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In connection with a study of the high temperature conductivity anomaly for the silver halides, several forms of chi square tests have been applied to determine the statistical validity of the results. First the usual procedure for obtaining a least squares fit of conductivity us. temperature is described. Next a discussion of experimental errors shows that uncertainties in resistance, sample dimensions, and temperature measurements may give an accumulated error of around 0.5 %, and that this per cent error should be independent of temperature to a good first approximation. A chi square test is then described for the distribution of deviations in an inter-mediate temperature range, and it is shown that in favorable cases the errors can be described satisfactorily by a normal error distribution with a mean of zero and a standard deviation of 0.39 % for AgCl and 0.43 % for AgBr. Finally a second form of chi square test is described to measure the goodness of fit of the fitting function based on the defect model for the conductivity. It is shown that the value of chi square increases rapidly as the upper end of the intermediate temperature region is extended into the higher temperature region where the anomaly occurs, thereby providing a rather clearly defined upper temperature limit for an acceptable fit. It is concluded that these uses of the chi square test have played an important role in the present study, and that similar uses should be valuable for analyzing the experimental results of other studies of diffusion and ionic conductivity.

Shape analysis of paleosecular variation (PSV) data, if it is properly used, can determine the statistical character of a PSV data set in many cases. Existing statistical techniques, such as the χ2 goodness of fit test, the ‘dipole moment,’ and eigenvalue methods, are reviewed with this point in mind. Two new parameters, K0 and Km, are introduced that can be used to derive a better estimate of angular standard deviation for a PSV data set, particularly if the data are most densely distributed near their mean. We have analyzed the shape of the distributions of 12 PSV data sets of the Brunhes and Matuyama epochs by means of these various statistical techniques. The results of our studies indicate that the minimum size of a PSV data set should be about 30 and that the shape analysis already mentioned should be used as one PSV data selection criterion.

In this article we present some empirical evidence which indicates that attempts to represent the probability distribution of the rates of return on common stocks by a member of the stable Paretian family of distributions, with 1 < α < 2, may be misleading and in fact may not produce an adequate fit to observed rates of return. We offer an alternative probability model for describing rates of return based on the hypothesized phenomenon of a changing variance. We test the “goodness of fit” of our model vis-a-vis a stable Paretian model for several series of rates of return. Finally we propose an extension of the stability test of Fama and Roll [6].

The treatment of censored data problems has been restricted almost exclusively to estimation of parameters. In this paper a goodness of fit test is presented for a completely specified distribution in which the single sample is censored at the kth ordered statistic (k < n), commonly known as Type II censored sample (David, 1970) , The technique is an extension of the Hartley and Pfaffenberger (1972) criterion, The approximate distribution of the test statistic is obtained via Pearson's Type VI and Type IV curves.

A new statistic TE for testing exponentiality is proposed. TE is sim¬ilar to Tikufs statistic T for testing normality and is the ratio of an estimator of scale parameter calculated from a sample with a proportion of observations censored to the estimator calculated from the whole sample. TE is shown to have Beta distribution. Besides, TE is both origin and scale invariant and is easy to compute. The power of T for non-exponential distributions is, on the whole, of the same magnitude as the power of Shapiro & Wilk statistic W-Exponential.

Modifications of the two-sided one-sample Kolmogorov-Smirnov “goodness-of-fit” test, for use with censored and truncated samples, are suggested. Tables of the distributions of the modified statistics are given. Applications to life testing and reliability estimation problems are discussed.

The controversy over contextual effects is reviewed from the standpoint of specification analysis. The uselessness of ad hoc calculations of "variance due to" is demonstrated, and more careful examinations of goodness of fit are advocated. In this regard, it is demonstrated that techniques involving the analysis of residuals can be powerful aids both to assessing the quality of parameter estimation and to allowing the data to reveal the unexpected. It is concluded that contextual effects are weak, but cannot be dismissed, and a new line of theorizing is advocated.

A new criterion for rotation to an oblique simple structure is proposed. The results obtained are similar to that obtained by Cattell and Muerle's maxplane criterion. Since the proposed criterion is smooth it is possible to locate the local maxima using simple gradient techniques. The results of the application of the Functionplane criterion to three sets of data are given. In each case a better fit to the subjective solution was obtained using the functionplane criterion than was reported for by Hakstian for the oblimax, promax, maxplane, or the Harris-Kaiser methods.

This paper describes a technique for testing for the goodness-of-fit of a set of observations on a random variable to a hypothesized distribution. The cell boundaries needed for partitioning the range of the random variable are randomly chosen using the computer, thus eliminating an arbitrary choice by the analyst. The power of this test procedure is compared to the test procedure using equal expected cell frequencies

0. Summary Monotone bounds, depending on the underlying multinomial probabilities and the sample size, are given for the chi-squared approximation to the distribution of Pearson’s statistic for goodness of fit for simple hypotheses. These bounds apply to the distribution of a single statistic and to the joint distribution of two statistics associated with the margins of a two-way table, in both the central and non-central cases. Let Xff=C(O --E)”IE be Pearson’s (1900) statistic for testing goodness of fit in terms of the observed (0) and the expected (E) cell frequencies in a sample of N independent trials j let {xl, . . ., xk) be the underlying mnltinomial probabilities j and denote by FN( -) and t)”( .) the actual cumulative distribution function (cdf) of X; and its limiting chi-squared ( x2) form, respectively, where v =k -1. Esseen (1945) studied the rate of convergence of FIN( .) to (J~( -) j in the central case he derived the uniform bound

In connection with a least-squares solution for fitting one matrix,A, to another,B, under optimal choice of a rigid motion and a dilation, Schonemann and Carroll suggested two measures of fit: a raw measure,e, and a refined similarity measure,e s , which is symmetric. Both measures share the weakness of depending upon the norm of the target matrix,B,e.g.,e(A,kB) ≠e(A,B) fork ≠ 1. Therefore, both measures are useless for answering questions of the type: “DoesA fitB better thanA fitsC?”. In this note two new measures of fit are suggested which do not depend upon the norms ofA andB, which are (0, 1)-bounded, and which, therefore, provide meaningful answers for comparative analyses.

The impact of observational errors on the dendrograms produced by the complete linkage and single linkage hierarchical grouping techniques was investigated. Data corresponding to basal taxonomies were perturbed by adding error to the ranks of the inter-specimen pair proximities. The goodness of fit of the dendrograms obtained from the perturbed data and that corresponding to the basal taxonomies were measured by means of the Goodman-Kruskal gamma coefficient. Empirical sampling distributions of the gamma coefficients indicated that the single linkage grouping technique was more sensitive to the type of data errors employed than the complete linkage technique.

It is shown that the Stone-Coles method is not a content method, but rather an alternative to the ordinary distance methods. As such, it is argued, it is of limited value.

Three Gompertz-based models were fitted to Ontario mortality data pertaining to the time period 1964–1968. These are the linear Gompertz, quadratic Gompertz and Makeham models. Significant departures from each model were observed for the unrestricted age range for both sexes and for each of nine broad cause of death categories. The quadratic Gompertz and Makeham models, however, generally gave significant improvements. For the 30–70 age range, the linear Gompertz model gave an adequate fit for most causes of death data but some very marked departures occurred for data relating to neoplasms in both sexes as well as for cardiovascular and digestive system data in males. This lack of fit was effectively removed with the introduction of the quadratic term in the Gompertz hazard. The Makeham generalization of the linear Gompertz model was less successful in improving the fit.

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, W 2, V, U 2, A 2—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.

Multiple matrix sampling (MMS) procedures were utilized to determine the necessary parameters of a Pearson Type I curve. Empirical norms distributions were approximated by both the Type I model and the negative hypergeometric model. Four existing ITED norms distributions, two subtests and two grades, were approximated by the MMS procedures. Two sampling designs for each test-grade combination were studied. Comparison of approximations obtained for the Type I curve and the negative hypergeometric curve supported the use of the Type I curve for determining test score distributions of large populations. (Author) 0

This paper gives a numerical procedure for estimating the parameters of the law of comparative judgement under a variety of conditions. A broad class of goodness-of-fit functions is treated, including weighted least-squares and maximum-likelihood criteria. Any response function that has the necessary derivatives can be employed. Various kinds of constraints may be imposed on the parameters of the model. Special attention is given to linear constraints; and to equality constraints, where any of the parameters may be assumed to be known in advance and any subset of the parameters may be assumed to take on either a specified or an unspecified common value. In the maximum-likelihood case, for large samples, approximate standard errors and confidence intervals for the estimated parameters may be obtained, and goodness of fit may be tested by the likelihood-ratio technique. Numerical examples are provided.

This study proposes and illustrates a method for estimating the degree to which a factor rotation to an hypothesized target represents an improvement over rotation to a random target. The original hypothesized target matrix is transformed so that the direction of targeted salients is randomly determined with the restriction that the new target is orthogonal to the original. When applied to a factor matrix of content and response style measures, the hypothesized target yielded a decidedly better fit to the data than did the random target.

SUMMARY The purposes of this paper are to provide some additional results on the comparison of tests for normality and to indicate in the light of a somewhat startling result the need for caution in the use of these tests. The most notable efforts at comparing tests for normality are those of Shapiro, Wilk & Chen (1968), Lilliefors (1967) and of an unpublished Stanford University report of M. A. Stephens. The efforts of these authors have, however, been con- fined to the case where both It and o2 are assumed unknown. In this article, we consider two cases, the first, where both the mean, ,u, and the variance, o2, of the normal distribution are assumed unknown, and the second, where only -2 is assumed unknown. It is the comparison of the results between these two cases which leads to the cautionary note. It is hoped that our results will help serve as a guide in the use of such tests.

The test-statistic T for testing normality (Tiku [8]) and the testE statistic TE for testing exponentiality (Tiku, Rai and Mead E[17]) are generalized to test normality and exponentiality of k independent random samples. The distributions of these generalized statistics and their power properties are studied.

This paper explores the possibility of applying the mathematical techniques of the Discrete Fourier Transformation (DFT) to the estimation of the true work curve of the Uchida-Kraepelin psychodiagnostic test, which has been widely used in Japan. Section 1 reviews the mathematical model of the work curve which was proposed by Kashiwagi (1962) on the basis of the principle of mechanical or electric oscillation. Section 2 discusses from the mathematical viewpoint those aspects of DFT and of spectral density that are necessary in the present analysis. 3 numerical examples based on 220 cases obtained at Japanese National Railways are presented in Section 3, and it is shown that the DFT estimates yield perfect goodness of fit to the true values based on Kashiwagi's item-test least squares regressional estimation model. Related problems and suggested approaches are discussed in Section 4.

Statistical fit of model predictions to empirical evidence is found to be an insufficient condition for establishing the validity of a planning model where the dynamic behavior is of particular importance. The paper describes a spectral analysis statistical test that can be used to validate the structure of a planning model by comparing the time series generated by the model with the actual time series of events for the real system under study. Validation of an ambulance simulation model is reported in which the model apparently was valid based on classical goodness of fit tests of aggregate data. However, following a spectral analysis of the simulation results, an entirely new method of generating incidents was found to be necessary. The resulting model then was able to duplicate realistically the essential cyclical nature of hourly demand for emergency care observed in the real system which periodically created excessive busy and idle periods not realized in the structure of the original model.

In a paper of the same title, Ewens has considered the problem of inferring whether the genotypic frequencies observed in a small sample are consistent with a particular population model, in which all types are selectively neutral. The present paper gives the general theory of such sampling schemes, when the sampling is from either a deterministic, or a stochastically varying, population. The theory is illustrated on a population model which was introduced by Karlin and McGregor. Various results are proved for both the population, and its sample. It is shown that Ewens' sampling theory applies to this population model, and conjectures as to why different population models may have the same sampling theory are made.

General algorithms for computing the likelihood of any sequence generated by an absorbing Markov-chain are described. These algorithms enable an investigator to compute maximum likelihood estimates of parameters using unconstrained optimization techniques. The problem of parameter identifiability is reformulated into questions concerning the behavior of the likelihood function in the neighborhood of an extremum. An alternative characterization of the concept of identifiability is proposed. A procedure is developed for deciding whether or not this definition is satisfied.

A stationary cast-iron button sliding against a rotating cast-iron disk generated a time-varying friction signal. The signal was recorded using digital computer data-acquisition techniques. Sixty runs were taken, using different values for various parameters (such as load, velocity, and temperature). The data were analyzed on a digital computer by two different techniques. The first was a time-series analysis: the Fourier transform of each run was taken and the power spectral density of the run was studied. The second technique was a standard statistical analysis using the Kolmogorov-Smirnov goodness-of-fit test. From the two analyses, some interesting conclusions were made: 1 – the friction behaves like a random process, 2 – friction may be treated as a constant signal with superimposed white noise, 3 – the instantaneous coefficient of friction is normally distributed, 4 – friction is influenced by load and velocity, and 5 – the mean value and standard deviation are functionally related.

The study deals with the problem of determining true dimensionality of data-with-error scaled by Kruskal's multidimensional scaling technique. Artificial data was constructed for 6, 8, 12, 16, and 30 point configurations of 1, 2, or 3 true dimensions by adding varying amounts of error to the true distances. Results show how stress is affected by error, number of points, and number of dimensions, and indicate that stress and the “elbow” criterion are inadequate for purposes of identifying true dimensionality when there is error in the data. The Wagenaar-Padmos procedure for identifying true dimensionality and error level is discussed. A simplified technique, involving a measure calledConstraint, is suggested.

Linear explosive stochastic difference equations have been studied by White [6], Anderson [1], Rao [3] and the author [4], [5], leading to the derivation of limit distributions of certain sample functions bearing on structural estimation and goodness of fit tests. In this paper the author discusses certain results of similar interest on the structural parametric estimators of a linear explosive model generating a pair of related stochastic processes.

Let {X 2 N1, ..., X 2 Nr} be statistics of the type X 2 = Σ({O} − E)2/ E for testing mUltiple hypotheses in a categorical array of N objects. This study is concerned with the large-sample properties of the joint distribution of {X 2 N1, ..., X 2 Nr} when these cannot be assumed to be independent. The limiting form of the joint distribution is given explicitly under quite general conditions for a large class of joint distributions involving Pearson's statistics. In the case of tests for goodness of fit of simple hypotheses, the rate of convergence to this limiting form is given precisely in terms of multidimensional Esseen type bounds.

This article presents a method of obtaining an analytical expression for the transfer function of a system in the form of a ratio of two polynomials of the complex variable, 8, from the frequency response- data. The method alleviates the deficiencies of some of the other methods previously reported in the literature. First, it guarantees that the denominator polynomial, when factorized, would not have right-half plane poles and, second, there is no need to know the degrees of the numerator and denominator polynomials, a priori. By systematically examining the time response of the synthesized transfer function, say to a unit step function, the method guarantees that the form and the coefficient values of the transfer function which is finally arrived at do give the best fit to the frequency response data. The minimum mean square error criterion is used to judge the goodness of fit.

The beta distribution is the model for frequency distributions of relative humidity observations. The standard chi-square (χ2) test measures the goodness of fit of the beta distribution model to the observed data base. Ninety distributions of daily, mean 5-, 10- and 15-day, and mean monthly relative humidity data sets from 19 U. S. stations form the basis for this report. At the 0.05 level of significance the tests reject only 5% of the mean monthly and 3% of the daily and the mean 5-, 10- and 15-day distributions. The hypothesis that relative humidity follows a beta distribution is not rejected.

A probabilistic, multidimensional version of Coombs' unfolding model is obtained by assuming that the projections of each stimulus and each individual on each axis are normally distributed. Exact equations are developed for the single dimensional case and an approximate one for the multidimensional case. Both types of equations are expressed solely in terms of univariate normal distribution functions and are therefore easy to evaluate. A Monte Carlo experiment, involving 9 stimuli and 3 subjects in a 2 dimensional space, was run to determine the degree of accuracy of the multidimensional equation and the feasibility of using iterative methods to obtain maximum likelihood estimates of the stimulus and subject coordinates. The results reported here are gratifying in both respects.

Thirteen profile similarity measures were compared, using generated data. Profiles were generated from sets of three standards by adding random and normally distributed error components to the profile points of the standards. The three standards within each set were vaned systematically, altering the elevation, scatter, and shape similarities between the standards. A correct classification occurred if the generated profile was most similar to the standard from which it was generated. Significant differences were found between the proportions of correct classifications for the 13 profile similarity measures under all conditions. Osgood and Suci's D and Cattell's rB were superior to or equal to all other measures under all conditions.

A new statistic T for testing for normality is proposed. T is easy to compute and against skew distributions, and symmetric distributions having large kurtosis, is generally more powerful than Shapiro & Wilk statistic W. T is also both origin and scale invariant. Besides, T tends to normality with increasing sample size.

Box and Pierce [3] showed that the residual autocorrelations computed from an autoregressive-moving average time series model (single variate) can be successfully utilized for testing the goodness of fit. Here we extend their results to multiple autoregressive time series models. It is shown that the residual autocorrelations can be approximately represented as a singular linear transform of the corresponding white noise autocorrelations and that they possess a multivariate singular normal distribution. Finally a simple, approximate chi-square statistic is proposed to test the fit.

Because the decision net approach yields complex nets as its data points, there are no techniques for combining or analyzing this data. Several procedures for analyzing consumer decision nets are proposed. Techniques for examining purely structural properties, net efficiency, goodness of fit and grouping problems, and attribute importance are applied to decision nets.

A least-squares solution for scaling the variables of a Guttman simplex is developed. The procedure yields a ratio scale, two varieties of interval scale, and orders the variables. A measure of the goodness of fit of the scale to the data is suggested. An example of the application of the method is given. The problem of non-positive correlations is discussed.

It is shown that the goodness of fit of observed spectra of asymmetric top molecules, when using the reduced Hamiltonian of Watson, can be affected by the choice of molecular z -axis. The spectra of carbonyl fluoride and sulphur dioxide are used to illustrate this effect, and the convergence properties of the Hamiltonian are related to the order of magnitude of the co-efficient s 111 used in the contact transformation.

In this paper we compare the asymptotic efficiencies of several tests that are available for testing uniformity on the circle. Since the problem of testing goodness of fit on the circle can be reduced to testing uniformity by a simple probability transformation, these comparisons are applicable also to the goodness of fit situation. The alternatives to uniformity considered here are the familiar circular normal distributions (CND's) with density \begin{equation*}\tag{1.1}g(\alpha) = \lbrack 2\pi I_0(\kappa)\rbrack^{-1} \exp\lbrack\kappa \cos \alpha\rbrack,\quad - \pi \leqq \alpha < \pi.\end{equation*} $0 \leqq \kappa < \infty$ is a parameter of concentration, larger values of $\kappa$ corresponding to more concentration towards the mean direction zero, and $I_0(\kappa)$ is the Bessel function of purely imaginary argument. When $\kappa = 0 (1.1)$ is the uniform density, so the null hypothesis is $H_0: \kappa = 0$. The test compared here are (i) Ajne's test A (ii) Watson's test W (iii) Rayleigh's test R (iv) Ajne's test N (v) Kuiper's test V (iv) Spacings test U. In subsequent sections each of these tests is briefly described and its Bahadur efficiency [4], [5] is computed, using large deviation results. We compare the local slopes of the test statistics, i.e. the slopes in the neighborhood of the hypothesis. On the basis of these comparisons, we fine that limiting efficiencies of the first three tests viz., Ajne's test A, Watson's W and Rayleigh's test based on R, are identical, while the other tests have lower asymptotic efficiencies. Further conclusions are given in Section 7. Finally in Section 8 a simple inequality between the Ajne's N and Kuiper's V, whose asymptotic performances are identical, is noted.

This study is concerned with the extension of the Bradley-Terry model for paired comparisons to situations which allow an expression of no preference. A new model is developed and its performance compared with a model proposed by Rao and Kupper. The maximum likelihood estimates of the parameters are found using an iterative procedure which, under a weak assumption, converges monotonically to the solution of the likelihood equations. It is noted that for a balanced paired comparison experiment the ranking obtained from the maximum likelihood estimates agrees with that obtained from a scoring system which allots two points for a win, one for a tie and zero for a loss. The likelihood ratio test of the hypothesis of equal preferences is shown to have the same asymptotic efficiency as that for the Rao-Kupper model. Two examples are presented, one of which introduces a set of data for an unbalanced paired comparison experiment. Initial applications of the test of goodness of fit suggest that the propos...

[1]. In his 1937 paper in this journal J. Neyman put forward his ψ2 K test of goodness of fit. This met the long-felt need of statisticians for a smooth test of fit but possessed two technical disadvantages which seem, so far, to have prevented its use in cases where it would be most appropriate. These were that, on the one hand, the small sample distributions of the test functions were not known, and on the other hand, the labour of their computation was excessively heavy for large samples.

A multistage multiple decision procedure is developed to test if a number of sets of independent variables are binomially distributed and if so is the case, to test if they are homogeneous with respect to the binomial parameter p. The homogeneity hypothesis is decomposed in a hierarchy of homogeneity hypotheses of the nested type. The binomial distributions are either untruncated or truncated in different ways. In the multiple procedure the goodness of fit test is included as an integrated part. For those test situations which cannot be considered as two-sample tests, chi-square tests are applied throughout. Small-sample and large-sample versions of all tests are given, also of the goodness of fit tests. Special merits of the procedure are its power to exclude deviating and heterogeneous sets of variables and to separate different sets of variables where the variables within each set are homogeneous with respect to p, but the different sets have different values of p. The significance level of th...

For each month a linear regression equation has been derived for the relation between the daily sums of circumglobal and global radiation based on measurements at De Bilt during a period of two years. The goodness of fit of this relation is proved by comparison of the standard deviations of the differences of the calculated and observed daily sums and of the standard deviation of the monthly mean of the global radiation or of this mean itself. Using the period of observations from 0 to 24 hrs. one gets results which are 52 per cent less accurate than those from a period of observations from 8 to 8 hrs. If one estimates the daily sums of the global radiation by means of a linear relation to the relative duration of sunshine the standard deviation of the differences between calculated and observed daily sums is 2.5 times as large as in the first case.

The main intent of this paper is to introduce a new statistical procedure for testing a complete sample for normality. The test statistic is obtained by dividing the square of an appropriate linear combination of the sample order statistics by the usual symmetric estimate of variance. This ratio is both scale and origin invariant and hence the statistic is appropriate for a test of the composite hypothesis of normality. Testing for distributional assumptions in general and for normality in particular has been a major area of continuing statistical research-both theoretically and practically. A possible cause of such sustained interest is that many statistical procedures have been derived based on particular distributional assumptions-especially that of normality. Although in many cases the techniques are more robust than the assumptions underlying them, still a knowledge that the underlying assumption is incorrect may temper the use and application of the methods. Moreover, the study of a body of data with the stimulus of a distributional test may encourage consideration of, for example, normalizing transformations and the use of alternate methods such as distribution-free techniques, as well as detection of gross peculiarities such as outliers or errors. The test procedure developed in this paper is defined and some of its analytical properties described in ? 2. Operational information and tables useful in employing the test are detailed in ? 3 (which may be read independently of the rest of the paper). Some examples are given in ? 4. Section 5 consists of an extract from an empirical sampling study of the comparison of the effectiveness of various alternative tests. Discussion and concluding remarks are given in ?6. 2. THE W TEST FOR NORMALITY (COMPLETE SAMPLES) 2 1. Motivation and early work This study was initiated, in part, in an attempt to summarize formally certain indications of probability plots. In particular, could one condense departures from statistical linearity of probability plots into one or a few 'degrees of freedom' in the manner of the application of analysis of variance in regression analysis? In a probability plot, one can consider the regression of the ordered observations on the expected values of the order statistics from a standardized version of the hypothesized distribution-the plot tending to be linear if the hypothesis is true. Hence a possible method of testing the distributional assumptionis by means of an analysis of variance type procedure. Using generalized least squares (the ordered variates are correlated) linear and higher-order

In the factor-analytic model, let some of the factors be known (i.e., the factor loadings are given in advance; they maye.g. be obtained from some previous analyses). However, their covariance matrix may, or may not, be known. The remaining factors (if any) are assumed to be uncorrelated among themselves and to the first set. For this model, the maximum likelihood equations are obtained and an iterative method for the solution is proposed.