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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]

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]

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]

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]

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]

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]

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]

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]

Departures from statistical independence are conjoined with an assessment of predictive accuracy' in a coefficient of association for nominal-level 2 x 2 contingency tables which is both interpretable as a PRE measure and consistent with research hypotheses of manifold forms. Measurement assumptions and operating characteristics o[ the measure are delineated; definitional and computational formulate are derived from classical probability theory; comparisons with other relevant statistics are made' and The test of significance is shown to be the traditional chi-square test. [ABSTRACT FROM AUTHOR]

This article analyzes the relationship between stochastic dominance (SD) and mean-semivariance (ES) criteria. This paper contains two main results. First, there are good conceptual bases for anticipating greater consistency between ES and second SD (SSD) rules than between mean-variance (EV) and SSD rules. In particular, with one possible and relatively minor exception (identical mean and semivariance, the efficient set of portfolios derived from the ES rule with semivariance calculated around a fixed point is a subset of the SSD efficient set.

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]

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]

Suppose a population containing items classified either as defectives or as non-defectives. This article presents a new group testing method to detect all defectives in the population. The proposed method applies either when the actual number of defectives {or an upper bound) is known or when its probability distribution is known. The method is surprisingly simple yet compares favorably with other existing methods by the number of tests required to detect all defectives. [ABSTRACT FROM AUTHOR]

It is shown that a test suggested by Bechhofer, Kiefer, and Sobel [1], for selecting the better of two binomial populations, can be formulated and solved when a maximum number of tests is specified. If is assumed that the probability of correctly selecting the better population is specified to be better than P[sup *] when the difference in success probabilities exceeds a specified change[sup *]. The populations are sampled equally and it is assumed that the trial ends either when the difference in the number of successes exceeds a calculated value of s, or when N tests have been made, whichever is sooner. [ABSTRACT FROM AUTHOR]

The probabilities of the Type I errors of the Welch approximate-t test and the Aspin-Welch test for the Behrens-Fisher problem have been calculated for selected sets of degrees of freedom and nominal significance levels. The results show satisfactory agreement to the desired level. Thus, in practice, one can just use the usual t-table to carry out the Welch approximate t-test without much loss of accuracy. [ABSTRACT FROM AUTHOR]

A step-by-step method is presented herein far partitioning a certain kind of hypothesis H about the m-way contingency table into (a) a series of hypotheses about marginal tables formed from the m-way table by ignoring one or more of table's m dimensions; and (b) a hypothesis about independence, conditional independence, or conditional equiprobability in the m-way table. This step-by-step method facilitates both the testing of H and the calculation of F, the estimated expected frequencies in the m-way table under FI. The method introduced herein for calculating F is easier to apply than the usual iterative-scaling method in many cases. [ABSTRACT FROM AUTHOR]

The derivation is presented of the asymptotic distribution of the number N of observations required to terminate the classical sequential probability ratio test (SPRT) with bounds a, b (a > 0 > b) of a simple parametric hypothesis H: theta = theta[sub 1] against a simple parametric alternative K: theta = theta[sub 1] + change(change > 0) when change tends to zero. it is shown that if the common probability density function (p.d.f.) of the basic random variables belongs to the Darmois-Koopman exponential class and if the true value of theta[sub 0] of theta is different from theta[sub 1], then as change arrow right 0, N change tends in probability to a constant A > 0 which is a specified multiple of a if theta[sub 0] > theta[sub 1] and of -b if theta[sub 0] < theta[sub 1]. Also there exists B > 0 which is finite and depends on a if theta[sub 0] < theta[sub 1] and on -b if theta[sub 0] < theta[sub 1] such that (N change - A)/B square root of change tends to N(0, 1) as change arrow right 0. Finally, sigma few numerical examples are worked to illustrate the magnitudes of A and B. [ABSTRACT FROM AUTHOR]

As is true of any statistical application, testing econometric statistical hypotheses involves constructing critical regions with prescribed probabilities of Type I error and determining probabilities of Type II error under alternative hypotheses. Such constructions and determinations presuppose knowledge of exact statistical distribution functions of econometric estimators and test statistics either directly, or indirectly, as in the case where a statistical distribution function that approximates the exact distribution with known margin of error is used. If this requisite information is not available, econometric statistical inference remains guesswork. Partly because of the complicated nature of systems of simultaneous econometric structural equations and the large numbers of structural constants required to characterize distribution functions of econometric statistics, however, only a few exact marginal distributions of econometric estimators and test statistics have been extracted so far. This article is intended as a contribution to the growing literature of econometric distribution theory and to assist the efforts of econometric statisticians in providing a rational basis for econometric statistical inference. [ABSTRACT FROM AUTHOR]

Let t[sub I]= x[sub I]/s, I = 1, 2 be noncentral Student t-variates with noncentrality parameters delta[sub I], where x[sub 1], x[sub 2] and s are independent. The simultaneous distribution of t[sub 1] and t[sub 2] is studied and the power function of the studentized maximum and minimum modulus tests are tabulated for significance level 0.05, degrees of freedom 1, 2, 5, 10, 20, 50, infinity and delta[sub 1], delta[sub 2] = 0(1)5. Asymptotically exact approximations to these power functions are discussed. The marginal power of the maximum modulus test is also tabulated. [ABSTRACT FROM AUTHOR]

The lack of memory property of the exponential distribution plays an important part in the branch of applied probability. This property assumes the information regarding the probability distribution. In the present paper we give a characteristic property of the exponential distribution based on the means of the conditional distributions. Considering a random variable T with finite mean and such that P(T > O) > 0, and denoting by y a positive number such that P(T > y) > 0, we show that T has an exponential distribution if and only if the mean of the conditional distribution, given T > y, exceeds the mean of the unconditional distribution by the quantity y for all y. Also given in the paper is a similar characterization for the geometric distribution. Since the information concerning the expected values is easily accessible, we expect these properties to be useful in dealing with the practical problems. [ABSTRACT FROM AUTHOR]

For Bernoulli random variables sequential tests of the simple hypotheses p = p[SUB o] vs. p = p[sub 1] are considered. In particular a comparison is made between the performance of sequential probability ratio tests (SPRTs) and curtailed single sampling plans (CSSPs), when both tests have the same error probabilities. In [1] it is shown that the CSSP has a strong optimality property. Nevertheless, if one admits randomized SPRTs, the latter are better. Numerical examples are considered in detail for p[sub o] = 1/2 and the CSSP with n = 2. The method of finding the randomized SPRTs with the correct error probabilities is indicated. These rules have positive probability of deciding without taking any observations. [ABSTRACT FROM AUTHOR]

There is not much in the literature on tolerance limits for discrete distributions, since generally one will find it difficult to derive explicit expressions for these limits. Distribution-free tolerance limits for discrete cases were discussed. If the form of the distribution function is known, one should try to determine the formula of the tolerance limits for the particular family of distributions to which it belongs. Otherwise, an important information is not utilized. In applications for sampling inspection reliability, inventory control, the observed random variables are very often assumed to have a binomial, negative-binomial or Poisson distribution. In these cases one can obtain explicit expressions for tolerance limits, which have desirable optimum properties. In the present article we derive uniformly most accurate upper tolerance limits for families of distributions with the monotone likelihood ratio property. For these families there exist uniformly most accurate upper confidence limits. Furthermore, the distribution functions which belong to monotone likelihood ratio families are stochastically ordered. Thus, monotone relationship exists in these cases between tolerance limits and confidence limits. This relationship is utilized to derive the formulae of uniformly most accurate tolerance limits. The general theory is presented in Section 2. Section 3 is devoted to uniformly most accurate upper confidence limits.

Cochran's Q test for differences between related-sample percentages or proportions has generally been incorrectly presented in secondary sources. The most common mistake results from failure to recognize that rows containing only 1's or only O's, i.e., only successes or only failures, do not affect the value of Q. The F test, however, is affected by such rows. The probabilities from the x[sup 2] and F approximations are compared with the exact probabilities in three sets of data. A rule of thumb, based on extensive study of the distribution of Q in small samples, is given as an aid in judging when the x[sup 2] approximation is satisfactory for practical purposes. [ABSTRACT FROM AUTHOR]

The article presents information on a nomogram for the "Student"-fisher t test. A nomogram is given for estimating the probability (P) for a given value of the "Student"-Fisher t test. W.S. Gosset, an employee of the Guiness brewing company in Dublin, published papers in 1908 in which he correctly solved three problems: the probable error of a mean, the distribution of the mean divided by its estimated standard deviation and the distribution of the estimated correlation coefficient between independent variates. Later "Student" and economist R.A. Fisher calculated tables of the relevant t distribution and Fisher gives a table of t and probabilities, corresponding to various degrees of freedom. Fisher and F. Yates, scholar provide in addition a column for P. It seemed that presentation of the P, degrees of freedom, t relationship in the form of a nomogram would be advantageous. It makes possible a fairly exact estimate of probabilities less than 0.0001 and makes it possible to get an estimate of P for any value of t from 1 to 65, instead merely of selected values.

A double sample test for a hypothesis concerning the mean mu of a normal random variable with unknown variance sigma[sup 2] is presented in this note. This test is an alternative to the usual single sample t test and is made by taking samples at two stages. After the first sample has been observed the hypothesis is rejected, accepted or a second sample is taken. This double sample test is a modification of a double sample test suggested by D. Owen. However, for the test being suggested here if the second sample is taken the information from the first sample is used in making the final decision. A table is included which is a tabulation of the rejection and acceptance points for various sample sizes and alpha = .05 and alpha = .01. Additional tables are included which give some power values and expected sample sizes for the new test and Owen's test. These power values are compared with power values of various single sample t tests. Conclusions about desirable sample sizes are presented for both the new test and Owen's test. [ABSTRACT FROM AUTHOR]

The principle of minimum discrimination information estimation is employed to obtain RBAN (Regular Best Asymtotically Normal) estimates of the cell frequencies of an r x r contingency table under hypotheses of either symmetry or marginal homogeneity. For the latter, a convergent iterative procedure is given to compute the estimates. The associated minimum discrimination information statistics are distributed asymptotically as chi[sup 2] under the null hypotheses. The procedures differ from those previously presented in the literature and permit of extension to multidimensional contingency tables. An example is given. [ABSTRACT FROM AUTHOR]

The subsample values of a statistic t are the values of t for subsets of the whole sample. Subsample values may be used as indicators of variability of t. For real valued statistics t, estimating a parameter theta, the subsample values are defined to form a set of typical values if the intervals between the ordered subsample values each include theta with equal probability. The typical value property holds exactly in some cases and approximately in others. [ABSTRACT FROM AUTHOR]

Consider the problem of testing the hypothesis H[sub 0]: theta Is an element of W[sub 0] against the alternative H[sub 1]: theta Is an element of W[sub 1] when a certain subset E[sub 0] subset W[sub 1] is given. This subset E[sub 0] is such that the risk of accepting Ho will be significant economically only if theta Is an element of E[sub 0]. For this problem a test obtained by generalising the equal probability test of Arrow (1960) is suggested. Applications of the test procedure to the problems of combining several independent tests and of comparing several populations with a standard are discussed. [ABSTRACT FROM AUTHOR]

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

The article focuses on an inequality for a class of bivariate chi-square distributions. A recurrent problem in simultaneous inference is discussed in the article. While the joint distributions of test statistics may be known in form, these frequently depend on parameters, values of which are unknown. In such cases it is not possible to give simultaneous procedures, which hold exactly with specified probability; instead, it is expedient to give approximate procedures for which useful probability bounds can be established. Considerable effort has been expended in this direction, particularly in connection with simultaneous inference regarding means. The study is concerned with the construction of two-sided simultaneous confidence limits for variances in the bivariate case. In the p-variate normal case, researcher C.G. Khatri has given a lower bound to the simultaneous confidence coefficient associated with one-sided lower confidence limits and with additional assumptions regarding the covariance matrix, Khatri has similarly bounded the simultaneous confidence coefficient in the case of one-sided upper limits; however, two-sided limits was not examined.

This note gives a systematic method of computing the probability P[sub n](beta) of the event [Multiple line equation(s) cannot be represented in ASCII], where F[sub n] and F are the empirical and theoretical cumulative distribution functions and beta is a function on the interval (0, 1). The result is used to determine not only the significance level but also the exact power for an appropriate testing situation. When beta is a linear function, P[sub n](beta) may represent the probability distribution of the one-sided Kolmogorov-Smirnov statistic D[sup +, sub n]=sup[sub -Infinity < x < Infinity] [F[sub n(x) - F(x)]. Numerical examples are given in section 2, and they show that the method is extensively useful for practical applications. [ABSTRACT FROM AUTHOR]

The problem of iterated tests of hypotheses has been treated from the standpoint of multiple decision theory. Under the assumption that there is a UMP test (not necessarily similar) for each component hypothesis, a procedure has been developed which ensures an upper bound for various probabilities of misclassification and which maximizes the probabilities of correct classification in some class. [ABSTRACT FROM AUTHOR]

The article discusses a class of modified distribution free measures of goodness of fit. A random variable with the continuous probability distribution function is defined. There are various measures of the difference between a specified continuous distribution function and the sample distribution function. The discrepancies between the continuous and the empirical distribution function are jointly distribution free. They can also be expressed as any function and may be regarded as a distribution free statistic of goodness of fit. Even the sum of all deviations can be regarded as a one-sided test statistic. Short-cut methods can be defined using the statistics for one-sided tests and for two-sided tests. To get the exact null distributions, one can make use of the fact that the discrepancies are unchanged by any move of the sample point within an interval. Thus one can reduce the problems to the classical urn model. By means of a computer the exact null distribution and critical values for small number is calculated.

An investigation of stratification is reported for a universe with strong positive skewness. The choice of stratification variable, the method of constructing strata, the sample allocation, and the number of strata are considered. Comparisons are made among four types of allocation in combination with the corresponding optimum stratifications. For four simpler methods of stratification, equal allocation and allocation proportional to stratum aggregates are contrasted. Gain from stratification are examined for several estimation variables. Illustrations utilize data for the 1961 universe of United States nonfederal general medical hospitals for short-term care. However, the finding apply to the stratification of other populations having similar distribution functions. [ABSTRACT FROM AUTHOR]

A number of examples which involve restricted alternatives are given; each of these, except one, deals with normal distributions. The examples demonstrate the following: (a) one way to construct a simple test when the alternative hypothesis is restricted, (b) a method by which the additional restrictions can be justified, and (c) a procedure that can be used to defend the original model. An interesting relationship between Bartholomew's chi[sup 2] and a test statistic, which is used in this paper, is pointed out. Finally, a certain distribution-free problem and a solution of it are proposed. [ABSTRACT FROM AUTHOR]

After a brief motivating discussion of approaches to significance tests in Section 1, the concept of expected significance level (ESL) is defined and its basic properties are presented in Section 2. Section 3 discusses the relationship between ESL and the familiar criterion of power. While a complete description of power for each size alpha is more informative than ESL, it is argued that ESL is often a reasonable compromise between high ideals and practicability. Section 4 gives two estimation methods for Monte Carlo simulation of ESL and shows how to estimate variances and covariances for the second of these methods. The first method leads to the familiar Wilcoxon two-sample statistic which is clearly a sample analogue of ESL. The second and more apposite method leads to a modification of the Wilcoxon statistic. [ABSTRACT FROM AUTHOR]

Two exact tests are presented for testing the hypothesis that the residuals from a least squares regression are homoscedastic. The results can be used to test the hypothesis that a linear [ratio] model explains the relationship between variables as opposed to the alternative that the ratio [linear ] specification is correct. The first test is parametric and uses the F-statistic. The second test is nonparametric and uses the number of peaks in the ordered sequence of unsigned residuals. In conclusion, the results of some experimental calculations of the powers of the tests are discussed. [ABSTRACT FROM AUTHOR]

This paper deals with the asymptotic relative efficiency (A.R.E.) and the asymptotic power of certain tests. These tests are distribution-free goodness-of-fit tests based on the empirical distribution function. They are the one and two sided Kolmogorov-Smirnov tests and a new one based on the absolute values of the order statistics. The method of obtaining the A.R.E. is due to Quade [5] and is outlined in Section 1. In this paper we are concerned with the case of scalar alternatives. The asymptotic power for the Kolmogorov-Smirnov tests is obtained in Section 2. Section 3 deals with a new test which allows a more facile use of a one-sided statistic while utilizing the same information normally included in a two-sided statistic. In Section 4, three examples are considered to illustrate the superiority of this new test. [ABSTRACT FROM AUTHOR]

In the case of moving average schemes or what are termed the m-dependent stochastic processes, the successive observations in the series are not stochastically independent. As a result, the usual sign test, the test based on the rank correlation tau (for randomness against trend alternatives), and the test proposed by Moore and Wallis [9] and further developed by Goodman and Grunfeld [2] (for the comovement between two time series) are not valid and require some modifications. Accordingly, the modified forms of these non-parametric tests are studied here in detail, and their applications considered. [ABSTRACT FROM AUTHOR]

The need for data compression, a consequence of the demands made on the telemetry system of a space vehicle, prompts consideration of the use of sample quantiles in estimating population parameters and obtaining tests of goodness of fit for large samples. In this paper optimal unbiased estimators of the mean and standard deviation are given using up to twenty quantiles when the parent population is normal. Moreover, the estimators are relatively insensitive to deviations from normality. A distribution-free goodness-of-fit test is presented based on the sum of the squares of four quantiles after an orthogonal transformation to independent normal deviates. If a frequency function is of the form f(x; p) = pf[sub 1](x) + (1 - p) f[sub 2](x), 0 < p < 1, where f, and f[sub 2] are normal frequency functions, the distribution is likely to be bimodal. Another goodness-of-fit test is obtained using four quantiles, which is likely to have considerable power with a null hypothesis of normality and the alternative hypothesis of bimodality. The "data compression ratios" obtained with the use of a quantile system can be on the order of 100 to 1. [ABSTRACT FROM AUTHOR]

This article presents information regarding parole prediction variables and the time factor in violations by burglars. Parole prediction studies have shown that it is possible to determine, before an inmate is released from prison under parole supervision, what are his relative chances for making a successful extramural adjustment. The original data were set up in 23 contingency tables with the number of months before violation as the independent variable in each case and one of the 23 parole prediction factors as the dependent variable. The Chi-square test was undertaken to test the probability that differences between the actual and the theoretical frequencies would occur by chance. In these instances where the probability that the differences were due to chance was very small the coefficient of mean square contingency was determined in order to measure the amount of the relationship that exists between the variables. Whether the relationship is a positive or a negative, one was determined by inspection of the data and the signs of the differences.

In hypothesis testing, the level of significance, α, often performs a screening as well as an evaluating function. Results which are not "significant" at some preset value of α are simply not reported. Unfortunately, this value of α is usually arbitrarily set at .05 or .01 and bears no relation to the needs of potential users of the study. This paper suggests an approach to making the selection of α more responsive to these potential applications. [ABSTRACT FROM AUTHOR]

The article discusses the problem of estimating an optimal interval containing the location of the maximum of a unimodal function by simultaneous experimentation for the case when an arbitrary probability density function for the distribution of the maximum is given. A particular type of problem is examined. Estimate the location x* of the maximum of an unknown unimodal function z(x). The estimate is taken as an interval D in which the maximum is located. Simultaneously observations of the function values at N points are performed before making the estimate. The objective is to distribute the observations in an optimal way. The resultant optimal strategy has the following properties: When the number of experiments is even, the optimal strategy selects an interval D, containing the maximum x*, for which the maximal a priori probability of containing x* is minimized. When the probability density function for the distribution of x* is uniform, the optimal strategy coincides with the mini-max search. When the number of experiments is odd, a search pattern results that always yields more information than an even search with one less experiment.

This paper presents derivations of expressions for the probability of killing a randomly located point target and for the expected coverage of an area target, of variable density. Particular emphasis is placed upon a general situation in which the phase distribution is not necessarily uniform. General distributions are employed to describe the phase, radial, and component behavior of the target and from these functions the distribution of a quadratic form is derived that provides the expected coverage or kill probability. The targeting conditions include a fixed lethal radius (kill-no-kill) when the center of gravity of the weapon impact points coincides with the center of the target density and a variable lethal radius when a weapon bias is present. [ABSTRACT FROM AUTHOR]

Research can be defined as a process of translation, or reproduction in which a communicable model or representation of the subject of research is the goal. In applied research, the goal is to formulate a statement which represents the operations necessary to bring about some desired result. What makes such research different from engineering design is the degree of certainty and completeness with which the model can be formulated at the beginning of the project. The means of achieving the desired result are represented in a network of hypotheses of greater or less probability. Some steps can be specified with near certainty, but others remain vague and uncertain until more information about the physical processes involved can be acquired. When the model attains certainty, and all operations can be specified, the project is completed. The uncertainties are reduced by formulating and testing hypotheses. This process can be described as a cycle of four phases: predicting in the model that a certain operation will produce a desired result; performing the operation; comparing the observed result with the prediction; working out the implications of the finding for the model, and hence for the next step in research. This cycle makes research a self-corrective activity. Despite its tentative nature, the model provides the research worker with criteria by which he can evaluate his progress at every step. Incorrect predictions are revealed, and dictate changes in the model.

The implications of replacing the assumption of normal distribution of the parameters (A, b, c) by chi-square and other related nonnegative distributions are discussed here in the framework of chance-constrained linear programming. [ABSTRACT FROM AUTHOR]

A "blowup factor" is defined for the measurement of the effect on forecast error variance of two types of misspecification which may be implicit in the choice of a particular adaptive forecasting scheme: (1) misspecification of the number of non-zero parameters of the stationary linear stochastic process generating the observed time series, and (2) misspecification arising from postulating stationarity when in fact the generating process is non-stationary in mean. [ABSTRACT FROM AUTHOR]