - 1, f^{(m)} \in \mathscr{L}_p, \|f^{(m)}\|_p \leqq M\}$, where $\|\cdot\|_p$ is the norm in $\mathscr{L}_p, m$ is a positive integer and $p$ is a real number, $p \geqq 1$. Let $\{\hat{f}_n(x)\}, n = 1, 2,\cdots$ be any sequence of estimates of a density at the point $x$ where $\hat{f}_n(x)$ depends on $n$ independent observations from some density $f \in W_p^{(m)}(M)$. It is…
This paper considers estimates of multidimensional density functions based on a bounded and bandlimited weight function. The asymptotic behavior of quadratic functions of density function estimates useful in setting up a test of goodness of fit of the density function is determined. A test of independent is also given. The methods use a Poissonization of sample size. The estimates considered are appropriate if interested in estimating density functions or determining local…
A method is presented for making statistical inferences about the upper tail of a distribution function. It is useful for estimating the probabilities of future extremely large observations. The method is applicable if the underlying distribution function satisfies a condition which holds for all common continuous distribution functions.
A formulation is given and a procedure is proposed for constructing a confidence interval for a certain ordered location or scale parameter and for simultaneously selecting all populations having parameters equal or larger than this ordered parameter with a preassigned minimal probability. The well-known indifference-zone formulation of the ranking problem is obtained as a special case as is the problem of interval estimation of an ordered parameter.
For a given sufficiently regular distribution $F$ two efficient location estimators are given. One is a linear combination of order statistics, called $L(F)$, and the other is an estimator derived from a rank test, called $R(F)$. The asymptotic variance of both estimators is then compared for various underlying distributions $H$ and it is shown that the asymptotic variance of $R(F)$ is never larger than the one of $L(F)$.
Harrison D. Weed · Ralph A. Bradley · Z. Govindarajulu·刊期 1974-11-01 · 卷 2（6）·The Annals of Statistics已编入正刊
若干单样本序贯秩检验的停止时间AI 解读
Two models for modified, one-sample, sequential probability ratio tests based on Lehmann alternatives are considered, one developed by Weed and Bradley and one by Govindarajulu. It is shown how they are related. Sure termination of the SPRT's is established under very general conditions.
In the estimation of a location parameter the Hodges-Lehmann estimator is known to have some "robust" properties, but it is very "expensive" for large sample sizes. By using the linearity of a special rank statistic we can find a linearized version which requires only $O(n \log n)$ operations.
We consider functions $\alpha(\bullet)$ and $\hat{\alpha}(\bullet)$ on a finite set $S$ which correspond to a function $M(\bullet)$ on the nonempty subsets of $S$ which has the Cauchy mean value property (i.e., $M(A + B)$ is between $M(A)$ and $M(B)$ whenever $A$ and $B$ are nonempty disjoint subsets of $S$). $\hat{\alpha}(\bullet)$ is isotone with respect to a partial ordering on $S$ and is equal to $\alpha(\bullet)$ when $\alpha(\bullet)$ is isotone. It is shown that $\hat{\alpha}(\bullet)$…
The conditions $|\phi_k| \leqq 1$ for all $k = 1,2, \cdots$ and $|\phi_k| = 1$ implies $\phi_{k+1} = \phi_k$ are both necessary and sufficient for a sequence of real numbers $\{\phi_k; k = 1,2, \cdots\}$ to be the partial autocorrelation function for a real, discrete parameter, stationary time series. If all partial autocorrelations beyond the $p$th are zero, the series is an autoregression. If all beyond the $p$th have magnitude unity, the series satisfies a homogeneous…
Consider the problem of estimating the common mean of two normal distributions. Two new unbiased estimators of the common mean are offered for the equal sample size case. Both are better than the sample mean based on one population for sample sizes of 5 or more. A slight modification of one of the estimators is better than either sample mean simultaneously for sample sizes of 10 or more. This same estimator has desirable large sample properties and an explicit simple upper…
The distribution $\Psi(\mathbf{x}; Z, M) = \operatorname{const}. \exp(\mathrm{tr} (ZM^T \mathbf{xx}^T M))$ on the unit sphere in three-space is discussed. It is parametrized by the diagonal shape and concentration matrix $Z$ and the orthogonal orientation matrix $M. \Psi$ is applicable in the statistical analysis of measurements of random undirected axes. Exact and asymptotic sampling distributions are derived. Maximum likelihood estimators for $Z$ and $M$ are found and…
Let $\{X_n\}$ be a sequence of i.i.d. random variables, each with probability density function $p(x \mid \theta, \xi)$ subject to certain regularity conditions. Here, $\theta$ is an $s$-dimensional vector of nuisance parameters, and $\xi \in (-r, r)$ is the parameter under test. The first $N$ members of the sequence $\{X_n\}$ are to be used for testing the hypothesis, $H_0: \xi = 0$, against the alternative, $H_1: \xi\neq 0$, while $\theta$ remains unspecified. The…
We consider procedures for statistical inference based on the smallest $r$ observations from a random sample. This method of sampling is of importance in life testing. Under weak regularity conditions which include the existence of a q.m. derivative for the square root of the ratio of densities, we obtain an approximation to the likelihood and establish the asymptotic normality of the approximation. This enables us to reach several important conclusions concerning the…
Consider the problem of estimating the location parameter $\theta \in R^d$ based on a sample of size $n$ from $(\theta + X, Y)$, where $X$ is a $d$-dimensional random vector, $Y$ is a random element of some measure space, and $(X, Y)$ has a known distribution. Let $\mathscr{J}^-$ denote the corresponding inverse Fisher information matrix. We show that there is always an invariant estimator $\hat{\theta}_n$ such that $\mathscr{L}(n^{\frac{1}{2}}(\hat{\theta}_n - \theta)) \rightarrow N(0, \mathscr{J}^-)$…
The theory of periodic smoothing splines is presented, with application to the estimation of periodic functions. Several theorems relating the order of the differential operator defining the spline to the saturation (order of bias) of the estimator are proven. The linear operator which maps a function to its periodic continuous smoothing spline approximation is represented as a convolution operator with a given convolution kernel. This operator is shown to be the limit of a…
For a wide class of stationary random sequences possessing a spectral density function, the variance of the best linear unbiased estimator for the mean is seen to depend asymptotically only on the behavior of the spectral density near the origin. Asymptotically efficient estimators based only on this behavior may be chosen. For spectral densities behaving like $\lambda^\nu$ at the origin, $\nu > -1$ a constant, the minimum variance decreases like $n^{-\nu-1}$, where $n$ is…
In case the joint density $f$ of $X = (X_1, \cdots, X_n)$ is Schur-concave (is an order-reversing function for the partial ordering of majorization), it is shown that $P(X \in A + \theta)$ is a Schur-concave function of $\theta$ whenever $A$ has a Schur-concave indicator function. More generally, the convolution of Schur-concave functions is Schur-concave. The condition that $f$ is Schur-concave implies that $X_1, \cdots, X_n$ are exchangeable. With exchangeability, the…
A random process called the Dirichlet process whose sample functions are almost surely probability measures has been proposed by Ferguson as an approach to analyzing nonparametric problems from a Bayesian viewpoint. An important result obtained by Ferguson in this approach is that if observations are made on a random variable whose distribution is a random sample function of a Dirichlet process, then the conditional distribution of the random measure can be easily…
Some generalizations of Dupac's dynamic stochastic approximation have been worked out to the more general cases of time variation. Sufficient conditions for convergence in the mean square and with probability one are given in case of deterministic trend and convergence with a bound is proved for the random trend case, using the estimation scheme $x_{n+1} = g_n(x_n) + a_n(\alpha - y_{n+1}(g_n(x_n)))$. This estimation procedure seems to be of practical use to a variety of…
The main result in this paper concerns the limiting behavior of normalized cumulative sums of induced order statistics obtained from $n$ independent two-dimensional random vectors, as $n$ increases indefinitely. By means of a Skorokhod-type embedding of these cumulative sums on Brownian Motion paths, it is shown that under certain conditions the sample paths of these normalized sums converge in a certain sense to a process obtained from the Brownian Motion by a…
It is shown that among all cumulative distribution functions passing through $k \geqq 2$ given points there is a unique one with minimal Fisher information; it is obtained by a curious type of spline interpolation. This answers some questions raisd by D. G. Kendall and J. W. Tukey.
Statistical properties of a variant of the histospline density estimate introduced by Boneva-Kendall-Stefanov are obtained. The estimate we study is formed for x in a finite interval, x∈[a,b]=[0,1] say, by letting F^n(x),xϵ[0,1] be the unique cubic spline of interpolation to the sample cumulative distribution function Fn(x) at equi-spaced points x=jh,j=0,1,⋯,l+1,(l+1)h=1, which…
Let Wp(m)(M)={f:f(ν)abs.cont.,ν=0,1,⋯,m−1,f(m)∈Lp,∥f(m)∥p≦M}, where ∥⋅∥p is the norm in Lp,m is a positive integer and p is a real number, p≧1. Let {f^n(x)},n=1,2,⋯ be any sequence of estimates of a density at the point x where f^n(x) depends on n independent observations from some density f∈Wp(m)(M). It is…
For a given sufficiently regular distribution F two efficient location estimators are given. One is a linear combination of order statistics, called L(F), and the other is an estimator derived from a rank test, called R(F). The asymptotic variance of both estimators is then compared for various underlying distributions H and it is shown that the asymptotic variance of R(F) is never larger than the one of L(F).
In the estimation of a location parameter the Hodges-Lehmann estimator is known to have some "robust" properties, but it is very "expensive" for large sample sizes. By using the linearity of a special rank statistic we can find a linearized version which requires only O(nlogn) operations.
We consider functions α(∙) and α^(∙) on a finite set S which correspond to a function M(∙) on the nonempty subsets of S which has the Cauchy mean value property (i.e., M(A+B) is between M(A) and M(B) whenever A and B are nonempty disjoint subsets of S). α^(∙) is isotone with respect to a partial ordering on S and is equal to α(∙) when α(∙) is isotone. It is shown that α^(∙)…
The conditions ∣ϕk∣≦1 for all k=1,2,⋯ and ∣ϕk∣=1 implies ϕk+1=ϕk are both necessary and sufficient for a sequence of real numbers {ϕk;k=1,2,⋯} to be the partial autocorrelation function for a real, discrete parameter, stationary time series. If all partial autocorrelations beyond the pth are zero, the series is an autoregression. If all beyond the pth have magnitude unity, the series satisfies a homogeneous…
The distribution Ψ(x;Z,M)=const.exp(tr(ZMTxxTM)) on the unit sphere in three-space is discussed. It is parametrized by the diagonal shape and concentration matrix Z and the orthogonal orientation matrix M.Ψ is applicable in the statistical analysis of measurements of random undirected axes. Exact and asymptotic sampling distributions are derived. Maximum likelihood estimators for Z and M are found and…
Maxmin C(α) Tests Against Two-Sided Alternatives
Let {Xn} be a sequence of i.i.d. random variables, each with probability density function p(x∣θ,ξ) subject to certain regularity conditions. Here, θ is an s-dimensional vector of nuisance parameters, and ξ∈(−r,r) is the parameter under test. The first N members of the sequence {Xn} are to be used for testing the hypothesis, H0:ξ=0, against the alternative, H1:ξ=0, while θ remains unspecified. The…
Asymptotic Results for Inference Procedures Based on the r Smallest Observations
We consider procedures for statistical inference based on the smallest r observations from a random sample. This method of sampling is of importance in life testing. Under weak regularity conditions which include the existence of a q.m. derivative for the square root of the ratio of densities, we obtain an approximation to the likelihood and establish the asymptotic normality of the approximation. This enables us to reach several important conclusions concerning the…
Consider the problem of estimating the location parameter θ∈Rd based on a sample of size n from (θ+X,Y), where X is a d-dimensional random vector, Y is a random element of some measure space, and (X,Y) has a known distribution. Let J− denote the corresponding inverse Fisher information matrix. We show that there is always an invariant estimator θ^n such that L(n21(θ^n−θ))→N(0,J−)…
For a wide class of stationary random sequences possessing a spectral density function, the variance of the best linear unbiased estimator for the mean is seen to depend asymptotically only on the behavior of the spectral density near the origin. Asymptotically efficient estimators based only on this behavior may be chosen. For spectral densities behaving like λν at the origin, ν>−1 a constant, the minimum variance decreases like n−ν−1, where n is…
In case the joint density f of X=(X1,⋯,Xn) is Schur-concave (is an order-reversing function for the partial ordering of majorization), it is shown that P(X∈A+θ) is a Schur-concave function of θ whenever A has a Schur-concave indicator function. More generally, the convolution of Schur-concave functions is Schur-concave. The condition that f is Schur-concave implies that X1,⋯,Xn are exchangeable. With exchangeability, the…
Some generalizations of Dupac's dynamic stochastic approximation have been worked out to the more general cases of time variation. Sufficient conditions for convergence in the mean square and with probability one are given in case of deterministic trend and convergence with a bound is proved for the random trend case, using the estimation scheme xn+1=gn(xn)+an(α−yn+1(gn(xn))). This estimation procedure seems to be of practical use to a variety of…
The main result in this paper concerns the limiting behavior of normalized cumulative sums of induced order statistics obtained from n independent two-dimensional random vectors, as n increases indefinitely. By means of a Skorokhod-type embedding of these cumulative sums on Brownian Motion paths, it is shown that under certain conditions the sample paths of these normalized sums converge in a certain sense to a process obtained from the Brownian Motion by a…
It is shown that among all cumulative distribution functions passing through k≧2 given points there is a unique one with minimal Fisher information; it is obtained by a curious type of spline interpolation. This answers some questions raisd by D. G. Kendall and J. W. Tukey.