Let $X$ and $Y$ be two random variables with continuous distribution functions $F$ and $G$ and means $\mu$ and $\xi$. In a linear model, the crucial property of the contrast $\Delta = \xi - \mu$ is that $X + \Delta =_\mathscr{L} Y$, where $= _\mathscr{L}$ denotes equality in law. When the linear model does not hold, there is no real number $\Delta$ such that $X + \Delta = _\mathscr{L} Y$. However, it is shown that if parameters are allowed to be function valued, there is…
Estimation of the $k$th derivative of a df by means of the $k$th-order difference quotients of the empiric df is investigated. In particular, consistency conditions are given, the asymptotic bias, variance, and mean-squared error of the estimator are computed, and means of minimizing the latter are discussed.
Minimum variance unbiased estimates of the inverse Gaussian distribution function for all possible cases are given. A direct relationship is established between its density function and the normal density function, which throws more light on its salient features and possibly on its application in statistical inference. It is shown that the estimates are very similar in nature to those of the normal distribution and can be evaluated from the normal and Student's t…
Asymptotically best linear unbiased estimates (ABLUE) of the location and scale parameters (when both are unknown) of the Cauchy distribution, based on k order statistics selected from a large sample, are considered. It is shown that the joint asymptotic relative efficiency (JARE) of the two estimates is maximized when the sample quantiles are equally spaced, and expressions are derived for the coefficients used in these estimates. In estimating the scale parameter using a…
An upper bound is given for the distribution function of the largest latent root and a lower bound for the distribution function of the smallest latent root for a sample covariance matrix from a normal population. The bounds are expressed in terms of products of chi-squared probabilities.
R. SRINIVASAN · L. B. GODIO·刊期 1974 · 卷 61（1）·Biometrika已编入正刊
检验分布对称性的 Cramér–von Mises 型统计量AI 解读
A Cramér-von Mises type statistic is proposed for testing the symmetry of a continuous distribution function. Its small sample and asymptotic null distributions are derived. It is shown to be consistent against all nonsymmetric alternatives.
Let x 1j , x 2j , ···, xnj ; j=1, 2, ···, k be k independent random samples drawn, respectively, from the distributions with cdf's Fj (x). Let F* j (x) denote the empirical distribution function of the jth sample. Conover [3] and Wolf [6] derive the null distribution for the statistic sup x;i < k [F* i (x) – F* i+1(x)]. This article presents tables of exact probabilities for this statistic for k = 3(1)10, and n = 5(1)20. An approximation based on Conover's limiting…
Consider a problem in which a certain statistic X has a specified distribution function F(x) if a given hypothesis H is true, and suppose that the hypothesis H is evaluated by calculating the tail area 1 – F(x) corresponding to the observed value x of the statistic X. Examples are given in which this tail area is equal to the posterior probability that H is true and in which it is equal to the likelihood ratio comparing H to a certain class K of alternatives. The purpose of…
The general stochastic approximation procedure: $X_{n+1} = X_n - a_n c_n^{-1}h(Y_n),\quad n = 1,2, \cdots$ is considered, where $h$ is a Borel measurable transformation on the random observations $Y_n$. Under some mild requirements on $h$ and on the error random variables, the asymptotic properties, the a.s. convergence and the asymptotic normality are studied. The analysis is confined to the case where the error random variables are (conditionally) distributed according to…
In this paper the minimax estimators of a cumulative distribution function $F$ is obtained for four types of loss functions. The result is quite general in that no restrictions are imposed on the unknown $F$. Moreover, the estimates do not depend upon the weight function used in the definition of the loss functions. It is also shown that the sample distribution function is minimax under one of these types of loss functions.
Let $F_n(x)$ be the empirical distribution function based on a random sample of size $n$ from a continuous symmetric distribution with center $\theta$. As a nonparametric estimator of $\theta$, we propose $a^\ast$ where $a^\ast$ is chosen so as to minimize the function $h$ where $h(a) = \max_x |F_n(x) + F_n((2a - x)^-) - 1|$. In this paper we present an algorithm for constructing the interval of all $a$ which minimize $h$. We show that if $a^\ast$ is chosen as the center of…
The theory of rank tests has been developed primarily for continuous random variables. Recently the asymptotic theory of linear rank tests has been extended to include purely discrete random variables under the null hypothesis of randomness (including the two-sample and $k$-sample problems) and under contiguous alternatives, for the two methods of assigning scores known as the average scores method and the randomized ranks method. In this paper the theory of rank tests is…
The classification statistic $W$ is used to classify an observasion as coming from one of two multivariate normal populations with common covariance matrix and different means when these parameters are estimated from two samples, one from each population. The distribution of $W$ depends on the Mahalanobis distance between the populations, $\alpha$. When the sample sizes approach infinity, the limiting distribution of $(W - \frac{1}{2}\alpha)/\alpha^{\frac{1}{2}}$ is the…
A class of statistics is considered, which are based on order-statistics and have properties analogous to Student's $t$. They can be used to estimate a required quantile of a random variable or to test hypotheses about a quantile; they are simple to compute, and can be calculated when not all sample values are available, e.g. for censored samples. The limiting distributions of these statistics are derived and shown to be independent of the distribution of the underlying…
This article considers the problem of testing a completely specified continuous symmetric distribution against alternatives which are also symmetric about the same point. The symmetry is utilized in obtaining a new distribution-free statistic of the Kolomogorov-Smirnov type which can be used to halve the width of the Kolomogorov-Smirnov confidence band for the unknown distribution function.
A simple statistical model is developed to assess the implications of random measurement errors in reported per capita incomes and of the bias caused by the use of official exchange rates for (1) regional per capita incomes and (2) regional income inequality. It appears that regional per capita incomes can be estimated with a fair degree of accuracy, provided that the measurement errors of the reported per capita incomes of individual countries are not too highly positively…
A simple statistical model is developed to assess the implications of random measurement errors in reported per capita incomes and of the bias caused by the use of official exchange rates for (1) regional per capita incomes and (2) regional income inequality. It appears that regional per capita incomes can be estimated with a fair degree of accuracy, provided that the measurement errors of the reported per capita incomes of individual countries are not too highly positively…
The problems of asymptotically optimum spacing of sample quantiles in connection with asymptotically best linear unbiased estimation of the location parameter of an extreme value distribution of Type I or the scale parameter of an extreme value distribution of Type II or III, are essentially the same. Previous results for the scale parameter of an exponential distribution can therefore be used.
D. L. Hanson · Gordon Pledger · F. T. Wright·刊期 1973-05-01 · 卷 1（3）·The Annals of Statistics已编入正刊
单调回归的一致性研究AI 解读
For each $t$ in some subset $T$ of $N$-dimensional Euclidean space let $F_t$ be a distribution function with mean $m(t)$. Suppose $m(t)$ is non-decreasing in each of the coordinates of $t$. Let $t_1, t_2,\cdots$ be a sequence of points in $T$ and let $Y_1, Y_2,\cdots$ be an independent sequence of random variables such that the distribution function of $Y_k$ is $F_{t_k}$. Estimators $\hat{m}_n(t; Y_1,\cdots, Y_n)$ of $m(t)$ which are monotone in each coordinate of $t$ and…
Suppose $g$ is a probability density on $R = (-\infty, +\infty)$ with a continuous derivative and with $I(g) = \int (g'/g)^2g < + \infty$. Suppose $\{\varepsilon_n\}$ is a sequence of positive numbers converging to zero and $V_i, Z_i$ are random variables, $Z_i \rightarrow 0$ with probability one and $V_n$ is conditionally (given $V_1, \cdots, V_{n-1}, Z_1, \cdots, Z_n$) distributed according to $g$. Estimates $h_n$ of $g'/g$ are constructed, which are based on $V_1 + Z_1, \cdots, V_n + Z_n$…
We consider the partial order on the unit square; $s_1 = (x_1, y_1) \ll s_2 = (x_2, y_2)$ if and only if $x_i \leqq y_i$ for $i = 1, 2$, and say that a real-valued function $f$ is isotone if $s_1 \ll s_2$ implies that $f(s_1) \leqq f(s_2)$. Suppose that for each point, $s$, in the unit square we have a distribution with median $m(s)$ and $m(s)$ is isotone. In this paper we propose an isotone estimator for $m$ which we denote by $\hat{m}$ and give an algorithm for computing $\hat{m}$…
Let $Z_1, Z_2,\cdots$, be independent and identically distributed random variables and $\{c_{ijn}\}$ real numbers; put $T_n = \sum^n_{i,j = 1} c_{ijn}Z_iZ_j$. This paper gives conditions under which the distribution of $T_n - ET_n$ converges to the distribution of $\sum \Upsilon_m(Y_m^2 - 1)$ with $\{\Upsilon_m\}$ a real sequence and $Y_1, Y_2,\cdots$ independent $N(0, 1)$ random variables. The results are applied to the calculation of the asymptotic distributions of test…
Let X and Y be two random variables with continuous distribution functions F and G and means μ and ξ. In a linear model, the crucial property of the contrast Δ=ξ−μ is that X+Δ=LY, where =L denotes equality in law. When the linear model does not hold, there is no real number Δ such that X+Δ=LY. However, it is shown that if parameters are allowed to be function valued, there is…
Estimation of the kth Derivative of a Distribution Function
Estimation of the kth derivative of a df by means of the kth-order difference quotients of the empiric df is investigated. In particular, consistency conditions are given, the asymptotic bias, variance, and mean-squared error of the estimator are computed, and means of minimizing the latter are discussed.
The general stochastic approximation procedure: Xn+1=Xn−ancn−1h(Yn),n=1,2,⋯ is considered, where h is a Borel measurable transformation on the random observations Yn. Under some mild requirements on h and on the error random variables, the asymptotic properties, the a.s. convergence and the asymptotic normality are studied. The analysis is confined to the case where the error random variables are (conditionally) distributed according to…
In this paper the minimax estimators of a cumulative distribution function F is obtained for four types of loss functions. The result is quite general in that no restrictions are imposed on the unknown F. Moreover, the estimates do not depend upon the weight function used in the definition of the loss functions. It is also shown that the sample distribution function is minimax under one of these types of loss functions.
Let Fn(x) be the empirical distribution function based on a random sample of size n from a continuous symmetric distribution with center θ. As a nonparametric estimator of θ, we propose a∗ where a∗ is chosen so as to minimize the function h where h(a)=maxx∣Fn(x)+Fn((2a−x)−)−1∣. In this paper we present an algorithm for constructing the interval of all a which minimize h. We show that if a∗ is chosen as the center of…
Rank Tests for One Sample, Two Samples, and k samples Without the Assumption of a Continuous Distribution Function
The theory of rank tests has been developed primarily for continuous random variables. Recently the asymptotic theory of linear rank tests has been extended to include purely discrete random variables under the null hypothesis of randomness (including the two-sample and k-sample problems) and under contiguous alternatives, for the two methods of assigning scores known as the average scores method and the randomized ranks method. In this paper the theory of rank tests is…
An Asymptotic Expansion of the Distribution of the Studentized Classification Statistic W1
The classification statistic W is used to classify an observasion as coming from one of two multivariate normal populations with common covariance matrix and different means when these parameters are estimated from two samples, one from each population. The distribution of W depends on the Mahalanobis distance between the populations, α. When the sample sizes approach infinity, the limiting distribution of (W−21α)/α21 is the…
Limiting Distributions of Statistics Similar to Student's t
A class of statistics is considered, which are based on order-statistics and have properties analogous to Student's t. They can be used to estimate a required quantile of a random variable or to test hypotheses about a quantile; they are simple to compute, and can be calculated when not all sample values are available, e.g. for censored samples. The limiting distributions of these statistics are derived and shown to be independent of the distribution of the underlying…
For each t in some subset T of N-dimensional Euclidean space let Ft be a distribution function with mean m(t). Suppose m(t) is non-decreasing in each of the coordinates of t. Let t1,t2,⋯ be a sequence of points in T and let Y1,Y2,⋯ be an independent sequence of random variables such that the distribution function of Yk is Ftk. Estimators m^n(t;Y1,⋯,Yn) of m(t) which are monotone in each coordinate of t and…
Suppose g is a probability density on R=(−∞,+∞) with a continuous derivative and with I(g)=∫(g′/g)2g<+∞. Suppose {εn} is a sequence of positive numbers converging to zero and Vi,Zi are random variables, Zi→0 with probability one and Vn is conditionally (given V1,⋯,Vn−1,Z1,⋯,Zn) distributed according to g. Estimates hn of g′/g are constructed, which are based on V1+Z1,⋯,Vn+Zn…
We consider the partial order on the unit square; s1=(x1,y1)≪s2=(x2,y2) if and only if xi≦yi for i=1,2, and say that a real-valued function f is isotone if s1≪s2 implies that f(s1)≦f(s2). Suppose that for each point, s, in the unit square we have a distribution with median m(s) and m(s) is isotone. In this paper we propose an isotone estimator for m which we denote by m^ and give an algorithm for computing m^…
Let Z1,Z2,⋯, be independent and identically distributed random variables and {cijn} real numbers; put Tn=∑i,j=1ncijnZiZj. This paper gives conditions under which the distribution of Tn−ETn converges to the distribution of ∑Υm(Ym2−1) with {Υm} a real sequence and Y1,Y2,⋯ independent N(0,1) random variables. The results are applied to the calculation of the asymptotic distributions of test…