Partial prior knowledge is quantified by an interval $I(L, U)$ of $\sigma$-finite prior measures $Q$ satisfying $L(A) \leq Q(A) \leq U(A)$ for all measurable sets $A$, and is interpreted as acceptance of a family of bets. The concept of conditional probability distributions is generalized to that of conditional measures, and Bayes theorem is extended to accommodate unbounded priors. According to Bayes theorem, the interval $I(L, U)$ of prior measures is transformed upon…
A technique is introduced to prove limit theorems for simple linear rank statistics by means of an approximating statistic based on sequential ranks. This approximation is shown to be close enough to prove asymptotic normality of simple linear rank statistics under the null hypothesis and to obtain a bound on the rate of convergence to normality when the score function is unbounded. In addition, a law of the iterated logarithm and an invariance principle are given for simple…
It is suggested that problems in a reliability context may be handled by a Bayesian nonparametric approach. A stochastic process is defined whose sample paths may be assumed to be increasing hazard rates by properly choosing the parameter functions of the process. The posterior distribution of the hazard rates is derived for both exact and censored data. Bayes estimates of hazard rates and $\operatorname{cdf's}$ are found under squared error type loss functions. Some…
The results of Wald on the consistency of the maximum likelihood estimate are extended. Applications are made to mixture distributions and to clustering when the number of clusters is not known.
This paper establishes the asymptotic optimality (in the sense of Robbins) of the empirical Bayes distribution function created from the Bayes rule relative to the Dirichlet process prior with unknown parameter $\alpha(.)$. It will follow that the same result applies to the estimation of the mean of a distribution function.
Suppose we have to decide on the basis of appropriately drawn samples which of $k$ treatment populations are "good" compared to either given control values or to a control population from which an additional sample is available. The unknown parameters are assumed to vary randomly according to a prior distribution about which we only have the partial knowledge that it is contained in a given class $\Gamma$ of priors. Though we derive in both cases (under the assumption of…
This paper shows that Hotelling's $T^2$-test for testing $\mu = 0$ in the one-sample problem is robust against departures from normality in the following sense. It is still UMPI in a broad class of distributions, and the null distribution under any member of the class is the same as that under normality.
We extend Bickel's tests for heteroscedasticity to include wider classes of test statistics and fitting methods. The test statistics include those based on Huber's function, while the fitting techniques include Huber's Proposal 2 for robust regression.
Let $U_N$ be a one sample $U$-statistic with kernel $h$ of degree two, such that $Eh(X_1, X_2) = \vartheta$ and $\operatorname{Var} E\lbrack h(X_1, X_2)\mid X_1 \rbrack > 0$. It is shown that for a studentized $U$-statistic $\sup_x|P(N^{1/2}S^{-1}_N(U_N - \vartheta) \leqslant x) - \Phi(x)| = O(N^{-1/2})$ as $N \rightarrow \infty$, where $N^{-1}S^2_N = 4N^{-1}(N - 1)(N - 2)^{-2}\sum^N_{i=1} \lbrack (N - 1)^{-1} \sum_{j\neq i} h(X_i, X_j) - U_N \rbrack^2$ is the jackknife…
With the ranking after alignment procedure, a conditional permutational argument is introduced which gives rise to a conditional permutation distribution for the test statistic in randomized block designs. In this paper, the almost sure and uniform convergence of this distribution is derived for a wide class of test statistics as the number of blocks goes to infinity.
Let $\{X_n, n \geq 1\}$ be a sequence of independent identically distributed random variables with mean $\mu$ and unknown variance $\sigma^2$. We want to estimate $\mu$ by $\bar{X}_n$ with a loss function of $\sigma^{2 \delta - 2}(\bar{X}_n - \mu)^2 + \lambda n$, where $\delta > 0$ and $\lambda \rightarrow 0 +$. For $n_\lambda = o(\lambda^{-1/2})$ and $n_\lambda(\log \lambda)^{-1} \rightarrow -\infty$ as $\lambda \rightarrow 0+$, set $T = \inf\{n \geq n_\lambda: n^{-1} \sum^n_{i=1} (X_i - \bar{X}_n) + b_n \leq \lambda^{1/\delta} a_n\}$…
We are concerned with the admissibility of nonlinear estimators of the form $(aX + b)/(cX + d)$ in the one-parameter exponential family, in estimating $g(\theta)$ with quadratic loss. Our method will be reminiscent of that of Karlin who gave sufficient conditions for admissibility of linear estimators $aX$ in estimating the mean in the one-parameter family. Our results generalize those of Ghosh and Meeden who studied admissibility of $aX + b$ for estimating an arbitrary…
In the classical theory of competing risks (as well as in many reliability models and incomplete data problems) it is assumed that (A1) the risks (i.e., the random variables of interest) are independent and that (A2) death does not result from simultaneous causes. Employing our probabilistic solution to a related problem in probability modelling, we obtain strongly consistent estimators for the unobservable marginal distributions of interest. These estimators are analogous…
Let $f$ be a density possessing some smoothness properties and let $X_1,\cdots, X_n$ be independent observations from $f$. Some desirable properties of orthogonal series density estimates $f_{n,m,\lambda}$ of $f$ of the form $f_{n,m,\lambda}(t) = \sum^n_{\nu = 1} \frac{\hat{f}_\nu}{(1 + \lambda\nu^{2m})} \phi_\nu(t)$ where $\{\phi_\nu\}$ is an orthonormal sequence and $\hat{f}_\nu = (1/n)\sum^n_{j=1} \phi_\nu(X_j)$ is an estimate of $f_\nu = \int \phi_\nu(t)f(t) dt$, are…
Given a sample of independent random variables $Z_1, Z_2, \cdots, Z_n$ with identical distribution $p$ on a compact metric space $(M, d)$, a measure of central tendency is a sample centroid (of order $r > 0$) defined as a point $\hat{X}_n$ in $M$ satisfying $\frac{1}{n} \sum^n_{i=1} d^r(\hat{X}_n, Z_i) = \inf_{x \in M} \frac{1}{n} \sum^n_{i=1} d^r(x, Z_i).$ A (population) centroid of $Z$ is any point $x^\ast$ in $M$ such that $\int_M d^r(x^\ast, z) dp(z) = \inf_{x \in M} \int_M d^r(x, z) dp(z).$…
The Bayesian bootstrap is the Bayesian analogue of the bootstrap. Instead of simulating the sampling distribution of a statistic estimating a parameter, the Bayesian bootstrap simulates the posterior distribution of the parameter; operationally and inferentially the methods are quite similar. Because both methods of drawing inferences are based on somewhat peculiar model assumptions and the resulting inferences are generally sensitive to these assumptions, neither method…
Let $X_1, \cdots, X_n$ be i.i.d. $P(X > u) = F(u)$ and $Y_1, \cdots, Y_n$ be i.i.d. $P(Y > u) = G(u)$, where both $F$ and $G$ are unknown continuous distributions. For $i = 1, \cdots, n$ set $\delta_i = 1$ if $X_i \leq Y_i$ and 0 if $X_i > Y_i$ and $Z_i = \min \{X_i, Y_i\}$. One way to estimate $F$ from the observations $(Z_i, \delta_i) i = 1, \cdots, n$ is by means of the product limit (PL) estimator, $F^\ast_n$ (Kaplan-Meier, [8]). In this paper it is shown that $F^\ast_n$…
For the Cox regression model, the partial likelihood functions involve linear combinations of induced order statistics. Some invariance principles pertaining to such linear combinations of induced order statistics are studied and the theory is incorporated in the formulation of some repeated significance tests (for the hypothesis of no regression) based on these partial likelihoods.
Strong consistency and asymptotic normality are established for the maximum partial likelihood estimate of the regression parameter in Cox's regression model. Estimates are also derived for the underlying cumulative hazard function and survival distribution. We establish the asymptotic normality of these estimates and calculate the limiting variances.
In this paper influence curves for censored data estimators are considered. The influence curve of the Kaplan-Meier estimate of survival time is calculated, and it is shown that the influence curve provides an alternative derivation of the asymptotic variance of this estimate. A chain rule for influence curves is established and is used to calculate the influence curves of censored data estimators that are functions of the Kaplan-Meier estimate. The robustness properties and…
Smoothing of the data by averaging is suggested in order to study the maximum. The maximum of the smoothed data is approximated by that of a Gaussian sample and thus is more robust against outliers.
In a multivariate "errors in variables" regression model, the unknown mean vectors $\mathbf{u}_{1i}: p \times 1, \mathbf{u}_{2i}: r \times 1$ of the vector observations $\mathbf{x}_{1i}, \mathbf{x}_{2i}$, rather than the observations themselves, are assumed to follow the linear relation: $\mathbf{u}_{2i} = \alpha + B\mathbf{u}_{1i}, i = 1,2,\cdots, n$. It is further assumed that the random errors $\mathbf{e}_i = \mathbf{x}_i - \mathbf{u}_i, \mathbf{x}'_i = (\mathbf{x}'_{1i}, \mathbf{x}'_{2i}), \mathbf{u}'_i = (\mathbf{u}'_{1i}, \mathbf{u}'_{2i})$…
A sampling strategy consists of a sampling design and an estimator. Of special importance are strategies combining simple random sampling with the sample mean as an estimator. We consider this class of strategies and the smaller class of strategies with simple random sampling of (almost) fixed sample size. The convexity of the underlying loss function does not imply the completeness of the smaller class. However the strict convexity of the loss function together with the…
This paper is concerned with basic problems of statistical inference. The thesis is in three parts: (1) that inference is a procedure whereby one passes from a population (or sample) to a new individual; (2) that this connection can be established using de Finetti's idea of exchangeability or Fisher's concept of a subpopulation; (3) in making the connection use must be made of the appropriate probability. These three principles are used in a variety of situations and the…
Partial prior knowledge is quantified by an interval I(L,U) of σ-finite prior measures Q satisfying L(A)≤Q(A)≤U(A) for all measurable sets A, and is interpreted as acceptance of a family of bets. The concept of conditional probability distributions is generalized to that of conditional measures, and Bayes theorem is extended to accommodate unbounded priors. According to Bayes theorem, the interval I(L,U) of prior measures is transformed upon…
It is suggested that problems in a reliability context may be handled by a Bayesian nonparametric approach. A stochastic process is defined whose sample paths may be assumed to be increasing hazard rates by properly choosing the parameter functions of the process. The posterior distribution of the hazard rates is derived for both exact and censored data. Bayes estimates of hazard rates and cdf′s are found under squared error type loss functions. Some…
This paper establishes the asymptotic optimality (in the sense of Robbins) of the empirical Bayes distribution function created from the Bayes rule relative to the Dirichlet process prior with unknown parameter α(.). It will follow that the same result applies to the estimation of the mean of a distribution function.
Γ-Minimax Selection Procedures in Simultaneous Testing Problems
Suppose we have to decide on the basis of appropriately drawn samples which of k treatment populations are "good" compared to either given control values or to a control population from which an additional sample is available. The unknown parameters are assumed to vary randomly according to a prior distribution about which we only have the partial knowledge that it is contained in a given class Γ of priors. Though we derive in both cases (under the assumption of…
A Robustness Property of Hotelling's T2-Test
This paper shows that Hotelling's T2-test for testing μ=0 in the one-sample problem is robust against departures from normality in the following sense. It is still UMPI in a broad class of distributions, and the null distribution under any member of the class is the same as that under normality.
The Order of the Normal Approximation for a Studentized U-Statistic
Let UN be a one sample U-statistic with kernel h of degree two, such that Eh(X1,X2)=ϑ and VarE[h(X1,X2)∣X1]>0. It is shown that for a studentized U-statistic supx∣P(N1/2SN−1(UN−ϑ)⩽x)−Φ(x)∣=O(N−1/2) as N→∞, where N−1SN2=4N−1(N−1)(N−2)−2∑i=1N[(N−1)−1∑j=ih(Xi,Xj)−UN]2 is the jackknife…
Let {Xn,n≥1} be a sequence of independent identically distributed random variables with mean μ and unknown variance σ2. We want to estimate μ by Xˉn with a loss function of σ2δ−2(Xˉn−μ)2+λn, where δ>0 and λ→0+. For nλ=o(λ−1/2) and nλ(logλ)−1→−∞ as λ→0+, set T=inf{n≥nλ:n−1∑i=1n(Xi−Xˉn)+bn≤λ1/δan}…
We are concerned with the admissibility of nonlinear estimators of the form (aX+b)/(cX+d) in the one-parameter exponential family, in estimating g(θ) with quadratic loss. Our method will be reminiscent of that of Karlin who gave sufficient conditions for admissibility of linear estimators aX in estimating the mean in the one-parameter family. Our results generalize those of Ghosh and Meeden who studied admissibility of aX+b for estimating an arbitrary…
Let f be a density possessing some smoothness properties and let X1,⋯,Xn be independent observations from f. Some desirable properties of orthogonal series density estimates fn,m,λ of f of the form fn,m,λ(t)=∑ν=1n(1+λν2m)f^νϕν(t) where {ϕν} is an orthonormal sequence and f^ν=(1/n)∑j=1nϕν(Xj) is an estimate of fν=∫ϕν(t)f(t)dt, are…
Given a sample of independent random variables Z1,Z2,⋯,Zn with identical distribution p on a compact metric space (M,d), a measure of central tendency is a sample centroid (of order r>0) defined as a point X^n in M satisfying n1∑i=1ndr(X^n,Zi)=infx∈Mn1∑i=1ndr(x,Zi). A (population) centroid of Z is any point x∗ in M such that ∫Mdr(x∗,z)dp(z)=infx∈M∫Mdr(x,z)dp(z).…
Let X1,⋯,Xn be i.i.d. P(X>u)=F(u) and Y1,⋯,Yn be i.i.d. P(Y>u)=G(u), where both F and G are unknown continuous distributions. For i=1,⋯,n set δi=1 if Xi≤Yi and 0 if Xi>Yi and Zi=min{Xi,Yi}. One way to estimate F from the observations (Zi,δi)i=1,⋯,n is by means of the product limit (PL) estimator, Fn∗ (Kaplan-Meier, [8]). In this paper it is shown that Fn∗…
In a multivariate "errors in variables" regression model, the unknown mean vectors u1i:p×1,u2i:r×1 of the vector observations x1i,x2i, rather than the observations themselves, are assumed to follow the linear relation: u2i=α+Bu1i,i=1,2,⋯,n. It is further assumed that the random errors ei=xi−ui,xi′=(x1i′,x2i′),ui′=(u1i′,u2i′)…