ROBERT I. JENNRICH·刊期 1983 · 卷 70（1）·Biometrika已编入正刊
不等删失下对数秩置换检验的表现AI 解读
This note studies the extent to which the type I error of the permutation test based on the log rank score departs from its nominal value in the presence of unequal right censoring. For the asymptotic case, situations are identified that lead to conservative and to nonconservative behaviour, and the amount of conservatism is computed for some specific constant hazard situations. A small Monte Carlo study is used to support the relevance of the asymptotic results in a finite…
Summary Asymptotic approximations for the tail probabilities of certain statistics with complex moment generating functions are obtained by the usual methods of large deviation theory. It is pointed out that this approximation should be adequate over the entire range but of particular use for extreme tails. Approximations to the significance levels for one and two sample permutation tests are obtained and these tests are inverted to obtain approximate confidence intervals.…
Historical samples of military, naval, and merchant marine heights suffer from undercounts of short people, undercounts highly irregular compared to the truncated distributions for which standard methods are available. Our knowledge is too imprecise for parametric models of the undercount process but it suffices for intuitive description of the deviations from normality with which estimators should cope. Two new estimators are developed. The first adapts maximum likelihood…
Let $X_1, X_2, \cdots, X_n$ be i.i.d random variables with d.f. $F$. Suppose the $\{\hat{T}_n = \hat{T}_n(X_1, X_2, \cdots, X_n); n \geq 1\}$ are real-valued statistics and the $\{T_n(F); n \geq 1\}$ are centering functionals such that the asymptotic distribution of $n^{1/2}\{\hat{T}_n - T_n(F)\}$ is normal with mean zero. Let $H_n(x, F)$ be the exact d.f. of $n^{1/2}\{\hat{T}_n - T_n(F)\}$. The problem is to estimate $H_n(x, F)$ or functionals of $H_n(x, F)$. Under…
Two methods are proposed for setting confidence intervals for the median survival time. It is shown how these are related to bootstrap methods for censored data, and some numerical comparisons are presented.
The regression and correlation models are considered. It is shown that the bootstrap approximation to the distribution of the least squares estimates is valid, and some error bounds are given.
Efron's "bootstrap" method of distribution approximation is shown to be asymptotically valid in a large number of situations, including $t$-statistics, the empirical and quantile processes, and von Mises functionals. Some counter-examples are also given, to show that the approximation does not always succeed.
In the non-lattice case it is shown that the bootstrap approximation of the distribution of the standardized sample mean is asymptotically more accurate than approximation by the limiting normal distribution. The exact convergence rate of the bootstrap approximation of the distributions of sample quantiles is obtained. A few other convergence rates regarding the bootstrap method are also studied.
This article concerns setting standard errors and confidence intervals for the parameters of an unknown distribution when the data is subject to right censoring. The bootstrap, which is an elaboration of the jackknife, provides a general method for answering such questions. The validity of bootstrap methods is investigated using real data, computer simulations, and, in the final section, brief theoretical considerations.
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…
Two methods are proposed for setting confidence intervals for the median survival time. It is shown how these are related to bootstrap methods for censored data, and some numerical comparisons are presented.
We discuss several nonparametric methods for attaching a standard error to a point estimate: the jackknife, the bootstrap, half-sampling, subsampling, balanced repeated replications, the infinitesimal jackknife, influence function techniques and the delta method. The last three methods are shown to be identical. All the methods derive from the same basic idea, which is also the idea underlying the common parametric methods. Extended numerical comparisons are made for the…
Let $V_n$ be the standardized sum of squares of the means of $r + 1$ random samples of sizes $s_0, s_1, \cdots, s_r$, where $n = s_0 + s_1 + \cdots + s_r$, taken without replacement from $n$ numbers. Then using an approximation to the characteristic function of the means, an asymptotic expansion is obtained for the distribution of $V_n$ with first term being the distribution function of $\chi^2_r$ and with error of approximation generally of smaller order than $1/n$. When…
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…
Cox (1972) proposed a regression model for life tables and derived certain hypothesis testing procedures. If there is no censoring and we wish to test a simple null hypothesis, his test statistic is similar to a permutation test. This relation is explored and it is shown how to modify any rank test to allow for censored data. Cox's test and the log rank test are compared and it is concluded that if no censoring occurs the nonparametric test is to be preferred if the sample…
Cox (1972) proposed a regression model for life tables and derived certain hypothesis testing procedures. If there is no censoring and we wish to test a simple null hypothesis, his test statistic is similar to a permutation test. This relation is explored and it is shown how to modify any rank test to allow for censored data. Cox's test and the log rank test are compared and it is concluded that if no censoring occurs the nonparametric test is to be preferred if the sample…
Standard techniques of nonparametric statistics are related to likelihood procedures. In particular, rank tests and permutation tests in the regression problem are shown to be directly related to score function tests based on marginal and conditional likelihoods, respectively. The problem of estimating a population percentile is also examined from the viewpoint of marginal likelihood. These calculations tend to bring nonparametric procedures more closely in line with…
Standard techniques of nonparametric statistics are related to likelihood procedures. In particular, rank tests and permutation tests in the regression problem are shown to be directly related to score function tests based on marginal and conditional likelihoods, respectively. The problem of estimating a population percentile is also examined from the viewpoint of marginal likelihood. These calculations tend to bring nonparametric procedures more closely in line with…
Permutation tests are applied to the maximal t statistic in two-sample, completely randomized designs, and in two-sample, matched-pair designs, where the response is a vector and the experimenter is interested in one specific direction in each component. Recommendations for subsampling are also included for cases when the number of permutations required to complete the test is prohibitive. These procedures are then applied to data from two medical trials.
The problem of statistical comparison of two distributions, continuous as well as discrete, is considered. Very slight and reasonable modifications of traditional parameteric models, e.g. `normal distributions with equal variances', are shown to result in permutation tests, only. Fisher's permutations test is shown to have optimum properties which mean a good merit for its practical use. Further, an accurate method of determining the $p$-value of Fisher's test is proposed.
We study several combinatorial problems related to the choice of groups of subsets to be used in the subsampling procedure of Hartigan (J. Amer. Statist. Assoc. 1969). The complete separation property is shown to guarantee asymptotic efficiency relative to $t$. Problems associated with the relative variance criterion are discussed.
Stepwise regression is a member of a class of statistical methods which may be called opportunistic. The statistician is naturally concerned that an artifact should not be reported so he needs a stopping rule that is geared to the way in which variables are selected. A sampled permutation test is outlined which offers such a stopping rule for forward stepping.
The sampling procedure involving subsampling of nonrespondents is discussed. A rule for selecting a subsample of nonrespondents is proposed under which the subsampling fraction is not kept constant, but varied according to the sample nonresponse rates. Using this sampling rule, the variance of the estimator of the population mean is independent of the unknown rate of nonresponse in the population. This rule provides a procedure for determining the initial sample size and…
The size and power of Student's t-test are discussed under weaker than normal conditions. It is shown that assuming only a symmetry condition for the null hypothesis leads to effective bounds on the dispersion of the t-statistic. (The symmetry condition is weak enough to include all cases of independent but not necessarily identically distributed observations, each symmetric about the origin.) The connection between Student's test and the usual non-parametric tests is…
Let X1,X2,⋯,Xn be i.i.d random variables with d.f. F. Suppose the {T^n=T^n(X1,X2,⋯,Xn);n≥1} are real-valued statistics and the {Tn(F);n≥1} are centering functionals such that the asymptotic distribution of n1/2{T^n−Tn(F)} is normal with mean zero. Let Hn(x,F) be the exact d.f. of n1/2{T^n−Tn(F)}. The problem is to estimate Hn(x,F) or functionals of Hn(x,F). Under…
Efron's "bootstrap" method of distribution approximation is shown to be asymptotically valid in a large number of situations, including t-statistics, the empirical and quantile processes, and von Mises functionals. Some counter-examples are also given, to show that the approximation does not always succeed.
Let Vn be the standardized sum of squares of the means of r+1 random samples of sizes s0,s1,⋯,sr, where n=s0+s1+⋯+sr, taken without replacement from n numbers. Then using an approximation to the characteristic function of the means, an asymptotic expansion is obtained for the distribution of Vn with first term being the distribution function of χr2 and with error of approximation generally of smaller order than 1/n. When…
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…
The problem of statistical comparison of two distributions, continuous as well as discrete, is considered. Very slight and reasonable modifications of traditional parameteric models, e.g. `normal distributions with equal variances', are shown to result in permutation tests, only. Fisher's permutations test is shown to have optimum properties which mean a good merit for its practical use. Further, an accurate method of determining the p-value of Fisher's test is proposed.
We study several combinatorial problems related to the choice of groups of subsets to be used in the subsampling procedure of Hartigan (J. Amer. Statist. Assoc. 1969). The complete separation property is shown to guarantee asymptotic efficiency relative to t. Problems associated with the relative variance criterion are discussed.