Consider a time-continuous nonhomogeneous Markovian stochastic process $V$ having state space $A^0$. Let $A \subset A^0$ and let $P_{Aij}(\tau, t)$ be the $i \rightarrow j$ transition probability of the Markovian stochastic process $V_A$ arising in the hypothetical situation where states $A^0 - A$ have been eliminated from the state space of $V$. Based upon the concept of Kaplan and Meier's product-limit estimator, a nonparametric estimator $\hat{P}_{Aij}(\tau, t)$ is…
The rate of convergence of the point estimators for the parameter based on the sample mean of i.i.d. random vectors is obtained. The result is used to prove the Bahadur's efficiency of the regular best asymptotically normal estimator when the underlying distribution is in the exponential family. An example is given to show that if the distribution is not in the exponential family, then the regular best asymptotically normal estimator is not necessarily efficient in Bahadur's…
In this paper Pitman's method of constructing and comparing tests based on statistics which are asymptotically normal under the null hypothesis and the local alternatives is extended to sequential tests of statistical hypotheses. The asymptotic normality assumption in Pitman's theory is replaced in its sequential analogue by the weak convergence of normalized processes formed from these statistics under the null hypothesis and the local alternatives. Uniform invariance…
This paper considers the general linear regression model $Y_i = \sum_j \beta_j x_{ij} + \varepsilon_i$, and studies the problem of testing hypotheses about some of the $\beta$'s while regarding others as nuisance parameters. The test criteria discussed, which are based on ranks of residuals, are shown to be asymptotically distribution-free.
Asymptotic expansions are established for the power of distribution-free tests in the two-sample problem. These expansions are then used to obtain deficiencies in the sense of Hodges and Lehmann for distribution-free tests with respect to their parametric competitors and for the estimators of shift associated with these tests.
An asymptotic expansion is obtained for the distribution function of the standardized mean of a sample of $s$ observations taken randomly without replacement from a finite population of $n$ numbers. The expansion is given to order $1/n$ and agrees with the formal Edgeworth expansion. The proof of the result is obtained using an approximation to the characteristic function of the standardized sum.
In this note it is shown how to some extent the Edgeworth expansion for the distribution function of Kendall's $\\tau$ can be established by using a well-known general result on such expansions.
It is shown that the problem of asymptotic efficiency (in the Wolfowitz-sense) admits a satisfactory solution for the class of strongly asymptotically median unbiased estimators. The concept of strongly asymptotic median unbiasedness furthermore links up the classical approach (concerning estimator-sequences whose distribution functions converge) with Pfanzagl's concept of median unbiasedness.
This note extends a result of Efron and Morris on domination of the maximum likelihood estimator for the mean of a multivariate normal distribution. We show that this result and our extension follow from a certain differential inequality. In a certain class of estimators having a unique unbiased estimator for the quadratic risk we find necessary and sufficient conditions for risk estimate dominance of a particular set of estimators. We show that, in the sense of risk…
Random variables $X$ and $Y$ are mutually completely dependent if there exists a one-to-one function $g$ for which $P\lbrack Y = g(X)\rbrack = 1.$ An example is presented of a pair of random variables which are mutually completely dependent, but "almost" independent. This example motivates considering a new concept of dependence, called monotone dependence, in which $g$ above is now required to be monotone. Finally, this monotone dependence concept leads to defining and…
Let $q_n(y), 0 < y < 1,$ be a quantile process based on a sequence of i.i.d. rv with distribution function $F$ and density function $f.$ Given some regularity conditions on $F$ the distance of $q_n(y)$ and the uniform quantile process $u_n(y),$ respectively defined in terms of the order statistics $X_{k:n}$ and $U_{k:n} = F(X_{k:n}),$ is computed with rates. As a consequence we have an extension of Kiefer's result on the distance between the empirical and quantile processes,…
A sequence of medians of posterior distributions is approximately median unbiased of order $o(n^{-1}) \operatorname{iff}$ the prior density is equal to the square root of Fisher's information function. It is shown that in this case the sequence of medians of posterior distributions is even an optimum sequence of estimates within the class of all estimator sequences being approximately median unbiased of order $o(n^{-1}).$ The result is proved by showing equivalence with an…
Stochastic approximation procedures were shown by Sakrison to become asymptotically efficient estimators when used to minimize the Kullback-Leibler information, if certain conditions hold. Further results in this direction were obtained by Nevel'son and Has'minskij. This paper gives, first, alternative conditions for convergence and, secondly, shows that, under weaker conditions, asymptotic optimality is obtained by a modified stochastic approximation procedure. The modified…
Asymptotically optimal empirical Bayes squared error loss estimation procedures are developed for three families of continuous distributions, uniform $(0, \theta), \theta > 0,$ uniform $\lbrack \theta, \theta + 1), \theta$ arbitrary, and a location parameter family of gamma distributions. The approach taken is to estimate the Bayes estimator directly. However, for the $\lbrack \theta, \theta + 1)$ case, it is shown that the indirect approach of applying the Bayes estimator,…
One-sample linear rank tests are considered for the case where the observations are not independent but come from an autoregressive process. It is proposed to apply the tests under these circumstances to certain transformations of the observations, rather than to the observations themselves. Then the tests have asymptotically the same properties as under independence, both under the hypothesis and under contiguous location alternatives. In particular, they are asymptotically…
A reduced $U$-statistic (of order 2) is defined as the sum of terms $f(X_i, X_j),$ where $f$ is a symmetric function, $(X_1, \cdots, X_N)$ are independent and identically distributed (i.i.d.) random variables (rv's), and $(i,j)$ are drawn from a restricted, though balanced, set of pairs. (A $U$-statistic corresponds to summation over all $(i, j)$ pairs.) A limit normal distribution is found for the reduced $U$-statistic, and it follows that estimates based on reduced $U$-stat…
An asymptotic expression is given for the $\log$ error probability of a sequential test based on a random walk. This may be used to compute limiting relative efficiencies of such tests. The results are illustrated for the one-sided normal testing problem with an asymptotic Bayes test due to Schwarz. Some numerical comparisons are given for five sequential tests of a normal mean.
Consider a random sample from an absolutely continuous multivariate distribution. Let $\mathscr{J}$ be a class of sets which are not too long and thin. A point $\mathbf{\theta}_n$ chosen from a minimum volume set $S_n \in \mathscr{J}$ containing at least $r = r(n)$ of the data may be used as an estimate of the mode of the distribution. In this paper, it is shown that $\mathbf{\theta}_n$ converges almost surely to the true mode under very minor conditions on $\{r(n)\}$ and…
Several problems involving multivariate generalized Bayes estimators are investigated. First, a characterization of admissible estimators as generalized Bayes estimators is developed for certain multivariate exponential families and quadratic loss. The problem of verifying whether or not an estimator is generalized Bayes is also considered. Next, an important class of estimators for a multivariate normal mean is considered. (The class includes many minimax, empirical Bayes,…
Unbiased risk estimators are derived for estimators in certain classes of equivariant estimators of multinormal matrix means, $\xi,$ and regression coefficients $\beta.$ In all cases the covariance matrix is unknown. The underlying method, a multivariate version of that of James and Stein (1960), uses zonal polynomial expansions for the distributions of noncentral statistics. This gives, in one case, the required generalization of the Pitman-Robbins representation of…
Let $X_1, \cdots, X_n$ be i.i.d. $F_0$ and let $Y_1, \cdots, Y_n$ be independent (and independent also of $X_1, \cdots, X_n$) random variables. Then assuming that $F$ is distributed according to a Dirichlet process with parameter $\alpha,$ the authors obtained the Bayes estimator $\hat{F}_\alpha$ of $F$ under the loss function $L(F, \hat{F}) = \int (F(u) - \hat{F}(u))^2 dw(u)$ when $X_1, \cdots, X_n$ are censored on the right by $Y_1, \cdots, Y_n,$ respectively, and when it…
This paper provides an empirical Bayes approach to the problem of nonparametric estimation of a distribution (or survival) function when the observations are censored on the right. The results use the notion of a Dirichlet process prior introduced by Ferguson. The paper presents a generalization to the case of right censored observations of the rate result of an empirical Bayes nonparametric estimator of a distribution function of Korwar and Hollander in the uncensored case.…
This paper proves the apparently outstanding conjecture that the maximum likelihood estimate (m.l.e.) "behaves properly" when jackknifed. In particular, under the usual Cramer conditions (1) the jackknifed version of the consistent root of the m.l. equation has the same asymptotic distribution as the consistent root itself, and (2) the jackknife estimate of the variance of the asymptotic distribution of the consistent root is itself consistent. Further, if the hypotheses of…
Consider a time-continuous nonhomogeneous Markovian stochastic process V having state space A0. Let A⊂A0 and let PAij(τ,t) be the i→j transition probability of the Markovian stochastic process VA arising in the hypothetical situation where states A0−A have been eliminated from the state space of V. Based upon the concept of Kaplan and Meier's product-limit estimator, a nonparametric estimator P^Aij(τ,t) is…
This paper considers the general linear regression model Yi=∑jβjxij+εi, and studies the problem of testing hypotheses about some of the β's while regarding others as nuisance parameters. The test criteria discussed, which are based on ranks of residuals, are shown to be asymptotically distribution-free.
An asymptotic expansion is obtained for the distribution function of the standardized mean of a sample of s observations taken randomly without replacement from a finite population of n numbers. The expansion is given to order 1/n and agrees with the formal Edgeworth expansion. The proof of the result is obtained using an approximation to the characteristic function of the standardized sum.
In this note it is shown how to some extent the Edgeworth expansion for the distribution function of Kendall's tau can be established by using a well-known general result on such expansions.
Random variables X and Y are mutually completely dependent if there exists a one-to-one function g for which P[Y=g(X)]=1. An example is presented of a pair of random variables which are mutually completely dependent, but "almost" independent. This example motivates considering a new concept of dependence, called monotone dependence, in which g above is now required to be monotone. Finally, this monotone dependence concept leads to defining and…
Let qn(y),0<y<1, be a quantile process based on a sequence of i.i.d. rv with distribution function F and density function f. Given some regularity conditions on F the distance of qn(y) and the uniform quantile process un(y), respectively defined in terms of the order statistics Xk:n and Uk:n=F(Xk:n), is computed with rates. As a consequence we have an extension of Kiefer's result on the distance between the empirical and quantile processes,…
A sequence of medians of posterior distributions is approximately median unbiased of order o(n−1)iff the prior density is equal to the square root of Fisher's information function. It is shown that in this case the sequence of medians of posterior distributions is even an optimum sequence of estimates within the class of all estimator sequences being approximately median unbiased of order o(n−1). The result is proved by showing equivalence with an…
Asymptotically optimal empirical Bayes squared error loss estimation procedures are developed for three families of continuous distributions, uniform (0,θ),θ>0, uniform [θ,θ+1),θ arbitrary, and a location parameter family of gamma distributions. The approach taken is to estimate the Bayes estimator directly. However, for the [θ,θ+1) case, it is shown that the indirect approach of applying the Bayes estimator,…
Reduced U-Statistics and the Hodges-Lehmann Estimator
A reduced U-statistic (of order 2) is defined as the sum of terms f(Xi,Xj), where f is a symmetric function, (X1,⋯,XN) are independent and identically distributed (i.i.d.) random variables (rv's), and (i,j) are drawn from a restricted, though balanced, set of pairs. (A U-statistic corresponds to summation over all (i,j) pairs.) A limit normal distribution is found for the reduced U-statistic, and it follows that estimates based on reduced U-stat…
An asymptotic expression is given for the log error probability of a sequential test based on a random walk. This may be used to compute limiting relative efficiencies of such tests. The results are illustrated for the one-sided normal testing problem with an asymptotic Bayes test due to Schwarz. Some numerical comparisons are given for five sequential tests of a normal mean.
Consider a random sample from an absolutely continuous multivariate distribution. Let J be a class of sets which are not too long and thin. A point θn chosen from a minimum volume set Sn∈J containing at least r=r(n) of the data may be used as an estimate of the mode of the distribution. In this paper, it is shown that θn converges almost surely to the true mode under very minor conditions on {r(n)} and…
Unbiased risk estimators are derived for estimators in certain classes of equivariant estimators of multinormal matrix means, ξ, and regression coefficients β. In all cases the covariance matrix is unknown. The underlying method, a multivariate version of that of James and Stein (1960), uses zonal polynomial expansions for the distributions of noncentral statistics. This gives, in one case, the required generalization of the Pitman-Robbins representation of…
Let X1,⋯,Xn be i.i.d. F0 and let Y1,⋯,Yn be independent (and independent also of X1,⋯,Xn) random variables. Then assuming that F is distributed according to a Dirichlet process with parameter α, the authors obtained the Bayes estimator F^α of F under the loss function L(F,F^)=∫(F(u)−F^(u))2dw(u) when X1,⋯,Xn are censored on the right by Y1,⋯,Yn, respectively, and when it…
If fn(x) is any estimator of the density f(x), it is proved that the mean integrated square error is no better than O(n−1).