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In this article we propose two minimum chi-square estimators for a bivariate probit model. We call one estimator the Full Information and the other Limited Information Minimum Chi-Square because the first takes account of all the a priori information while the second does not. Both estimators are shown to be consistent. Moreover, the first is shown to be asymptotically as efficient as the maximum likelihood estimator and yet is computationally much simpler. For illustration, estimates are computed for the data used by Ashford and Sowden [1970.] [ABSTRACT FROM AUTHOR]

Monte Carlo methods were used to study the performance of the largest-root test, two trace-type tests and three determinantal tests in the fixed-effects MANOVA model when certain assumptions are violated. Results indicated that for protection against nonnormality and heterogeneity of covariance matrices, the largest-root test should be avoided, while the Pillai-Bartlett trace test may be recommended as the most robust of the MANOVA tests, with adequate power to detect true differences in a variety of situations. [ABSTRACT FROM AUTHOR]

When data are transformed to satisfy a spherical normal linear model, the mean θ of a variate in the original scale is a function of the mean μ and variance σ² of a normal variate. We consider several approximate confidence interval methods for θ, including a new method based on exact confidence intervals for linear functions of μ and σ² Monte Carlo estimates of coverage probabilities demonstrate the suitability of the new method for applications involving a wide range of data transformations, parameter values and sample sizes. [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]

Necessary and sufficient conditions for admissibility are given for test procedures based on a preliminary test of significance. Three types of problems are studied-testing the normal mean, fixed effects models of the analysis of variance and random effects models. Admissibility in this instance is equivalent to the intuitive and practical condition that acceptance regions of the procedures have convex sections in certain variables, while other variables are fixed, It is easy to check when the conditions hold. A discussion of optimality properties of these and other types of pooling procedures is given. [ABSTRACT FROM AUTHOR]

This article offers a practical guide to goodness-of-fit tests using statistics based on the empirical distribution function (EDF). Five of the leading statistics are examined-those often labelled D, W², V, U², A²-and three important situations: where the hypothesized distribution F(x) is completely specified and where F(x) represents the normal or exponential distribution with one or more parameters to be estimated from the data. EDF statistics are easily calculated, and the tests require only one line of significance points for each situation. They are also shown to be competitive in terms of power. [ABSTRACT FROM AUTHOR]

This article presents a data analysis approach consisting of a synthesis of experimental design methods and spectral methods of time series analysis which is appropriate in exploratory situations where the re- cording process generates a long sequence of correlated observations. [ABSTRACT FROM AUTHOR]

The use of Patnaik's [28] chi-square approximation to the noncentral chi-square distribution and the Wilson-Hilferty [321 transformation of chi-square to approximate normality are explored as a simple, efficient means of finding k-order-statistic confidence bounds on parameters and reliability of the one- and two-parameter negative exponential distributions. An important implication of the results leads to obtaining simple closed-form approximations to percentiles of the beta distribution either with integer or noninteger parameters. The general methodology is also used to approximate confidence or prediction intervals for the time of the kth exponential failure based on r

Three statistics for testing additivity and making detailed inferences on interactions in a two-way setup are considered. The first method uses the well-known sum of squares for testing and yields Scheffe bounds for all contrasts; the second introduces a maximum characteristic root statistic and Roy bounds, but is restricted to product-type contrasts; the last is limited to tetrad differences and its statistic is the maximum of all such differences. Bounds on percentage points for this maximum are provided in the appendix. Uses of the three methods in inferring various types of interaction contrasts are discussed and illustrated. [ABSTRACT FROM AUTHOR]

Following the results of Bhat and Nagnur [2], the locally asymptotically most stringent test of the hypothesis of no preference in paired comparison experiments is derived for the models of Rao and Kupper [9] and Davidson [3]. It is shown that the asymptotic relative efficiency of these tests when compared to the test based on the likelihood ratio statistic is unity. [ABSTRACT FROM AUTHOR]

It is generally recognized that when deciding on a significance level for a test, its power must also be taken into account. Determination of the optimal significance level, however, is not a straightforward task. It is pointed out here that by use of critical significance level and power we can achieve the usual objective of making a reject-or-accept decision without explicitly determining the optimal significance level. [ABSTRACT FROM AUTHOR]

A method is given for obtaining sequential tests of composite hypotheses for a certain class of distributions. Approximate properties of the tests are derived and the method illustrated. [ABSTRACT FROM AUTHOR]

Mean square errors associated with median response estimates are compared for the normal and logistic distributions, based on both maximum likelihood and minimum chi square analyses. In broad perspective there is little difference among the mean square errors for these four estimates, regardless of sample size and stimulus level spacing. The first three of the median response estimates require an iterative numerical solution, whereas the minimum chi square analysis for the logistic distribution yields an explicit expression for the associated median response estimate. This explicit estimation expression may be particularly convenient when nonuniform spacing of stimulus levels is involved in testing. [ABSTRACT FROM AUTHOR]

A test for the hypothesis of zero second-order interaction is proposed. The test is obtained by making use of conditional reference sets which are derived in Zelen [17]. This permits the derivation of an exact conditional test of significance for testing the hypothesis of zero second order interaction. The techniques used, in approximating the large sample distribution of this test, are extensions of those due to Gart [5] and Zelen [17]. The test is applied to a numerical example. [ABSTRACT FROM AUTHOR]

In this article the problem of testing for homogeneity of k (k > 3) independent negative binomial distributed random variables with parameters r[sub i], p (for i = 1, ..., k) is studied. The alternative hypotheses which are considered are a p-mixture of negative binomial distributions and a positive trend of the p's. Several cases may be distinguished: (1)p known, (2) p unknown and r[sub 1] =...= r[sub k],(3)p unknown and r[sub 1], ..., r[sub kappa] possibly unequal. The proposed tests are based on explicit optimality criteria. [ABSTRACT FROM AUTHOR]

Hypothesis test procedures for tests of separate families of hypotheses are defined. These procedures are then compared using Monte Carlo samples as data for five pairs of invariant distributions. The comparisons are made by calculating an approximate relative efficiency for each procedure with respect to the likelihood ratio procedure, which for three pairs of distributions considered is uniformly most powerful among the class of invariant procedures, [ABSTRACT FROM AUTHOR]

The problem considered is how to compare time-to-response data for different groups when the group membership of an individual can be arbitrarily varied during a study. Modified life tables can be constructed which reflect such changes in an individual's status, and associated measures of relative risk and statistical significance calculated. This is illustrated with survival data for heart-transplant patients, for which a patient can transfer from the nontransplanted to the transplanted group. Alternative procedures are given in which distinctive groups are defined for each transplant day. [ABSTRACT FROM AUTHOR]

The asymptotic method described in an earlier article (Bowman and Shenton [2]) is applied to derive the power of the square root of b[sub 1] test of normality against skewed alternatives. In particular, normal mixtures of two components and Type A distributions, each depending on four parameters, are considered. Comparisons of the assessment of percentage points of square root of b[sub 1] using Cornish-Fisher expansions based on the normal and Monte Carlo simulations are given. [ABSTRACT FROM AUTHOR]

Cochran's Q-test, proposed to test the equality of three proportions for correlated observations, may have a larger asymptotic significance level than the nominal one, unless the admissible family of distributions is restricted to ensure that under the hypothesis the correlations between the observations are also equal. For a large class of such restricted families, optimal C[sup 2](alpha) tests of the hypothesis are obtained, and the family for which the Q-test is an optimal C[sub 2](alpha) test is identified. The use of Q for testing a related hypothesis, suggested by Madansky, is also discussed: it is shown that the Q-test is not consistent against all alternatives to Madansky's hypothesis, while the classical chi[sup 2]-test provides an optimal C[sub 4](alpha) test for it. [ABSTRACT FROM AUTHOR]

Procedures for discriminating between models from separate families of hypotheses are examined, with principal emphasis on procedures invariant under location and scale transformations. The discrimination procedures considered are compared using Monte Carlo samples as data for five pairs of invariant distributions. These comparisons are made with respect to the best invariant procedure, a procedure requiring no knowledge of sampling distributions, using approximate relative efficiencies calculated from the Monte Carlo results. On the basis of these efficiencies and the computational complexities of the procedures, suggestions are made for their use. [ABSTRACT FROM AUTHOR]

We consider the problem of drawing inferences on the difference of the population means when an incomplete sample from a bivariate normal population is available. Whereas Mehta and Gurland [7] have considered this problem from the sampling theory point of view, we tackle here the same problem from a Bayesian approach. [ABSTRACT FROM AUTHOR]

Procedures are given for testing hypotheses and obtaining confidence intervals for the reliability of systems of k independent parallel components consisting of m < k component types. Assume that there are alpha[sub I] components of the ith component type, I = 1, 2 ..... m and that p, is the failure probability of the ith component type. Then the probability of failure of the system is rho = II[sup m, sub l=1] p[sup alpha, sub l]. If m independent sequences of Bernoulli trials are conducted, one for each component type, techniques which may be employed for statistical inference concerning p are given. These techniques are suitable under those conditions for which the binomial distribution may be satisfactorily approximated by the Poisson distribution. [ABSTRACT FROM AUTHOR]

A Monte Carlo study of confidence limits for the slope of the major axis of a bivariate normal distribution confirms that, when imaginary limits are interpreted as corresponding to an infinite interval covering all possible values of the parameter, the confidence interval behaves satisfactorily under repeated sampling. The excess of the actual over the nominal significance level is negligible even if samples are small and correlation is moderate. The probability of detecting a relationship correctly is never smaller for the major axis than for ordinary regression. imaginary and exclusive confidence limits do not create problems in practice. [ABSTRACT FROM AUTHOR]

A family of confidence intervals with endpoints that are simple averages of sample order statistics is defined. In this family the interval with shortest expected length is selected and compared to other confidence intervals for the same parameter. [ABSTRACT FROM AUTHOR]

The asymptotic non-null distribution is obtained for the modified form of the Pearson chi-square statistic studied by Dahiya and Gurland [3]. By utilizing this result the power is obtained for specific alternative distributions in testing for normality, This enables recommendations to be made as to the number of class intervals to be employed in applying the aforementioned modification of the Pearson chi-square test of normality. [ABSTRACT FROM AUTHOR]

Procedures for testing the difference of means are obtained in sampling from a bivariate normal distribution with covariance matrix Sigma when some of the observations on one of the variables are missing. A UMP test procedure is obtained when Sigma is known. When Sigma is not known, exact test procedures may be obtained by discarding partial data. To make use of all available data, approximate test procedures are proposed. These procedures are compared to the exact tests, obtained by discarding partial data, using Monte Carlo methods. [ABSTRACT FROM AUTHOR]

Suppose that in regression problems the simultaneous confidence intervals of the S-method are used only when a preliminary F-test rejects the null hypothesis that the regression parameters are zero. (Some of Scheffe's work emphasizes this usage [12, p. 87; 13, p. 66].) The probability of coverage should then be conditioned on the rejection. That for sufficiently large critical values and at least two degrees of freedom for error the conditional probability of simultaneous coverage is always smaller than the unconditional probability is established in this article. Also included are Monte Carlo studies of the discrepancy and interpretation. [ABSTRACT FROM AUTHOR]

We derive an asymptotically efficient method of estimating the coefficients of systems of seemingly unrelated regression equations when a first-order vector autoregressive error specification is hypothesized. This error specification is shown to contain several commonly used specifications as special cases. A Monte Carlo experiment was performed to examine the finite sample performance of our estimator in comparison with several other estimators. The general results were that the new estimator performed relatively well. [ABSTRACT FROM AUTHOR]

Analogous conditional and unconditional confidence interval procedures for the two parameter Weibull distribution are discussed. Results of an empirical study are given which suggest that, in many practical situations, the two procedures considered lead to near-equivalent results. Ramifications of the empirical results are also considered. [ABSTRACT FROM AUTHOR]

Procedures are developed for specifying confidence intervals, from possibly censored data, for any characteristic of a two-parameter Weibull population. Although the methods are formally Bayesian, they have exact relative frequency interpretation for uncensored and Type II censored data. Some simulation results are given to indicate relative frequency behavior for Type I censoring, and for comparison to other methods. [ABSTRACT FROM AUTHOR]

Theory of testing hypotheses (see [1]) derives sufficient conditions for the existence of optimum tests. These conditions often require that the class of distributions in question be ordered in some way, e.g., stochastically ordered, monotone likelihood ratio, Polya of some order, or exponential class. This article gives four simple examples which show that the sufficient ordering restrictions cannot be weakened to less stringent ordering restrictions and which thus give some insight into the structure and use of such orderings. A fifth example shows that ordering properties are not necessary for the existence of optimal tests. [ABSTRACT FROM AUTHOR]

Two tables of modest size and three relatively simple formulas provide all that is required to obtain percentage points of the Chi[sup 2] distribution at 25 significance levels with relative accuracy of at least .99995; and to obtain Poisson confidence limits of such accuracy at 13 confidence levels. For a small number of degrees of freedom or observed number of events, where the formulas are weak, the tables give exact Chi[sup 2] values or Poisson limits. [ABSTRACT FROM AUTHOR]

Sequential search procedures are described for determining an optimal dosage in the following biomedical problem. People are subjected to a certain chemotherapeutic treatment. The optimal dosage is the maximal dose for which the proportion of patients whose toxicity level does not cross the allowable limit is gamma. We discuss Bayesian and non-Bayesian sequential procedures for searching the optimal dosage, assuming a linear regression between toxicity and dosage, and normal conditional distribution of the toxicity level at each dose, with a known variance. [ABSTRACT FROM AUTHOR]

This article presents Bayesian estimators of the parameters of the Weibull hazard function in the restoration process case. These estimators are used in a model which optimizes the interval between machine overhauls. Since the properties of the model are not in closed form, a simulation experiment is used to evaluate the effectiveness of the model. The simulation results show that the model does yield lower machine operating costs than the non-Bayesian approach. The effectiveness of the model could be increased by improvements in the quality of the prior estimates used in Bayesian estimation. [ABSTRACT FROM AUTHOR]

The small sample properties of several distributed lag estimators are examined under conditions of misspecified lag structure, such as fitting a geometric estimator to a second order lag distribution. Evaluation includes analyses of the estimated average lags and the adjustment rates. Lag structure misspecification is found to yield large biases in the average lag and adjustment rate, especially compared to the biases resulting from a correctly specified but poorly designed estimator. The test statistics are found to be unreliable and often misleading indicators of lag structure misspecification. [ABSTRACT FROM AUTHOR]

Both the definition of economic activity and the definition of periodicity seem to determine whether there are long tides in economic affairs. Spectral analysis of growth rates finds no long waves in the U.S., U.K., and Sweden, but this is consistent with the appearance of long classical (upcross) swings in the deviations from trend of several raw data series. These upcross oscillations may not be independent components of economic evolution but merely distorted reflections of trend growth processes. [ABSTRACT FROM AUTHOR]

American census-taking procedures had evolved slowly from the inception of the decennial census in 1790 until 1840, when a new high seemed evident in the many additional categories included in the survey. On publication, however, numerous errors were discovered in the figures, precluding the usefulness of the census data. Paradoxically, the discovery of these errors in the sixth census was to become the catalyst for reforms that would spearhead the government's use of modern statistical procedures. [ABSTRACT FROM AUTHOR]

The present model is applicable when pi[sub Y], the population proportion with a nonsensitive attribute Y, is not known in advance and two samples are required to estimate pi[sub A], the population proportion with a sensitive attribute A. It consists of using the sensitive question and two nonsensitive alternate questions in each sample, yielding two unbiased estimates of pi[sub A]. The optimum estimator, a weighted average of these two estimates, is described and its minimum variance calculated. The efficiency of this estimator is compared with Moors' optimized version of the standard two sample single alternate question model, and with the one sample single alternate question model when pi[sub Y] is known. [ABSTRACT FROM AUTHOR]

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 correlated. The measurement of regional inequality is characterized by considerable uncertainty. [ABSTRACT FROM AUTHOR]

The article focuses on prestige patterns in scholarship and science. To throw some light on this question a questionnaire was recently sent to 1,275 individual chosen at random from the membership rosters of twelve different learned societies. The homogeneity of the returns was determined by the Chi square test. The sample was checked for homogeneity with respect to academic rank and section where the respondent was employed at the time he made out his questionnaire and at the time he was listed in the roster of his society. It is Interesting to note that history and economics are the only two fields in which one individual achieved a ranking frequency considerably above that of any other man listed. In every field individuals were listed prominently for their administrative ability, personal leadership in establishing schools and stimulating followers, for important offices held and for other characteristics not directly relevant to research in the limited sense.

Stating a hypothesis is becoming a common procedure in geographic writing, but the meaning and function of the term "hypothesis" have a wide array of interpretation. Some geographers use the term to pose fairly specific, directional relationships between phenomena, others use it to state more general relationships, and still others equate it with question, explanation, conclusion, contention, assumption and model. The confused state of affairs is interpreted as the result of scientific faddism and pretentiousness. [ABSTRACT FROM AUTHOR]

Explores the use of statistical tests for decision-making in regional geographical construction in the United States. Chi square; Variance analysis; Relationship between regional constructs and the areal dispersion of selected traits within regional frames.

An F-test of linear hypotheses is compared with Bonferroni t-tests. The individual confidence intervals from Bonferroni t-tests are uniformly shorter than S-intervals implied by the F-test. Power curves are constructed for a few specific alternative hypotheses as functions of the correlation between regressors for the special case of two hypotheses and two regressors. The power of the two procedures is similar when the correlation is small. For highly correlated regressors, however, the power of the Bonferroni method is generally inferior. Thus, if regressors can be controlled to be uncorrelated, the Bonferroni method is clearly superior; otherwise neither method dominates. [ABSTRACT FROM AUTHOR]

This article shows that information partitioning in contingency tables is a very general property of exponential families, One partitioning scheme occurs when the hypotheses of interest are stated as linear expressions in the natural parameters, A second partitioning scheme is possible when the hypotheses are stated as linear in the parameter which is the expectation of the statistic in the exponent. [ABSTRACT FROM AUTHOR]

Following Riedwyl [8] we generalize some one-sample statistics of the Cramer-von Mises type so that they can be used to test grouped data for goodness of fit. We prove that under the null hypothesis the asymptotic distributions of these statistics, when suitably standardized, coincide with the corresponding classical statistics if the ratio of the sample size to the number of groups (k) remains constant. For k fixed the asymptotic distributions are given under the null hypothesis, and it is shown how to obtain them under any alternative. Some results for finite sample sizes are also derived, [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]

This article tests the hypothesis of a variable lag of shipments of capital goods behind new orders, applied to three major Canadian industries producing mainly to order. This hypothesis, confirmed by earlier studies on American data, is corroborated on the Canadian data. Several supplementary conclusions are also drawn, including a suggestion that constraining the weights in the Almon lag distribution to zero at the beginning and end of the distribution may introduce distortion into the resulting estimates. [ABSTRACT FROM AUTHOR]

The combined use of direct estimation (in saturated models) and indirect testing (of unsaturated models), proposed in [7, 8] for the analysis of a given multidimensional contingency table, is extended here to the analysis of a set of T multidimensional tables (T is greater than or equal to 2). For this set of tables, a three-fold classification of models is introduced: (1)models that assume "complete homogeneity" among the tables; (2) models that allow "complete heterogeneity" among the tables; and (3) models that allow "partial heterogeneity." Stepwise procedures proposed in [8] for model selection are extended. "Guided" and "unguided" selection methods and "multidirectional" methods are introduced. For illustrative purposes, a set of two 4-way tables is analyzed. [ABSTRACT FROM AUTHOR]

In paired comparisons of a treatment and control, it frequently happens that the two members of each pair can be classified into distinguishable types. The completely randomized design, which assigns the members of each pair at random to treatment and control, then may by chance assign the treatment primarily to subjects of one type and thereby confound treatment and type. This difficulty can be avoided by restricting the randomization. We find that such restriction is desirable by analyzing several of the standard tests (Wilcoxon, t, and tests for dichotomous response) for efficiency and deficiency. [ABSTRACT FROM AUTHOR]