Measures of location (without assumption of symmetry) are defined as functionals satisfying certain equivariance and order conditions. Three classes of such measures are discussed whose estimators are respectively linear functions of order statistics, $R$-estimators and $M$-estimators. It is argued that such measures can be compared in terms of the (asymptotic) efficiencies of their estimators. Of the three classes considered, it is found that trimmed expectations (and…
An overview is given of an approach to the definition of descriptive characteristics of populations and to their estimation. The emphasis is on the robustness and efficiency of the estimators. Detailed summaries will be found in successive papers of the series dealing with the problems of location, scale and kurtosis.
A simple general approach to inference about the tail behavior of a distribution is proposed. It is not required to assume any global form for the distribution function, but merely the form of behavior in the tail where it is desired to draw inference. Results are particularly simple for distributions of the Zipf type, i.e., where $G(y) = 1 - Cy^{-\alpha}$ for large $y$. The methods of inference are based upon an evaluation of the conditional likelihood for the parameters…
The rate at which the mean square error decreases as sample size increases is evaluated for general $L^1$ kernel estimates and for the Fourier integral estimate for a probability density function. The estimates are then compared on the basis of these rates.
Let nS be an $m\times m$ matrix having the Wishart distribution $W_m(n,\Sigma)$. For large n and simple latent roots of $\Sigma$, it is known that the latent roots of S are asymptotically independently normal. In this paper an expansion, up to and including the terms of order $n^-1$, is given for the joint density function of the roots of S in terms of normal density functions. Expansions for the marginal distributions of the roots are also given, valid when the…
Let N be the stopping time of the sequential t-test, based on the i.i.d. sequence $Z_1,Z_2,...$, for testing that the ratio of mean to standard deviation in a normal population equals $\gamma_1$ agaist the alternative that it equals $\gamma_2$. Let P be the actual distribution of the $Z_i$ (not necessarily normal). It is proved that if $\gamma_1^2\neq\gamma_2^2$ and P is an arbitrary unbounded distribution, then there exist constants c > 0 and $\rho<1$ such that $P(N > n) < c\rho^n, n =1,2,\cdots$…
Sequential tests for one-sided hypotheses are compared, asymptotically, with non-sequential counterparts. An analog of Pitman efficiency is obtained, as is another comparison that has no purely non-sequential analog. With these methods of comparison, the limiting relative efficiency of the sequential test is never less than one and for most parameter values, it is infinite. An asymptotic notion of minimal relative efficiency is also considered.
A simple expression is developed for the difference between the least squares and minimum variance linear unbiased estimators obtained in linear models in which the covariance operator of the observation vector is nonsingular. Bounds and series expansion for this difference are obtained, and bounds for the efficiency of least squares estimates are also obtained.
Let $S(b) = \Sigma r\sigma r^2Xr^2(nr,br^2)$ be a positive linear combination of independent noncentral chi-square random variables. This note derives two representations for the tail probabilities P[S(b) >x], a Taylor series in the noncentrality parameters and a limiting form of this series for large x. An application of the latter result to statistical tests of Cramer-von Mises type is discussed.
Moore (1971) derives the limiting distribution of chi-square statistics for testing goodness of fit to k-variate parametric families, where the cell boundaries are allowed to be functions of the estimated parameter values. The only point at which random cells, multivariate observations etc. require a deviation from methods of proof given by Cramer (1946) is in the proof of the asymptotic negligibility of the remainder terms. Attention is drawn to an alternative proof of this…
The chi-square test, likelihood ratio test, and other goodness of fit tests in a family are shown to have monotonic power functions, and hence are unbiased, for testing a simple hypothesis that all cell probabilities are equal. The chi-square test is also shown to be Type D for this situation.
Exact limiting Bahadur efficiencies of linear rank tests for scale are derived using a minor extension of the results of Hoadley. Efficiency curves are derived for the linear rank statistics of Mood, Freund and Ansari, and Capon and Klotz relative to the F test for normal scale alternatives when the null hypothesis is that of common normality.
The asymptotic non-null distributions of four statistics, used for testing the hypothesis Ho: $\Sigma_1=\Sigma_2$ , are developed. These expansions are obtained by the partial differential equations method of Muirheard (Ann. Math. Statist. 41 1002-1010). The procedure permits a simple extension of the results of Pillai and Nagarsenker (J. Multivariate Anal. 2 96-114).
Recent results of Shorack (Ann. Math. Statist. (1972) 412-427) on the asymptotic normality of functions of order statistics are extended to stationary $\phi$-mixing processes and to a class of strong mixing processes. The results of this paper are based on the weak convergence of empirical processes relative to the metric $d_q$ as developed in Fears and Mehra (Ann. Statist. (1974) 586-596). Some remarks on trimmed and Winsorized means in the strong mixing case are also…
P. V. Rao · Eugene F. Schuster · Ramon C. Littell·刊期 1975-07-01 · 卷 3（4）·The Annals of Statistics已编入正刊
基于 Kolmogorov–Smirnov 统计量的位移与对称中心估计AI 解读
A point estimator and a set of confidence intervals based on the Kolmogorov-Smirnov statistic are proposed for the shift parameter in the two-sample problem. Asymptotic distibution of the etimator as well as asymptotic bounds for the lengths of the intervals are derived. The two-sample results are then adapted to the one-sample problem to define an estimator and a set of confidence intervals for the center of a symmetric population.
In many situations one is faced with the task of constructing a linear order, or ranking, of n objects based on data derived from a paired comparison experiment. Alternatively, one may desire to estimate the preference relation on the set of objects. Numerous criteria appear in the literature, and in practice, under which rankings or preferences may be selected. The main emphasis of this paper is on developing a general approach to maximum likelihood estimation of rankings…
In this paper, we shall represent a generalized Chernoff-Savage statistic as the sum of i.i.d. random variables plus a remainder term and analyze the order of magnitude of the remainder term. While Chernoff and Savage have proved that the remainder term, when suitably normalized, converges to O in probability, we obtain a stronger form of convergence in this paper. Our result gives an invariance principle and a law of the iterated logarithm for generalized Chernoff-Savage…
This paper introduces a new mixing condition for stationary processes which is weaker than $\phi$-mixing but stronger than strong mixing. Many processes arising in applications, e.g., first order autoregressive processes, obey the conditions. The main result is that the empiric cdf of a sample from such processes converges to a Gaussian process.
Ronald Mead · T. A. Bancroft · Chien-Pai Han·刊期 1975-07-01 · 卷 3（4）·The Annals of Statistics已编入正刊
不完全指定固定模型中方差分析检验的水平与功效AI 解读
Derivations are given of the size and power of a test of a hypothesis for an incompletely specified fixed linear model. Such a testing procedure involves a test of a hypothesis of main interest subsequent to a preliminary test to determine the inclusion or not of a term in the tentatively specified complete fixed linear model. An evaluation of the formulas of the size and power was made over a wide range of values of the parameters involved, and recommendations concerning…
In estimating a vector model, $\Sigma B(j)x(n-j)=\Sigma A(j)\epsilon(n-j), A(0)=I_r, E(\epsilon(m)\epsilon(n)')=\delta_{mn}K$ it is suggested that attention be confined to cases where $g(z) =\Sigma A(j)z^j, h(z)=\Sigma B(j)z^j$ have determinants with no zeroes inside the unit circle and have $I_r$ as greatest common left divisor and where $\1brack A(p)\vdots B(q) \rbrack$ is of rank r, where p, q are the degrees of g, h, respectively. It is shown that these conditions ensure…
The characteristic polynomial is derived for the information matrix $M$ for those main-effect and second-order models based on fractions of the $3^n$ factorial which render the model symmetric under permutation of the factors. Explicit formulas for the determinant of $M$ and the trace of $M^{-1}$ result.
This paper shows the asymptotic equivalence of the classical and symmetric optimality criteria for the finite state, arbitrary action compound decision problem.
Two nonparametric density estimators, based on Fourier series and the Fejer kernel, are presented. One of them $(\tilde{f}_N)$ is appropriate when the unknown density $f$ vanishes outside a known bounded interval; the other $(f_N^\sharp)$ is applicable without any assumptions about the support of $f$. The estimator $f_N^\sharp$ is of the type studied by Watson and Leadbetter (Sankhya A 26, 1964) and $\tilde{f}_N$ is almost of that type: both may be said to be of the "$\delta$…
Measures of location (without assumption of symmetry) are defined as functionals satisfying certain equivariance and order conditions. Three classes of such measures are discussed whose estimators are respectively linear functions of order statistics, R-estimators and M-estimators. It is argued that such measures can be compared in terms of the (asymptotic) efficiencies of their estimators. Of the three classes considered, it is found that trimmed expectations (and…
A simple general approach to inference about the tail behavior of a distribution is proposed. It is not required to assume any global form for the distribution function, but merely the form of behavior in the tail where it is desired to draw inference. Results are particularly simple for distributions of the Zipf type, i.e., where G(y)=1−Cy−α for large y. The methods of inference are based upon an evaluation of the conditional likelihood for the parameters…
The rate at which the mean square error decreases as sample size increases is evaluated for general L1 kernel estimates and for the Fourier integral estimate for a probability density function. The estimates are then compared on the basis of these rates.
Let nS be an m×m matrix having the Wishart distribution Wm(n,Σ). For large n and simple latent roots of Σ, it is known that the latent roots of S are asymptotically independently normal. In this paper an expansion, up to and including the terms of order n−1, is given for the joint density function of the roots of S in terms of normal density functions. Expansions for the marginal distributions of the roots are also given, valid when the…
Let N be the stopping time of the sequential t-test, based on the i.i.d. sequence Z1,Z2,..., for testing that the ratio of mean to standard deviation in a normal population equals γ1 agaist the alternative that it equals γ2. Let P be the actual distribution of the Zi (not necessarily normal). It is proved that if γ12=γ22 and P is an arbitrary unbounded distribution, then there exist constants c > 0 and ρ<1 such that P(N>n)<cρn,n=1,2,⋯…
Let S(b)=Σrσr2Xr2(nr,br2) be a positive linear combination of independent noncentral chi-square random variables. This note derives two representations for the tail probabilities P[S(b) >x], a Taylor series in the noncentrality parameters and a limiting form of this series for large x. An application of the latter result to statistical tests of Cramer-von Mises type is discussed.
On the Asymptotic Distributions of Some Statistics Used for Testing Σ1=Σ2
The asymptotic non-null distributions of four statistics, used for testing the hypothesis Ho: Σ1=Σ2 , are developed. These expansions are obtained by the partial differential equations method of Muirheard (Ann. Math. Statist. 41 1002-1010). The procedure permits a simple extension of the results of Pillai and Nagarsenker (J. Multivariate Anal. 2 96-114).
Recent results of Shorack (Ann. Math. Statist. (1972) 412-427) on the asymptotic normality of functions of order statistics are extended to stationary ϕ-mixing processes and to a class of strong mixing processes. The results of this paper are based on the weak convergence of empirical processes relative to the metric dq as developed in Fears and Mehra (Ann. Statist. (1974) 586-596). Some remarks on trimmed and Winsorized means in the strong mixing case are also…
This paper introduces a new mixing condition for stationary processes which is weaker than ϕ-mixing but stronger than strong mixing. Many processes arising in applications, e.g., first order autoregressive processes, obey the conditions. The main result is that the empiric cdf of a sample from such processes converges to a Gaussian process.
In estimating a vector model, ΣB(j)x(n−j)=ΣA(j)ϵ(n−j),A(0)=Ir,E(ϵ(m)ϵ(n)′)=δmnK it is suggested that attention be confined to cases where g(z)=ΣA(j)zj,h(z)=ΣB(j)zj have determinants with no zeroes inside the unit circle and have Ir as greatest common left divisor and where $\1brack A(p)\vdots B(q) \rbrack$ is of rank r, where p, q are the degrees of g, h, respectively. It is shown that these conditions ensure…
The characteristic polynomial is derived for the information matrix M for those main-effect and second-order models based on fractions of the 3n factorial which render the model symmetric under permutation of the factors. Explicit formulas for the determinant of M and the trace of M−1 result.
Two nonparametric density estimators, based on Fourier series and the Fejer kernel, are presented. One of them (f~N) is appropriate when the unknown density f vanishes outside a known bounded interval; the other (fN♯) is applicable without any assumptions about the support of f. The estimator fN♯ is of the type studied by Watson and Leadbetter (Sankhya A 26, 1964) and f~N is almost of that type: both may be said to be of the "δ…