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Several writers have studied the problem of obtaining confidence bands for a straight line regression under the requirement that the bands be at a specified confidence level, (1 - α) say, for values of the independent variable restricted to a pre-specified closed interval. The bands studied by other writers have been either bands of equal width throughout the interval or trapezoidal bands. This paper studies confidence bands of the classical hyperbolic type under the restriction noted above. Relevant distribution results differ according to whether the pre-specified interval on the independent variable is symmetric or asymmetric about the mean of the independent variable values in the experiment. In the former case the problem under study, is shown to be equivalent to a problem studied by Halperin et al (JASA, Sept. 1967) for which distribution theory and tables are already available. The symmetrical case is generalized to obtain confidence bands in multiple linear regression at level (1 - α) for an ellipsoidal region on the independent variables centered at the point of means of the independent variable values used in the experiment. For straight line regression some numerical comparisons are made with bands of the other types mentioned above. These comparisons suggest that the bands proposed in this paper are uniformly superior to bands of equal width in the sense of having smaller average width; the proposed bands appear to be superior to trapezoidal bands in the same sense for intervals of practical interest, i.e. within or reasonably close to the range on the independent variables used in the experiment. [ABSTRACT FROM AUTHOR]

An introduction to the papers published in the 1972 issue of the "Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability," Vol. VI, is presented.

This paper presents a reparameterization of the conventional log-linear regression model. This reparameterization is shown to be more reasonable than the conventional specification. The stochastic assumptions of this reparameterized model are derived from the expected value properties of the dependent variable. Comparison with the conventional method of fitting log-linear models shows the regression intercept and the variance of the regression intercept to be different. A minimum variance unbiased estimate and another consistent estimate of the intercept are derived. [ABSTRACT FROM AUTHOR]

Given N points distributed at random on [0,1), let np, be the size of the largest number of points clustered within an interval of length p. Previous work finds Pr (np ≥ n), for n> N/2, and for n < N/2, p=1/L, L an integer. The formula for the case p=1/L is in terms of the sum of L × L determinants and is not computationally feasible for large L. The present paper derives such a computational formula. [ABSTRACT FROM AUTHOR]

The model considered in this paper is simple linear regression (Ey[sub i] = beta[sub 1] + beta[sub 2] x[sub 1], i = 1, ..., n), and the problem is to make statistical inferences about an unknown value of x corresponding to one or more additional observed values of y. The maximum likelihood estimator x of x and the classical (l - alpha) 100% confidence set S for x have some undesirable properties. For example, x has infinite mean square error and P {S = (- infinity, + infinity)} > 0. The purpose of this paper is to demonstrate that insight and understanding, as well as a useful class of solutions, can be obtained by looking at the problem from a Bayesian point of view. A result which follows from a general Bayes solution is that the inverse estimator [4] is Bayes with respect to a particular informative prior. [ABSTRACT FROM AUTHOR]

This paper concerns a new Normal approximation to the beta distribution and its relatives, in particular, the binomial, Pascal, negative binomial, F, t, Poisson, gamma, and chi square distributions. The approximate Normal deviates are expressible in terms of algebraic functions and logarithms, but for desk calculation it is preferable in most cases to use an equivalent expression in terms of a function specially tabulated here. Graphs of the error arc provided. They show that the approximation is good even in the extreme tails except for beta distributions which are J or U shaped or nearly so, and they permit correction to obtain still more accuracy. For use beyond the range of the graphs, some standard recursive relations and some classical continued fractions are listed, with some properties of the latter which seem to be partly new. Various Normal approximations are compared, with further graphs. The new approximation appears far more accurate than the others. Everything an ordinary user of the approximation might want to know is included in this paper. The theory behind the approximation and most proofs are postponed to a second paper immediately following this one. [ABSTRACT FROM AUTHOR]

In this paper a method is developed for estimating the parameters in the multivariate normal distribution in which the missing observations are not restricted to follow certain patterns as in most previous papers. The large sample properties of the estimators are discussed. Equivalence with maximum likelihood estimators has been established for a subclass of problems. The results of some simulation studies are provided to support the theoretical development. [ABSTRACT FROM AUTHOR]

The 1967 Committee on Publications, chaired by David L. Wallace, found that many American Statistical Association members desired more review and survey papers. These have been hard to come by and so as the author's last act before leaving office, decided to provide a short survey paper on some related ideas in a field where nearly all of the statisticians sometimes work — that of contingency tables. These ideas are largely available in literature and yet they have not often been put together, though statisticians I. J. Good's monograph and Leo Goodman's many papers form good sources. But the author's paper is not intended as a review of the literature, only as a survey of one set of ideas about estimation in the analysis of contingency tables. The author fears that the first act of most social scientists upon seeing a contingency table is to compute chi-square for it. Sometimes this process is enlightening, sometimes wasteful, but sometimes it does not go quite far enough. The author collected 500 samples of writing published about 1961.

In this paper confidence bands are given for the entire line for simple linear regressions. The conventional bands that have been given in the past are curvilinear. In this paper we consider confidence bands that are straight lines. It is shown that under certain conditions the "width" of these straight line bands is less than the width of the conventional curvilinear bands. [ABSTRACT FROM AUTHOR]

An earlier paper [3] provides axioms for the seasonal adjustment of economic time series. Seasonally adjusted data should be minimum variance, unbiased, and linear in an appropriate sense. These axioms yield a unique method for seasonal adjustment. The seasonally adjusted data may be obtained by an application of ordinary least squares regression. The problem of seasonal adjustment of economic time series in [3] is not the same as the problem of seasonal adjustment of data for econometric analysis. The first problem is completely resolved by appeal to the axioms of minimum variance, unbiasedness, and linearity. The second problem requires formulation as a standard problem in econometrics: estimation of the parameters of a single equation in a system of simultaneous equations. Lovell [4] has proposed to solve the problem of seasonal adjustment of data for econometric analysis by applying ordinary least squares directly to a structural equation in a system of simultaneous equations. Ordinary least squares estimation of the parameters of a structural equation usually results in "least squares bias." However, under certain special assumptions such a procedure can be justified, as Wold [6] has pointed out. Under these assumptions Lovell's procedure is valid. In this paper the seasonal adjustment of data for econometric analysis is formulated as a general simultaneous equations problem. Conditions for identification of the parameters to be estimated and methods for constructing consistent and asymptotically unbiased and efficient estimates are derived. Extensions to multivariate problems of seasonal adjustment for econometric analysis are sketched. [ABSTRACT FROM AUTHOR]

The article presents information about Samuel S. Wilks, a great contributor to statistics. Wilks' behavior toward applications was peculiarly split, he encouraged his students to work on applications, not always an easy thing to do in a strongly theoretical mathematics department; he often told students about applications that he regarded as "neat" or "cute" or "clever"; the statistical colloquium he guided often had speakers on practical applications of mathematical statistics; he himself wrote some practical papers in statistics, but in the classroom he rarely discussed applications. Repeatedly, however, practical problems explicitly influenced both his own publications and those of his students and Wilks often used applications as motivation for the discussion of distribution theory, usually going well beyond the needs of the original problem. The papers to be discussed exhibit incompletely and fragmentarily a major influence on the work of the man and his students. Wilks watched the development of the statistical theory of order statistics closely. Indeed he wrote a masterful summary of the literature. The general area involves the study of the statistical properties of ordered measurements.

Various methods based on Student t variates have been suggested and used for obtaining simultaneous confidence intervals for several means, or for several contrasts among means. Determination of an overall confidence level for such intervals involves evaluating the probability mass of a multivariate t distribution over a hypercube centered at the origin, with sides paralleling the coordinate planes, or obtaining bounds for this probability mass. Since such distributions involve many nuisance parameters, an impossible number of tables would be necessary in order to make exact confidence intervals. In the virtual absence of tables, approximations and bounds become important. In this paper, an attempt has been made to investigate the adequacy of certain suggested approximations [2], [5], [8] by computing the exact distributions for some particular cases. These exact distributions have been compared with approximations. This paper is concerned with two-sided confidence intervals, rather than one-sided intervals. [ABSTRACT FROM AUTHOR]

In a previous paper [12], an attempt was made to show how spectral techniques could be used to compare the effects of two seasonal adjustment procedures on the series to which they were applied. The two procedures compared were: (a) the technique currently used by the Bureau of Labor Statistics for seasonally adjusting employment, unemployment, and labor force monthly statistics, and (b) the so-called "residual" method, proposed by Brittain [2], Samuelson [16], and others. Spectra of the original and the seasonally adjusted series and the cross spectrum of the two were used to aid in the assessment of whether either procedure removed more than could be considered seasonal, introduced spurious regularities, and/or distorted temporal relationships. It was concluded that both techniques removed more than seasonal effects from, and produced some temporal distortion in, the series to which they were applied. Neither method appeared to be superior to the other. It is the purpose of this paper to carry the previous analysis one step further and to compare the BLS procedure with a modified version of the regression method of seasonal adjustment suggested by Cowden [3] and Mendershausen [11], and recently revived by Harman [9, 10] in an exceptionally sophisticated form. In addition to Hannan's work along the lines suggested, Nettheim [13] and Rosenblatt [15] have made studies. Rosenblatt [15] has carried out analyses similar to those reported here. [ABSTRACT FROM AUTHOR]

This paper deals with estimating the location and the scale parameters of the extreme value distribution when the sample size is very large, using sample quantiles. An estimator is given for the location parameter when the scale parameter is known, based on one or more (up to 15) order statistics. Also given is an estimator for the scale parameter when the location parameter is known, built on two order statistics. An iterative procedure is utilized to estimate the parameters when both are unknown, using two order statistics. The problem of estimating the percentage point is also dealt with, and a comparison with Lieblein's method is included. [ABSTRACT FROM AUTHOR]

Let X[sub i1], X[sub i2],..., X[sub iota n], I=1, 2, be a pair of random samples from populations which are normally distributed with means theta[sub iota], and common known variance tau[sup 2]. The problem is to estimate the function psi(theta[sub 1], theta[sub 2]) = maximum (theta[sub 1], theta[sub 2]). In this paper we consider five different estimators (or sets of estimators) for psi(theta[sub 1], theta[sub 2]) and evaluate their biases and mean square errors. The estimators are (I) psi(X[sub 1], X[sub 2]), where X[sub I] is the sample mean of the ith sample; (ii) the analogue of the Pitman estimator, i.e. the a posteriori expected value of psi(theta[sub 1], theta[sub 2]) when the generalized prior distribution is the uniform distribution on two dimensional space; (iii) a class of estimators which are generalized Bayes with respect to generalized priors which are products of uniform and normal priors; (iv) hybrid estimators, i.e. those which estimate by (X[sub 1] + X[sub 2])/2 when |X[sub 1] -X[sub 2]| is small, and estimate by psi(X[sub 1], X[sub 2]) when |X[sub 1] - X[sub 2]| is large; (v) maximum likelihood estimator. The bias and mean square errors for these estimators are tabled, graphed, and compared. Also the invariance properties of these estimators are discussed with a rationale for restricting to invariant estimators. [ABSTRACT FROM AUTHOR]

Detailed examination of the lifetimes of components and assemblies by means of the usual sequential probability ratio tests is often not feasible because of the prohibitive time or cost involved in such an examination. Situations commonly arise, however, where extraneous information is available (perhaps in the form of experience of similar situations or concerning the reputation of the supplier of the components or assemblies) which reflects on the current, situation. Such information might be incorporated in a Bayesian analysis, but little work seems have been done in this area. This paper presents a possible sequential life test incorporating prior information, applied to the simplest situation of components with exponentially distributed lifetimes, tested individually. The results for conventional sequential lifetests are used as a rough yardstick against which to measure the properties of the Bayesian lifetest discussed in the paper. [ABSTRACT FROM AUTHOR]

Ia this paper we consider a general class of Markov processes with discrete time parameter {X[sub n]: n = 0, 1, 2, · · ·} where the transitions from one state to another are determined by: Pr[X[sub n+1] = h[sub i](χ)X[sub n] = χ = p[sub i] (k = 1, 2, · · ·, k) The functions h[sub j](·) are strictly increasing, and in general the Markov processes have dependent increments. In section 1 the model is described more precisely. Considering the sequence of distribution functions F[sub n](χ) = Pr[X[sub n] ≥ χ] (n = 0, 1, 2, x] (n = 0, 1, 2, · · · · · ) a numerical method is given by which upper and lower bounds can be determined for F[sub n](χ) (n ≤ n[sub 0]. Numerical results are given for some special cases, and the determination of an alarm level in a flame failure control system is described as an application. The results show that the upper and lower bounds can lie very close together. Ia this paper we consider a general class of Markov processes with discrete time parameter {X[sub n]: n = 0, 1, 2, · · ·} where the transitions from one state to another are determined by: Pr[X[sub n+1] = h[sub i](χ)X[sub n] = χ = p[sub i] (k = 1, 2, · · ·, k) The functions h[sub j](·) are strictly increasing, and in general the Markov processes have dependent increments. In section 1 the model is described more precisely. Considering the sequence of distribution functions F[sub n](χ) = Pr[X[sub n] ≥ χ] (n = 0, 1, 2, x] (n = 0, 1, 2, · · · · · ) a numerical method is given by which upper and lower bounds can be determined for F[sub n](χ) (n ≤ n[sub 0]. Numerical results are given for some special cases, and the determination of an alarm level in a flame failure control system is described as an application. The results show that the upper and lower bounds can lie... [ABSTRACT FROM AUTHOR]

This paper compares general and linear regression for the biextremal and Gumbel bivariate extreme models. The technique is based on the values of the correlation ratios and coefficients. The computations show that linear regression is a good approximation to the general one in both cases. [ABSTRACT FROM AUTHOR]

The decision-theoretic approach to point estimation involves the choice of an estimate to minimize the expected loss associated with the estimate. The purpose of this paper is to indicate the influence of risk aversion on point estimates for classes of payoff functions including the piecewise linear and quadratic payoff functions. Increased risk aversion results in a point estimate closer to zero for a quadratic pay. off function and a lower estimate with a piecewise linear payoff function, for example. [ABSTRACT FROM AUTHOR]

Exact small sample behavior in two-way contingency tables is investigated for Pearson's chi-square statistic (X²), Light and Margolin's C statistic and its related R² measure of association, Kuliback's minimum discrimination information statistic (2&lcirc;), and Goodman and Kruskal's Lambda. R² is shown to be identical to Goodman and Kruskal's tb, leading to a test for independence based on 4. In small samples from a product of multinomials model, the null distribution of C is better approximated by a x² distribution than is the null distribution of X²; both are considerably better approximated by a x² distribution than is the null distribution of 2&lcirc;. It is proved for tables with two columns and any number of rows that if the column totals are equal, then X² ≤ 2&lcirc;; thus, X² is more conservative than 2&lcirc;. Hence, use of 2&lcirc; should be avoided in testing independence in tables with small samples. [ABSTRACT FROM AUTHOR]

This paper establishes a procedure for estimating the various spectral parameters in an echo type situation where both the signal is received as well as lagged and attenuated forms of the signal. The lag parameters are assumed frequency dependent and the basic assumption made is that these lags are sufficiently large to cause significant variation in the spectrum over the narrow trend of frequencies considered. The asymptotic properties of the estimates of these parameters are established and some simulations and an analysis of real data are considered. [ABSTRACT FROM AUTHOR]

It is known that in the random effect model, exact F-tests are not obtainable from the analysis of variance table for testing the main effects in a three-way layout. Similarly exact, F-tests are not obtainable for the main effects and certain interactions in higher-way layouts. In this paper level-α tests obtainable from the analysis of variance table are given for testing these effects. Tables are provided to illustrate the size function and the power function of the tests. [ABSTRACT FROM AUTHOR]

When an experiment is run with a factorial layout and some of the factors are random effects, there may not be an exact test for some effect of interest. This paper considers three statistics that could be used to test that hypothesis using a ratio of linear combinations of independent Chi-square statistics. The common case, utilizing four Chi-square statistics, is examined for a number of configurations of nuisance parameters. Both the power and the probability of a type I error are used in the comparison. Two statistics appear to be equally good over a large region and, in certain situations, the statistics involving the subtraction of Chi-square statistics is shown to be more stable. [ABSTRACT FROM AUTHOR]

In experimental designs with randomized blocks, estimates of the inter-block variance often have few degrees of freedom since they depend only on the block averages (the inter-block information). These degrees of freedom are further reduced by the number of parameters appearing in the inter-block information. However, some parameters also appear in the intra-block information, and thus allow two independent estimators. The difference of these two estimators provides additional information about the inter-block variance. This paper shows how this extra information may be recovered to improve tests of hypotheses concerning inter-block parameters. [ABSTRACT FROM AUTHOR]

The problem of estimating the slope of a polynomial regression at a fixed point of the experimental region such that (a) the variance of the least-square estimate of the slope at the fixed point is a minimum and {b) the average variance of the least-square estimate of the slope is a minimum is discussed in this paper. In general these designs can be obtained using Kiefer-Wolfowitz [5] characterization of c-optimal designs, Federov [2] characterization of L-optimal designs, and Studden's [10] generalization of the Elfving Theorem [1]. After presenting a brief review of these characterization theorems, specific illustrations for the quadratic and cubic regressions are presented in detail. [ABSTRACT FROM AUTHOR]

In regression analysis with heteroscedastic and/or correlated errors, the usual assumption is that the covariance matrix Sigma of the errors is completely specified, except perhaps for a scalar multiplier. This condition is relaxed in this paper by assuming only that Sigma has a certain pattern; for example, that Sigma is diagonal or partitionable into a diagonal matrix of sub-matrices. The method used for estimating Sigma is the standard procedure of equating certain quadratic forms of the observations (in this case, squares and products of residuals from regression) to their expectations, and solving for the unknown variances and covariances. A numerical example illustrates the method. [ABSTRACT FROM AUTHOR]

In many multiple correlation and regression studies the values of the variables are difficult to measure accurately. This paper attempts to discuss the effects of such errors of measurements on the squared multiple correlation coefficient. With independent errors and a multivariate normal model, the effect is very roughly that R[sup 2] becomes reduced to about R[sup '2] = R[sup 2] g[sub y] g[sub w], where g[sub y] is the coefficient of reliability of y and g[sub w] is a weighted mean of the coefficients of reliability of the x's. If most intercorrelations among the true X's ace detrimental to R[sup 2], the value of R[sup '2] may exceed R[sup 2]g[sub y] g[sub w] by 10 percent to 20 percent. The situation in which errors are correlated with the true values and the effects of errors on the interpretation of regression coefficients are briefly discussed. [ABSTRACT FROM AUTHOR]

In this paper methods for obtaining approximations and bounds for the moments of order statistics from a continuous parent distribution are discussed. These bounds and approximations depend on the distribution function only through certain moments of order statistics in small samples. It is shown that for the Cauchy distribution bounds and approximations of all finite moments can be obtained. Some numerical calculations for normal and Cauchy distributions are also given. [ABSTRACT FROM AUTHOR]

This paper uses a family of bivariate extreme-value distributions to estimate the probability of a large exceedance of a random variable given that a certain other random variable not independent of the first has exceeded a certain value. A simple method of reasonably good efficiency is given for estimating a bivariate extreme-value distribution from independent bivariate samples. The method is used to analyze the performance of a spacecraft command receiver which has an indication of data quality so that commands likely to be in error can be rejected. [ABSTRACT FROM AUTHOR]

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 by D[sub 1], D[sub 2] and D[sub 3] in this abstract. D[sub 2] and D[sub 3] are applied to Kronecker products, D[sub 1] is applied to ordinary products. This is a matter of efficiency only. D[sub 1] is developed first. Transforming matrices into column vectors turns out to be very convenient. D[sub 2] and D[sub 3] are developed then. D[sub 3] is a generalisation of D[sub 1]. [ABSTRACT FROM AUTHOR]

This paper develops a theoretical framework for the unrelated question randomized response technique suggested by Walt R. Simmons. The statistical efficiency of this technique is compared with the Warner technique under situations of both truthful and untruthful responses. Methods of allocating the total sample to each of two subsamples required by the unrelated question approach are developed. Recommendations are made concerning choices of values for those parameters which can be assigned at the discretion of the investigator. [ABSTRACT FROM AUTHOR]

A method is proposed to eliminate trends from a sample from a nonstationary system of linear stochastic difference equations. The autocovariance matrix and the spectral density matrix of the detrended component of the sample are derived. The latter matrix turns out to have the same form as the spectral density matrix for a stationary system when expressed in terms of the roots of the system. Since the parameters of the system are assumed known throughout the paper, the problem of statistical inference does not arise. [ABSTRACT FROM AUTHOR]

When two correlation coefficients are calculated from a single sample, rather than from two samples, they are not statistically independent, and the usual methods for testing equality of the population correlation coefficients no longer apply. This paper considers the situation when the sample is from a multivariate normal distribution. Several possible large sample testing procedures are given, all based on Fisher's z-transformation. Power curves are given for each procedure and for seven values of the asymptotic correlation between the two sample correlation coefficients. [ABSTRACT FROM AUTHOR]

The quite general test for heteroskedasticity presented here regresses the absolute values of the residuals obtained by ordinary least-squares on some variable(s). Denoting the O.L.S. residuals by |u|, one obtains, for instance, a regression like |u| = a + bz + epsilon where z is a variable, a and b regression coefficients and epsilon the residuals of the new regression. We call the acceptance of a non-zero value for both a and b a case of "mixed heteroskedasticity", which we deem frequent in practice though neglected in handbooks. The paper also summarizes another test due to S. M. Goldfeld and R. E. Quandt and examines the powers of the two by using Monte-Carlo simulations: the new test seems to compare favourably, except perhaps in the case of large samples. [ABSTRACT FROM AUTHOR]

A progressive-censoring model arises from a life test of a sample of items in which one or more of the survivors may be removed from the test at the time of any failure. Such a model is often more realistic for actual failure data which must be analyzed by a statistician than one in which all survivors are assumed to be removed from test simultaneously. This paper deals with the situation in which the underlying failure-time distribution for the population sampled is the two-parameter Weibull distribution. The reliable life for the population is defined to be the 100(1-R) percent point of the failure-time distribution, where R is a specified population survival proportion, or reliability. An exact confidence bound on reliable life based on three observed ordered failure times is derived for this progressive-censoring model. The criterion used for selecting the order numbers of the three failure times upon which the bound is based depends upon computed values of the power function of the test associated with the bound. A table from which lower bounds can be obtained is given for R equal to .95, confidence level .90, sample size equal to 2, 3, . . . , 6, and all possible censorings. [ABSTRACT FROM AUTHOR]

In this paper some properties of the multivariate Beta distributions are discussed. Specifically it is shown that the density of the multivariate Beta I distribution can be decomposed as a product of independent ordinary Beta I densities and the Dirichlet densities, while the density of the multivariate Beta II distribution ss a product of independent ordinary Beta II densities and the inverted Dirichlet densities Some extentions of the results to the generalized multivariate Beta distributions are indicated. [ABSTRACT FROM AUTHOR]

Economic statisticians do not enjoy an untarnished reputation for accurate forecasting. We have managed, over the years, to come up with some memorable failures. While we have also had our share of successes, they are not as well remembered nor as numerous as we should like. Recently, however, we have begun to pay more attention to the record, and a substantial body of evidence on forecasting performance has accumulated. In this paper I propose to review this record, try to arrive at a balanced appraisal, and offer some suggestions for improvement. [ABSTRACT FROM AUTHOR]

This paper deals with certain aspects of variance-component estimation for the unbalanced two-way completely-random classification where the numbers of observations in the subclasses are treated as random variables not necessarily independent of some of the random effects of the model. General results are given on the expectations of two commonly-used estimators of the vector of variance components. Numerical approximations are presented for these expectations for one sub-family of the family of all possible joint distributions of the subclass numbers and the random effects. [ABSTRACT FROM AUTHOR]

The least squares estimator of a regression coefficient beta is vulnerable to gross errors and the associated confidence interval is, in addition, sensitive to non-normality of the parent distribution. In this paper, a simple and robust (point as well as interval) estimator of beta based on Kendall's [6] rank correlation tau is studied. The point estimator is the median of the set of slopes (Y[sub I]--Y[sub I])/(t[sub I]--t[sub I]) joining pairs of points with t[sub 1] is not equal to t[sub 1], and is unbiased. The confidence interval is also determined by two order statistics of this set of slopes. Various properties of these estimators are studied and compared with those of the least squares and some other nonparametric estimators. [ABSTRACT FROM AUTHOR]

This paper describes and derives the asymptotic behavior of the new Normal approximation introduced in Part I and of a family of Normal approximations based on roots, including the square root approximations of Fisher, and Freeman and Tukey, and the cube root approximations of Wilson and Hilferty, Camp, and Paulson. Various asymptotic comparisons are made, all of which rank the new approximation first, the cube root approximations second, and the other root approximations (and the ordinary Normal approximation) third. For instance, in the binomial case, if the tail probability is fixed as n arrow right infinity, the errors resulting from the foregoing approximations are generally of order n[sup -3/2], n[sup -1], and n[sup -1/2] respectively, while for a tail probability approaching 0, the relative error in approximating it approaches 0 if the corresponding standard Normal deviate is of smaller order than n[sup 1/2] n[sup 1/4], or n[sup 1/6] respectively, but not generally otherwise. These comparisons are deduced from far more detailed results which are also given. [ABSTRACT FROM AUTHOR]

This paper deals with proportional stratified sampling in the situation where the estimation variable X is difficult and expensive to observe, while the possibly erroneous stratification variable Y is easy and inexpensive to get at. The usually biased estimate [Multiple line equation(s) cannot be represented in ASCII text] is compared with the unbiased estimate [Multiple line equation(s) cannot be represented in ASCII text] where the P[sub I] are stratum weights and y[sub I] and x[sub I] are means of the units sampled from the I:th stratum. The two estimates are similar in that they utilize information from only those population units that make up the sample. While mu[sub a] is the more inexpensive estimate, mu[sub b] is usually preferable if one judges by the size of the mean square error, which, among other things, depends on the number of strata and the location of the stratum boundaries. In particular, the properties of mu[sub a] and mu[sub b] are discussed in connection with an example involving the bivariate normal distribution. [ABSTRACT FROM AUTHOR]

The numerous sources of variation among life tables are reducible to two. This paper finds expressions for these separately. The first source of variation is the rate of increase attributed to the observed population. A given underlying continuous curve of mortality by age will show somewhat different age-specific death rates according to the rate of increase of the population subject to the given mortality. At ages beyond about 15, for a given underlying mortality, the age-specific rates observed will be lower the more rapid the increase. Conversely, to infer the stationary age-specific rates from observed rates a larger quantity must be added the greater the increase to which the observed population is subject. The second source of difference is the form of the curve put through the (discrete) l[sub x], the number living to age x, to integrate and find [sub n]L[sub x], the number in the stationary population from age x to x + n. Some considerations are presented on the choice which must be made on each of these two matters, the life table being fully specified to the last decimal place once those choices are made. Finally, a recommended procedure for making life tables is provided, covering special devices for the early and terminal ages, using a new formula (6) which provides more rapid iteration for the ages from 5 to 80. [ABSTRACT FROM AUTHOR]

Subsets of the dependent variable of a linear model often conform to known constraints; for example, this situation arises with Engel curves and with models which estimate transition probabilities, market shares, or other classes of proportions. These models imply a variety of restrictions on the parameters, explanatory variables and residuals which must be treated explicitly in estimating and testing hypotheses about the parameters. This paper discusses these restrictions and develops an estimating procedure which yields consistent estimates which are asymptotically efficient, unbiased and normal. [ABSTRACT FROM AUTHOR]

In this paper we examine how well the asymptotic null distribution of a two sample test due to Lehmann approximates the small sample distribution of the test, compare the validity of this Lehman test with the validity of the two sample t test under the null hypothesis of equal means, and compare the power of this Lehmannn test with the power of the t test. Our general conclusion is that experimenters will prefer to use the t test when the underlying distribution is the scale contaminated compound normal distribution and the sample sizes are less than thirty. [ABSTRACT FROM AUTHOR]

An algorithm for the determination of the economic design of X-charts based on Duncan's model is described in this paper. This algorithm consists of solving an implicit equation in design variables n (sample size) and k (control limit factor) and an explicit equation for h (sampling interval). The use of this algorithm not only yields the exact optimum but also provides valuable information so that the sensitivity of the optimum loss-cost (L*) can be evaluated. Loss-cost contours are used to discuss the nature of the loss-cost surface and the effect of the design variables. The effect of two parameters, the delay factor (e), and the average time for an assignable cause to occur (1/lambda), on the optimum design is evaluated. Numerical examples are used for illustrations. [ABSTRACT FROM AUTHOR]

This paper describes the application of remote console computing in a graduate statistics seminar entitled "Machine Aided Statistical Modeling," which was offered at Harvard University in the spring semester 1965-1966. Three conceptually different computing systems are discussed, together with examples aimed at demonstrating the potential of on-line interactive computing in statistical research, practice, and education. [ABSTRACT FROM AUTHOR]

Let X[sub 1], ... X[sub r], be independent B(n,p) random variables. In this paper n is estimated with known p. The MLE is developed and shown to be consistent in a relative sense and asymptotically normal. Motivated by Y[sub i] = X[sub i]/q we consider Y[sub 1], ... ,Y[sub r] independent N(mu, mu) and examine three estimates of mu; (i) the MVUE, (ii) the MLE, and (iii) ?. These are shown to be asymptotically equivalent. [ABSTRACT FROM AUTHOR]

This paper presents three upper bounds for the variance of an estimator, based on observations selected with varying probabilities from a finite population, the elements of which are ranked with respect to the Y values. Accordingly, the usefulness of these bounds relates to the pre-enumeration analysis where one may well know the intended probabilities and joint probabilities corresponding to the sampling scheme but does not know the Y values. If, however, one can make a conservative guess at the largest Y value, one can use these bounds. Some examples are included to illustrate the theory. [ABSTRACT FROM AUTHOR]

In this paper, successive sampling procedures using multi-stage sampling design have been developed. It is found that it is generally advantageous to retain a fraction of the sample selected on previous occasions for improving the estimate of a mean on the most recent occasion. This type of partial replacement may sometimes be recommended for estimating a mean over all occasions. This is specially true if the experimenter is interested in not only obtaining an overall estimate for the entire period but also in separate estimates for each occasion. In a sampling enquiry repeated on three occasions it has been observed that for estimating the mean on the third occasion it would be preferable to repeat the same sample fraction from one occasion to the next, while for estimating the mean over all occasions the sample fraction repeated on the second occasion should not be repeated on the third but in its place a sub-sample from the sample selected afresh on the second occasion should be retained. [ABSTRACT FROM AUTHOR]

This paper deals with certain aspects of variance component estimation for the unbalanced one-way random classification where the number (N[sub I]) of observations in the ith class is treated as a random variable not necessarily independent of the class effect (A[sub iota]). It is assumed that in general P(N[sub I] = 0) > 0. The conditional expectations (given the number of observations in each class) of all estimators of the between variance component (sigma[sup 2, sub alpha]) belonging to a certain class of estimators are derived. A general expression is found for the expected value of that estimator of sigma[sup 2, sub alpha] yielded by analysis of variance of class means. The limit of this expression (as the number of classes arrow right Infinity) is given; and it is shown that, if the bivariate distribution function of A[sub I], N[sub I] belongs to a certain class of distribution functions, then this limit is less than sigma[sup 2, sub a]. Numerical approximations to the expected values of two estimators of sigma[sup 2, sub a] are presented for one subclass of such distribution functions. [ABSTRACT FROM AUTHOR]