For exponential families with density \begin{equation*}x \rightarrow C(\theta, \eta)h(x)\exp\lbrack a(\theta)T(x) + \Sigma^p_{i=1} a_i(\theta, \eta)S_i(x)\rbrack, (\theta, \eta) \in \Theta \times H, \Theta \subset \mathbb{R},\end{equation*} $a$ increasing and continuous, there exists for every sample size an estimator for $\theta$ which is--in the class of all median unbiased estimators--of minimal risk for any monotone loss function.
Let $X_1, \cdots, X_n$ be a random sample from an unknown $\operatorname{cdf} F$, let $y_1, \cdots, y_n$ be known real constants, and let $Z_i = \min(X_i, y_i), i = 1, \cdots, n$. It is required to estimate $F$ on the basis of the observations $Z_1, \cdots, Z_n$, when the loss is squared error. We find a Bayes estimate of $F$ when the prior distribution of $F$ is a process neutral to the right. This generalizes results of Susarla and Van Ryzin who use a Dirichlet process…
Let $Z$ be a random vector whose distribution is spherically symmetric about the origin. A random vector $X$ which is representable as the image of $Z$ under affine transformation is said to have an ellipsoidally symmetric distribution. The model of ellipsoidal symmetry is a useful generalization of multivariate normality. This paper proposes and studies some goodness-of-fit tests which have good asymptotic power over a broad spectrum of alternatives to ellipsoidal symmetry.
The objective in nonparametric regression is to infer a function $m(x)$ on the basis of a finite collection of noisy pairs $\{(X_i, m(X_i) + N_i)\}^n_{i=1}$, where the noise components $N_i$ satisfy certain lenient assumptions and the domain points $X_i$ are selected at random. It is known a priori only that $m$ is a member of a nonparametric class of functions (that is, a class of functions like $C\lbrack 0, 1\rbrack$ which, under customary topologies, does not admit a…
The supremum of the empirical distribution function $F_n$ centered at its expectation $F$ and standardized by division by its standard deviation has recently been shown by Jaeschke to have asymptotically an extreme-value distribution after a second location and scale transformation depending only on the sample size $n$. In this paper the studentized form of the above statistic, obtained by division by the estimated standard deviation, is shown to have the same large sample…
It is well known that the limit distribution of the supremum of the empirical distribution function $F_n$ centered at its expectation $F$ and standardized by division by its standard deviation is degenerate, if the supremum is taken on too large regions $\varepsilon_n < F(u) < \delta_n$. So it is natural to look for sequences of linear transformations, so that for given sequences of sup-regions $(\varepsilon_n, \delta_n)$ the limit of the transformed sup-statistics is…
The relation between fitting autoregression and periodogram is herein presented. More specifically, the asymptotic error covariance matrices of the estimates of the autoregressive parameters using Yule-Walker equations are expressed in terms of the periodogram. These expressions permit the immediate calculation of the error covariance matrices for various time series problems including the autoregressive spectral estimation and fitting autoregression to the data with…
A new asymptotic theory for studying the effect of dependence of the observations in experimental design for the linear model is developed. The uniform design is shown to be asymptotically optimal in a strong sense for estimating location and in a weaker sense for estimating the slope of a straight line regression. Numerical results supporting the asymptotics appear in a companion paper.
While usual sequential analysis deals with i.i.d. observations, this paper studies sequential tests for the dependent case of sampling without replacement from a finite population. A general weak convergence theorem is obtained and it is applied to the asymptotic analysis of the tests. Motivated by such applications as election predictions and acceptance sampling, the case of hypergeometric populations is studied in detail and a simple test with a triangular continuation…
We discuss the following problem: given a random sample $\mathbf{X} = (X_1, X_2, \cdots, X_n)$ from an unknown probability distribution $F$, estimate the sampling distribution of some prespecified random variable $R(\mathbf{X}, F)$, on the basis of the observed data $\mathbf{x}$. (Standard jackknife theory gives an approximate mean and variance in the case $R(\mathbf{X}, F) = \theta(\hat{F}) - \theta(F), \theta$ some parameter of interest.) A general method, called the…
Haberman's bound for a norm of the difference between the least squares and the best linear unbiased estimators in a linear model with nonsingular covariance structure is examined in the particular case when a vector norm involved is taken as the Euclidean one. In this frequently occurring case, a new substantially improved bound is developed which, furthermore, is applicable regardless of any additional condition.
We consider the estimation of parameters in the time series model $X(t) = \sum^q_{j = 1}a_jX(t - j) + \varepsilon(t) - \varepsilon(t - j) - \sum^p_{j = 1} c_j\{\varepsilon(t - j) - \varepsilon(t - j - 1)\}$ where the $\varepsilon(t)$ are independently identically distributed random variables with zero mean and variance $\sigma^2$. We compute the exact $\log$ likelihood of the model, propose and justify an asymptotic approximation of it. The latter will be used to derive…
It is shown that the bivariate density of the absolute normal distribution is totally positive of order 2. Necessary and sufficient conditions are given for the trivariate density of the absolute normal distribution to be totally positive of order 2 in pairs of arguments. These results are then used to show that certain generalized bivariate and trivariate $t, \chi^2$ and $F$ random variables are associated.
Modifications of the $t$-test are considered which are robust under certain violations of the independence assumption. The additional number of observations these modified tests require in order to obtain under independence the same power as the $t$-test, is obtained asymptotically.
As a natural, intuitive model for inferences about certain characteristics of finite populations, Bruce Hill has proposed a sequence of exchangeable variables $X_1, \cdots, X_{n + 1}$ which have distinct values with probability one and have the property that, conditional on $X_1, \cdots, X_n$, the next observation $X_{n + 1}$ is equally likely to fall in any of the $n + 1$ intervals determined by $X_1, \cdots, X_n$. Harold Jeffreys had previously assumed such a model (in the…
A methodology is presented for analyzing periodic autoregressions which is also applicable when inferring the second order properties of periodically correlated processes. In addition, capitalizing on the connection between periodic and multiple autoregressions, a method is set forth for analyzing the latter, which overcomes the usual requirements of a large number of both parameters and computer storage locations. This is achieved by introducing an orthogonal…
Let $x_1, \cdots, x_n$ be i.i.d. random variables with a distribution depending on the real parameter. Under what conditions is a generalized Bayes estimator independent of the choice of the even loss function? The known answer to this question is that this independence holds if the posterior density is symmetric and unimodal. The description of distributions and corresponding generalized prior densities on the real line, for which the posterior density is symmetric and…
An $n \times n$ matrix $H$ with all its entries $+1$ and $-1$ is Hadamard if $HH' = nI$. It is well known that $n$ must be 1, 2 or a multiple of 4 for such a matrix to exist, but is not known whether Hadamard matrices exist for every $n$ which is a multiple of 4. The smallest order for which a Hadamard matrix has not been constructed is (as of 1977) 268. Research in the area of Hadamard matrices and their applications has steadily and rapidly grown, especially during the…
A method of estimating the parameters of a linear regression model when the covariance matrix is an unknown diagonal matrix is investigated. It is assumed that the observations fall into $k$ groups with constant error variance for a group. The estimation is carried out in two steps, the first step being an ordinary least squares regression. The least squares residuals are used to estimate the covariance matrix and the second step is the calculation of the generalized least…
In generalized ridge estimation, the components of the ordinary least squares (OLS) regression coefficient vector which lie along the principal axes of the given regressor data are rescaled using known ridge factors. Generalizing a result of Swindel and Chapman, it is shown that, if each ridge factor is nonstochastic, nonnegative, and less than one, then there is at most one unknown direction in regression coefficient space along which ridge coefficients have larger mean…
A decision problem is characterized by a loss function $V$ and opinion $H$. The pair $(V, H)$ is said to be strongly stable iff for every sequence $F_n \rightarrow_\omega H, G_n \rightarrow_\omega H$ and $L_n\rightarrow V, W_n\rightarrow V$ uniformly, $\lim_{\varepsilon \downarrow 0} \lim \sup_{n\rightarrow \infty} \lbrack \int L_n(\theta, D_n(\varepsilon)) dF_n(\theta) - \inf_D \int L_n(\theta, D) dF_n(\theta)\rbrack = 0$ for every sequence $D_n(\varepsilon)$ satisfying $\int W_n(\theta, D_n(\varepsilon)) dG_n(\theta) \leqq \inf_D \int W_n(\theta, D) dG_n(\theta) + \varepsilon.$…
In order to obtain quantitative results about the influence of outliers on tests, their maximum size and minimum power over certain neighborhoods are evaluated asymptotically. The neighborhoods are defined in terms of $\varepsilon$-contamination and total variation, the tests considered are based on statistics $n^{-\frac{1}{2}} \sum^n_{i=1} IC(x_i), IC$ a rather arbitrary function. Furthermore, the unique $IC^\ast$ is determined that leads to a maximin test with respect to…
Consider a time-continuous nonhomogeneous Markovian process $V$ having state space $A^0$. For $A \subset A^0$ and $i, j \in A, P_{Aij}(\tau, t)$ is the $i \rightarrow j$ transition probability of the Markovian process $V_A$ which arises in the hypothetical situation where states $A^0 - A$ have been eliminated from the state space of $V$. Let $\hat{P}_{Aij}(\tau, t)$ be the generalized product-limit estimator of $P_{Aij}(\tau,t)$. It is shown that the vector consisting of…
For exponential families with density x→C(θ,η)h(x)exp[a(θ)T(x)+Σi=1pai(θ,η)Si(x)],(θ,η)∈Θ×H,Θ⊂R,a increasing and continuous, there exists for every sample size an estimator for θ which is--in the class of all median unbiased estimators--of minimal risk for any monotone loss function.
Let X1,⋯,Xn be a random sample from an unknown cdfF, let y1,⋯,yn be known real constants, and let Zi=min(Xi,yi),i=1,⋯,n. It is required to estimate F on the basis of the observations Z1,⋯,Zn, when the loss is squared error. We find a Bayes estimate of F when the prior distribution of F is a process neutral to the right. This generalizes results of Susarla and Van Ryzin who use a Dirichlet process…
Let Z be a random vector whose distribution is spherically symmetric about the origin. A random vector X which is representable as the image of Z under affine transformation is said to have an ellipsoidally symmetric distribution. The model of ellipsoidal symmetry is a useful generalization of multivariate normality. This paper proposes and studies some goodness-of-fit tests which have good asymptotic power over a broad spectrum of alternatives to ellipsoidal symmetry.
The objective in nonparametric regression is to infer a function m(x) on the basis of a finite collection of noisy pairs {(Xi,m(Xi)+Ni)}i=1n, where the noise components Ni satisfy certain lenient assumptions and the domain points Xi are selected at random. It is known a priori only that m is a member of a nonparametric class of functions (that is, a class of functions like C[0,1] which, under customary topologies, does not admit a…
The supremum of the empirical distribution function Fn centered at its expectation F and standardized by division by its standard deviation has recently been shown by Jaeschke to have asymptotically an extreme-value distribution after a second location and scale transformation depending only on the sample size n. In this paper the studentized form of the above statistic, obtained by division by the estimated standard deviation, is shown to have the same large sample…
It is well known that the limit distribution of the supremum of the empirical distribution function Fn centered at its expectation F and standardized by division by its standard deviation is degenerate, if the supremum is taken on too large regions εn<F(u)<δn. So it is natural to look for sequences of linear transformations, so that for given sequences of sup-regions (εn,δn) the limit of the transformed sup-statistics is…
We discuss the following problem: given a random sample X=(X1,X2,⋯,Xn) from an unknown probability distribution F, estimate the sampling distribution of some prespecified random variable R(X,F), on the basis of the observed data x. (Standard jackknife theory gives an approximate mean and variance in the case R(X,F)=θ(F^)−θ(F),θ some parameter of interest.) A general method, called the…
We consider the estimation of parameters in the time series model X(t)=∑j=1qajX(t−j)+ε(t)−ε(t−j)−∑j=1pcj{ε(t−j)−ε(t−j−1)} where the ε(t) are independently identically distributed random variables with zero mean and variance σ2. We compute the exact log likelihood of the model, propose and justify an asymptotic approximation of it. The latter will be used to derive…
Positive Dependence of the Bivariate and Trivariate Absolute Normal, t,χ2, and F Distributions
It is shown that the bivariate density of the absolute normal distribution is totally positive of order 2. Necessary and sufficient conditions are given for the trivariate density of the absolute normal distribution to be totally positive of order 2 in pairs of arguments. These results are then used to show that certain generalized bivariate and trivariate t,χ2 and F random variables are associated.
Modifications of the t-test are considered which are robust under certain violations of the independence assumption. The additional number of observations these modified tests require in order to obtain under independence the same power as the t-test, is obtained asymptotically.
As a natural, intuitive model for inferences about certain characteristics of finite populations, Bruce Hill has proposed a sequence of exchangeable variables X1,⋯,Xn+1 which have distinct values with probability one and have the property that, conditional on X1,⋯,Xn, the next observation Xn+1 is equally likely to fall in any of the n+1 intervals determined by X1,⋯,Xn. Harold Jeffreys had previously assumed such a model (in the…
Let x1,⋯,xn be i.i.d. random variables with a distribution depending on the real parameter. Under what conditions is a generalized Bayes estimator independent of the choice of the even loss function? The known answer to this question is that this independence holds if the posterior density is symmetric and unimodal. The description of distributions and corresponding generalized prior densities on the real line, for which the posterior density is symmetric and…
An n×n matrix H with all its entries +1 and −1 is Hadamard if HH′=nI. It is well known that n must be 1, 2 or a multiple of 4 for such a matrix to exist, but is not known whether Hadamard matrices exist for every n which is a multiple of 4. The smallest order for which a Hadamard matrix has not been constructed is (as of 1977) 268. Research in the area of Hadamard matrices and their applications has steadily and rapidly grown, especially during the…
A method of estimating the parameters of a linear regression model when the covariance matrix is an unknown diagonal matrix is investigated. It is assumed that the observations fall into k groups with constant error variance for a group. The estimation is carried out in two steps, the first step being an ordinary least squares regression. The least squares residuals are used to estimate the covariance matrix and the second step is the calculation of the generalized least…
A decision problem is characterized by a loss function V and opinion H. The pair (V,H) is said to be strongly stable iff for every sequence Fn→ωH,Gn→ωH and Ln→V,Wn→V uniformly, limε↓0limsupn→∞[∫Ln(θ,Dn(ε))dFn(θ)−infD∫Ln(θ,D)dFn(θ)]=0 for every sequence Dn(ε) satisfying ∫Wn(θ,Dn(ε))dGn(θ)≦infD∫Wn(θ,D)dGn(θ)+ε.…
In order to obtain quantitative results about the influence of outliers on tests, their maximum size and minimum power over certain neighborhoods are evaluated asymptotically. The neighborhoods are defined in terms of ε-contamination and total variation, the tests considered are based on statistics n−21∑i=1nIC(xi),IC a rather arbitrary function. Furthermore, the unique IC∗ is determined that leads to a maximin test with respect to…
Consider a time-continuous nonhomogeneous Markovian process V having state space A0. For A⊂A0 and i,j∈A,PAij(τ,t) is the i→j transition probability of the Markovian process VA which arises in the hypothetical situation where states A0−A have been eliminated from the state space of V. Let P^Aij(τ,t) be the generalized product-limit estimator of PAij(τ,t). It is shown that the vector consisting of…