摘要:

Let X(1)≤X(2)≤ … ≤X(n)be an ordered sample of a random variableXwhich has continuous probability distribution functionF(x), and letFn(x) be the corresponding empirical distribution function. The following three statistics, introduced by A. Rényi, are considered:

ISSN:0162-1459    

DOI:10.1080/01621459.1969.10501019

出版商:Taylor & Francis Group    

年代:1969

数据来源: Taylor

ISSN:0162-1459    

DOI:10.1080/01621459.1969.10501020

出版商:Taylor & Francis Group    

年代:1969

数据来源: Taylor

摘要:

In the early years of the eighteenth century A. De Moivre, N. Bernoulli, and P. R. Montmort found three solutions for the following problem of games of chance: Two players, A and B, play a series of games. In each game, independently of the others, either A wins a counter from B with probabilitypor B wins a counter from A with probabilityq(p+q= 1). The series ends if either A wins a total number ofacounters from B or B wins a total number ofbcounters from A. What is the probability that A wins the series in at mostngames? Denote this probability byPn(a,b). In this paper simple and elementary proofs are given for the various formulas forPn(a,b). Furthermore, it is shown how these formulas can be applied in the theories of order statistics, random walks, storage, queues, Brownian motion, and dams.

ISSN:0162-1459    

DOI:10.1080/01621459.1969.10501021

出版商:Taylor & Francis Group    

年代:1969

数据来源: Taylor

摘要:

A lower bound is derived concerning a special form of the entropy function of information theory. It is applied to Hoeffding's bound for the probability of the deviation of the sample mean from its expected value and for the corresponding problem concerning the sample variance. Furthermore, applying the bound one can obtain a discard function for a binary symmetric channel, a lower bound for the total error probability in testing two simple hypotheses, given the number of observations, and a lower bound for the deviation of the empirical distribution from the true distribution.

ISSN:0162-1459    

DOI:10.1080/01621459.1969.10501022

出版商:Taylor & Francis Group    

年代:1969

数据来源: Taylor

摘要:

The problem considered in this paper is one of allocating a predetermined aggregate among its components. An example from econometrics is the allocation among various competing consumption items of a consumer's budget, the total size of which is usually taken as determined outside the allocation framework.

ISSN:0162-1459    

DOI:10.1080/01621459.1969.10501023

出版商:Taylor & Francis Group    

年代:1969

数据来源: Taylor

摘要:

This paper presents the exact sampling distributions of the ordinary and the two-stage least squares estimators of a structural parameter in a structural equation with two endogenous variables in a complete system of stochastic equations. The results show that the distributions of the two estimators are essentially similar to each other. It can also be seen that both distributions depend crucially upon the deviation of a regression coefficient of disturbance terms of two endogenous variables from a structural parameter, and that the first estimator possesses moments up to the orderN-2, while the second possesses them up to the orderK-1, whereNis the sample size andKis the number of exogenous variables excluded from the equation to be estimated. The small sample properties of the estimators are investigated by numerical evaluations of the density functions.

ISSN:0162-1459    

DOI:10.1080/01621459.1969.10501024

出版商:Taylor & Francis Group    

年代:1969

数据来源: Taylor

摘要:

In an earlier paper [5] the authors compared the power of the BLUS test with the probability of a correct decision of the Durbin-Watson bounds test. A method to compute the distribution of the Von Neumann ratio under the null hypothesis and under the alternative hypothesis was given. In the present paper the latter method is used to tabulate the BLUS-test statistic and to compute the exact significance points of the Durbin-Watson test for several examples. Powers of both tests are computed and compared. It appears that, for the cases considered, the power of the exact Durbin-Watson test exceeds that of the BLUS procedure, while the latter is greater than the probability of a correct decision in the Durbin-Watson bounds test.

ISSN:0162-1459    

DOI:10.1080/01621459.1969.10501025

出版商:Taylor & Francis Group    

年代:1969

数据来源: Taylor

摘要:

For the classical linear model, Theil and Koerts developed the so-called best linear unbiased scalar-variance-matrix (BLUS) estimator. In this note a theorem on the variance matrix of the BLUS estimation errors will be proved. A new derivation will be given for “price” and “efficiency” of the estimator, as defined by Koerts.

ISSN:0162-1459    

DOI:10.1080/01621459.1969.10501026

出版商:Taylor & Francis Group    

年代:1969

数据来源: Taylor

摘要:

Matrix differentiation, the procedure of finding partial derivatives of the elements of a matrix function with respect to the elements of the argument matrix, is the subject matter of this paper. The method followed here amounts to linking the differentials of the matrix function and the argument matrix, and then identifying matrices of partial derivatives. The basic assumption made is mathematical independence of the elements of the argument matrix. Matrix functions are divided into two categories: Kronecker matrix products and non-Kronecker (= ordinary) matrix products. Three definitions of partial derivatives matrices are used, to be denoted byD1,D2andD3in this abstract.D2andD3are applied to Kronecker products,D1is applied to ordinary products. This is a matter of efficiency only.D1is developed first. Transforming matrices into column vectors turns out to be very convenient.D2andD3are developed then.D3is a generalisation ofD1.

ISSN:0162-1459    

DOI:10.1080/01621459.1969.10501027

出版商:Taylor & Francis Group    

年代:1969

数据来源: Taylor

摘要:

The method of Sweep-Out is used to obtain computing formulas for calculating the least-squares estimator and its variance matrix in linear models, not necessarily of full rank, in which certain restrictions may hold on the actual parameters. A test for a specified value of parametric restrictions is also discussed.

ISSN:0162-1459    

DOI:10.1080/01621459.1969.10501028

出版商:Taylor & Francis Group    

年代:1969

数据来源: Taylor