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Learning the latent network structure from large scale multivariate point process data is an important task in a wide range of scientific and business applications. For instance, we might wish to estimate the neuronal functional connectivity network based on spiking times recorded from a collection of neurons. To characterize the complex processes underlying the observed data, we propose a new and flexible class of nonstationary Hawkes processes that allow both excitatory and inhibitory effects. We estimate the latent network structure using an efficient sparse least squares estimation approach. Using a thinning representation, we establish concentration inequalities for the first and second order statistics of the proposed Hawkes process. Such theoretical results enable us to establish the non-asymptotic error bound and the selection consistency of the estimated parameters. Furthermore, we describe a least squares loss based statistic for testing if the background intensity is constant in time. We demonstrate the efficacy of our proposed method through simulation studies and an application to a neuron spike train dataset. for this article are available online. [ABSTRACT FROM AUTHOR]

This article examines the use of ranked set sampling (RSS) with cluster randomized designs (CRDs), for potential improvement in estimation and detection of treatment or intervention effects. Outcome data in cluster randomized studies typically have nested structures, where hierarchical linear models (HLMs) become a natural choice for data analysis. However, nearly all theoretical developments in RSS to date are within the structure of one-level models. Thus, implementation of RSS at one or more levels of an HLM will require development of new theory and methods. Under RSS-structured CRDs developed to incorporate RSS at different levels, a nonparametric estimator of the treatment effect is proposed; and its theoretical properties are studied under a general HLM that has almost no distributional assumptions. We formally quantify the magnitude of the improvement from using RSS over SRS (simple random sampling), investigate the relationship between design parameters and relative efficiency, and establish connections with one-level RSS under completely balanced CRDs, as well as studying the impact of clustering and imperfect ranking. Further, based on the proposed RSS estimator, a new test is constructed to detect treatment effects, which is distribution-free and easy to use. Simulation studies confirm that in general, the proposed test is more powerful than the conventionalF-test for the original CRDs, especially for small or medium effect sizes. Two empirical studies, one using data from educational research (i.e., the motivating application) and the other using human dental data, show that our methods work well in real-world settings and our theory provides useful predictions at the stage of experimental design, and that substantial gains may be obtained from using RSS at either level. Supplementary materials for this article are available online. [ABSTRACT FROM AUTHOR]

The ranked-set sampling literature includes both inference procedures that rely on the assumption of perfect rankings and inference procedures that are robust to violations of this assumption. Procedures that assume perfect rankings tend to be more efficient when rankings are in fact perfect, but they may be invalid when perfect rankings fail. As a result, users of ranked-set sampling must decide between efficiency and robustness, and there is at present little to guide their decision. In this article we introduce three rank-based goodness-of-fit tests that may be consulted in making these decisions. Our strategy in producing these tests is to think of the judgment order statistic classes as separate samples, compute the ranks of the units from each sample within the combined sample, and use these ranks to test whether the judgment rankings are perfect. Consideration of both power and ease of use leads us to recommend use of a test that rejects when the concordance between the vector of mean ranks and its null expectation is small. Tables of critical values and appropriate asymptotic theory for applying this test are provided, and we illustrate the use of the tests by applying them to a biological dataset. [ABSTRACT FROM AUTHOR]

The main objective of statistics of extremes lies in the estimation of quantities related to extreme events. In many areas of application, such as statistical quality control, insurance, and finance, a typical requirement is to estimate a high quantile, that is, the value at risk at a level p (VaRp), high enough so that the chance of an exceedance of that value is equal to p, small. In this article we deal with the semiparametric estimation of VuRp for heavy tails. The classical semiparametric estimators of parameters characterizing the tail behavior of the underlying model F usually exhibit a high bias for low thresholds, that is, for large values of k, the number of top order statistics used for the estimation. We shall here deal with bias reduction techniques for heavy tails, trying to improve the performance of the classical high quantile estimators through the use of an adequate bias-corrected tail index estimator. The new high quantile estimators have a mean squared error smaller than that of the classical estimators, even for small values of k. They are, thus, alternatives to the classical estimators not only around optimal levels but also for other levels. The asymptotic distributional properties of the proposed classes of estimators are derived. The estimators are compared with alternative ones, not only asymptotically hut also for finite samples, through Monte Carlo techniques. An application to the analysis of different datasets in the field of finance is also provided. [ABSTRACT FROM AUTHOR]

The Pareto distribution is a simple model for nonnegative data with a power law probability tail. In many practical applications, there is a natural upper bound that truncates the probability tail. This article derives estimators for the truncated Pareto distribution, investigates their properties, and illustrates a way to check for fit. These methods are illustrated with applications from finance, hydrology, and atmospheric science. [ABSTRACT FROM AUTHOR]

In this article we focus on the detection of possible outliers based on the widely used boxplot procedures. The outliers in a set of data are defined to be a subset of observations that appear to be inconsistent with the remaining observations. We identify the outliers by constructing a boxplot with its lower fence (LF) and upper fence (UF) either (a) satisfying the requirement that if the given sample is outlier-free, then the probability that one or more of the sample data would fall outside the region (LF, UF) is equal to a prescribed small value a, or (b) taken to be the tolerance limits, derived from an outlier-free random sample, within which a specified large proportion fi of the sampled population would be asserted to fall with a given large probability y. Exact expressions that can be routinely used to evaluate the constants needed in the construction of the boxplot's outlier region for samples taken from the family of location-scale distributions are obtained for both procedures. This article shows that the commonly constructed boxplot is in general inappropriate for detecting outliers in the normal and especially the exponential samples. We recommend that the graphical boxplot be constructed based on the knowledge of the underlying distribution of the dataset and by controlling the risk of labeling regular observations as outliers. [ABSTRACT FROM AUTHOR]

For any two random variables X and Y with distributions F and G defined on (0, ∞), X is said to stochastically precede Y if P(X ≤ Y) ≥ 1/2. For independent X and Y, stochastic precedence (denoted by X ≤1P Y) is equivalent to E[G( X- )] < 1/2. The applicability of stochastic precedence in various statistical contexts. including reliability modeling, tests for distributional equality versus various alternatives, and the relative performance of comparable tolerance hounds, is discussed. The problem of estimating the underlying distribution(s) of experimental data under the assumption that they obey a stochastic precedence (sp) constraint is treated in detail. Two estimation, approaches, one based on data shrinkage and the other involving data translation, are used to construct estimators that conform to the sp constraint, and each is shown to lead to a root n-consistent estimator of the underlying distribution. The asymptotic behavior of each of the estimators is fully characterized. Conditions are given under which each estimator is asymptotically equivalent to the corresponding empirical distribution function or, in the case of right censoring, the Kuplan-Meier estimator. In the complementary cases, evidence is presented. both analytically and via simulation, demonstrating that the new estimators tend to outperform the empirical distribution function when sample sites are sufficiently large. [ABSTRACT FROM AUTHOR]

Do a random censorship and/or order restrictions (e.g., nonnegativity, monotonicity, convexity) affect estimation of a smooth density under mean integrated squared error (MISE)? Under mild assumptions, the known asymptotic results, which are concerned only with rates, answer "no." This answer, especially for censored data, contradicts practical experience and statistical intuition. So what can be said about constants of MISE convergence? It is shown that asymptotically (a) censorship does affect the constant, and this allows one to find a relationship between sample sizes of directly observed and censored datasets that implies the same precision of estimation, and (b) an order restriction does not affect the constant, and thus no isotonic estimation is needed. Intensive Monte Carlo simulations show that the lessons of the sharp asymptotics are valuable for small sample sizes. Also, the estimator developed is illustrated both onsimulated data and a dataset of lifetimes of conveyer blades used at wastewater treatment plants. [ABSTRACT FROM AUTHOR]

Discusses rank-based autoregressive order identification. Traditional and rank-based order-identification methods; Alternative strategy of order identification; Efficiency of rank-based methods with various score functions; Study of real-life series.

The generalized Pareto distribution (GPD) was introduced by Pickands to model exceedances over a threshold. It has since been used by many authors to model data in several fields. The GPD has a scale parameter (σ > 0) and a shape parameter (-∞ < k < ∞). The estimation of these parameters is not generally an easy problem. When k > 1, the maximum likelihood estimates do not exist, and when k is between ½ and 1, they may have problems. Furthermore, for k ≤-½, second and higher moments do not exist, and hence both the method-of-moments (MOM) and the probability-weighted moments (PWM) estimates do not exist. Another and perhaps more serious problem with the MOM and PWM methods is that they can produce nonsensical estimates (i.e., estimates inconsistent with the observed data). In this article we propose a method for estimating the parameters and quantiles of the GPD. The estimators are well defined for all parameter values. They are also easy to compute. Some asymptotic results are provided. A simulation study is carried out to evaluate the performance of the proposed methods and to compare them with other methods suggested in the literature. The simulation results indicate that although no method is uniformly best for all the parameter values, the proposed method performs well compared to existing methods. The methods are applied to real-life data. Specific recommendations are also given. [ABSTRACT FROM AUTHOR]

The Simes method for testing intersection of more than two hypotheses is known to control the probability of type I error only when the underlying test statistics are independent. Although this method is more powerful than the classical Bonferroni method, it is not known whether it is conservative when the test statistics are dependent. This article proves that for multivariate distributions exhibiting a type of positive dependence that arise in many multiple-hypothesis testing situations, the Simes method indeed controls the probability of type I error. This extends some results established very recently in the special case of two hypotheses. [ABSTRACT FROM AUTHOR]

When we have n independently and identically distributed observations, it is an interesting question how the Fisher information is distributed among order statistics. The recipe for the Fisher information in order statistics is easy, but the detailed calculation has been known to be complicated. An indirect approach, using a decomposition of the Fisher information in order statistics, simplifies the calculation. Some recurrence relations for the Fisher information in order statistics are derived that facilitate the calculation. The Fisher information in the first r order statistics is an r multiple integral, but it can be simplified to just a double integral by using the decomposition. A recurrence relation further simplifies the double integral to a sum of single integrals. An "information plot" is suggested, from which we can read at once the Fisher information in any set of consecutive order statistics for a parametric distribution. [ABSTRACT FROM AUTHOR]

One commonly observed response pattern in a one-factor design with ordered treatment levels is the umbrella or unimodal ordering, in which the response variable increases with an increase in the treatment level up to a point, then decreases with further increase in the treatment level. The turning point is called the peak of an umbrella ordering. In many practical settings, there are two or more different populations with umbrella orderings, and it is of interest to compare these populations. In this article, likelihood ratio tests are developed for testing whether two groups with umbrella orderings have the same peaks. Three cases are considered: variances known, variances unknown and different for each group, and variances unknown and the same for both groups. Extensions to randomized block designs and to a more general class of testing problems are discussed. [ABSTRACT FROM AUTHOR]

In a nonrandom sample or a nonrandomized comparison of treated and control groups, methods are developed for displaying the sensitivity of confidence intervals for population quantiles to the magnitude of the departure from random selection. Quantiles of a single distribution and for a shift in two distributions are both discussed. These confidence intervals are based on a new inequality for the distribution of the number of successes in a nonrandom sample or nonrandomized comparison. The inequality has additional applications. In particular, it greatly accelerates the computations in a sensitivity analysis for the Mantel-Haenszel statistic in a 2 × 2 × S contingency table. Examples include a nonrandom collection of blood samples from a town with comparatively high levels of radon gas, an observational study of chromosome damage from mercury, and a study of the drug allopurinol as a cause of skin rash. The proof of the inequality makes use of Savage's lattice for ranks and Holley's inequality, which gives sufficient conditions for stochastic ordering in a lattice. [ABSTRACT FROM AUTHOR]

In measuring visual acuity, the extremes of a set of normally distributed measures are of concern, together with one or more covariates. This leads to a model in which (X, Y1, Y2) are jointly normally distributed with Y, Y2 exchangeable and (X, Y,) having a common correlation. Inferential procedures are developed for correlations and linear regressions among X and the ordered Y values. This requires determination of the covariance matrix of X, Y(1) = min{Y1 , V2 } and Y(2) = max{Y1 , Y2 }. The inadequacy of certain estimates that ignore the nonnormality of {X, Y(1) , Y(2) } is also discussed. Although the bivariate case is emphasized because of the context of the visual acuity model, many' results are given for the more general multivariate case. [ABSTRACT FROM AUTHOR]

Statistical modeling of the largest k observations or the exceedances over a high threshold based on extreme value theory has produced powerful methods for drawing inferences about the tail behavior of a distribution; however, a method for selecting k has not been established. We suggest using a forward selection procedure based on a suitable goodness-of-fit statistic to search for the optimal k when the sample is of finite size and from a distribution in the domain of attraction of the Gumbel distribution. Two criteria are examined: generalized least squares and the Shapiro--Wilk goodness of fit. Simulation studies indicate that the latter is preferable in terms of variation. Extension of the selection procedure to distributions in the domains of attraction of the Fréchet and Weibull distributions is also discussed. The use of the proposed method is illustrated through two examples based on real data. [ABSTRACT FROM AUTHOR]

A new model for the analysis of lifetime data in the presence of a covariate is derived based on physical and statistical considerations. The model depends on five parameters that have dear physical interpretations. Two methods for the estimation of the parameters and of the quantiles are presented: one based on the order statistics and the other a regression estimator. The quantile estimators as well as the estimators of four of the five parameters are given in closed form. The fifth parameter can be estimated independently of the other parameters using either a closed form or a numerically simple algorithm. A simulation study shows that the regression estimators are better for estimating the parameters but the order statistics estimators perform better for the estimation of low quantiles. The methodology is also illustrated by an example of a real life data. [ABSTRACT FROM AUTHOR]

We apply general covariance identities and inequalities to some functions of multivariate normal variables. We recover, in particular, a recent covariance identity due to Siegel and provide simple estimates on the variance of order statistics. We also present some computations. [ABSTRACT FROM AUTHOR]

Model diagnostic procedures in failure time models using hazard-based residuals rely on the assumption that the residual vector closely resembles a random sample from a unit exponential distribution when the model holds. This article formally investigates the validity of this critical assumption by deriving and examining the properties of parametrically, semiparametrically, and nonparametrically estimated residuals for complete and right-censored data. The joint distribution of the residual vector is characterized, and the behavior of some tests for exponentiality when applied to the residuals is examined analytically and through Monte Carlo methods. Findings reveal that the critical assumption of approximate unit exponentiality of the residual vector may not be viable and, consequently, the model diagnostic procedures considered, which revolve on checking the approximate unit exponentiality of the residual vector (specifically, hazard plotting and the use of spacings and total-time-on-test statistics on the residual vector) may have serious defects. This is especially evident in situations where the failure time distribution is not exponential or when the residuals are obtained nonparametrically in the no-covariate model or semiparametrically in the Cox proportional hazards model. [ABSTRACT FROM AUTHOR]

An extension of the bivariate model suggested by Dale is proposed for the analysis of dependent ordinal categorical data The so-called multivariate Dale model is constructed by first generalizing the bivariate Plackett distribution to any dimensions Because the approach is likelihood based, it satisfies properties that are not fulfilled by other popular methods, such as the generalized estimating equations approach The proposed method models both the marginal and the association structure in a flexible way The attractiveness of the multivariate Dale model is illustrated in three key examples, covering areas such as crossover trials, longitudinal studies with patients dropping out from the study, and discriminant analysis applications The differences and similarities with the generalized estimating approach are highlighted [ABSTRACT FROM AUTHOR]

A study is made of some unusual location estimates first proposed in 1977 by J S Maritz, M Wu, and R G Staudte, Jr., who established some strong robustness properties of these estimates, such as redescending influence functions and, in some cases, full efficiency at the Cauchy model It is further shown here that the estimates, called M-W-S estimates, are derivable from a rank-based scheme founded on convex functions and hence are regular, are unique, and have associated tests and confidence intervals The tests belong to a family of signed-rank-type tests, and their distributions have nice combinatorial properties It is not the case, therefore, that a redescending influence function need imply all the computational irregularities possessed by redescending M estimates. In addition, M-W-S estimates are shown to have breakdown point of 5, a very strong robustness property. [ABSTRACT FROM AUTHOR]

A ranked set sample consists entirely of independently distributed order statistics and can occur naturally in many experimental settings, including problems in reliability When each ranked set from which an order statistic is drawn is of the same size, and when the statistic of each fixed order is sampled the same number of times, the ranked set sample is said to be balanced Stokes and Sager have shown that the edf Fn of a balanced ranked set sample from the cdf F is an unbiased estimator of F and is more precise than the edf of a simple random sample of the same size The nonparametric maximum likelihood estimator (MLE) F* of F is studied in this article Its existence and uniqueness is demonstrated, and a general numerical procedure is presented and is shown to converge to F*. If the ranked set sample is balanced, it is shown that the EM algorithm, with Fn as a seed, converges to the unique solution (F*) of the problem's self-consistency equations, the consistency of every iterate of the EM algorithm is also demonstrated. The modifications needed to obtain similar results in unbalanced cases are also discussed. Finally, the results of a simulation study are reported, which support the claim that the nonparametric maximum likelihood estimator, as approximated by an appropriate iterate of the EM algorithm, performs well in the unbalanced case where Fn is inapplicable and performs better than Fn in balanced cases where both estimators exist and can be compared [ABSTRACT FROM AUTHOR]

A formula is provided for the covariance of X1, with the minimum of the multivariate normal vector (Xl, X2, . . . . ,Xn). The resulting expression has an intuitive interpretation as the weighted average of the n covariances of X1, with X1, X2, . . . . ,Xn, where the ith weight equals the probability that Xi, is the minimum. The formula is surprising for several reasons. First, results involving extrema of order statistics in the presence of correlation are usually much more complex. Second, an attempt at a direct proof runs into difficulties involving the nontrivial distinction between a conditional expectation and an ordinary covariance. Finally, although these difficulties are genuine on a term-by-term basis, they cancel out when weighted and combined. The geometric interpretation in n-dimensional space is that although the vector of covariances of X1 with (X1, X2, . . . . Xn) is in general different from the vector of appropriate conditional expectations, these vectors always have the same projection onto the vector of probabilities that Xi, is the smallest, i = 1 , 2,. . . . ,n. The formula arises in analysis of commodity and financial futures contracts. [ABSTRACT FROM AUTHOR]

Ranked-set samples have been shown to lead to improved methods of estimation in parametric settings under specific distributional forms when actual measurement of the sample observations is difficult but ranking them is relatively easy. The earliest work with ranked-set data concentrated on estimating a population mean or variance. More recently, a ranked-set sample estimator of a cumulative distribution function was developed and used to obtain a simultaneous confidence interval for the function. In this article, we take the next logical step and use this ranked-set empirical distribution function to construct distribution-free competitors to the standard Mann-Whitney-Wilcoxon estimation and testing procedures. The appropriate null distribution tables for the associated test are presented for the case of perfect ranking. Asymptotic relative efficiency comparisons between the simple random sample Mann-Whitney-Wilcoxon procedures and their ranked-set analogues are discussed, and the results of a small-sample Monte Carlo simulation study of the same arc presented. [ABSTRACT FROM AUTHOR]

A quantile function based analog to Neyman's smooth tests for goodness-of-fit hypotheses is developed. These procedures possess asymptotic and small-sample power properties similar to those for smooth tests. They are, however, computationally simple, applicable to both complete and Type H censored data, and easily focused in the direction of any specific set of alternatives. These results are used to develop a new test for exponentiality. Only location/scale null hypotheses will be considered. The ideas are easily adapted to other types of nulls, however. Neyman first proposed smooth tests for the goodness-of-fit hypothesis. As stated by Kopecky and Pierce and by Rayner and Best, the basic idea behind these tests is to embed the null density in a larger exponential family. The procedure is then based upon the likelihood-ratio test statistic for testing that the additional parameters equal zero. Smooth type tests may be viewed as a compromise between omnibus test procedures, with generally low power in all directions, and procedures whose power is focused in the direction of a specific alternative. It is easily shown that any of the smooth type tests are inconsistent for some alternatives. Further, as stated by Koziol, one of the main criticisms of these tests has been that the larger exponential families employed seem to be selected more for mathematical convenience than as statistically reasonable alternative distributions. Results of Monte Carlo studies by Miller and Quesenberry and others, how- ever, indicate that smooth tests perform well for a wide range of alternatives. The proposed tests are developed by embedding the quantile function associated with the null hypothesis into a larger linear model and then using typical regression techniques to test that the coefficients of the added terms in this model vanish. These statistics are interpreted and asymptotically have many of the properties of the corresponding regression tests. Further, as with Neyman's smooth tests, if the alternative model is properly selected, the resulting procedure provides a good omnibus test. [ABSTRACT FROM AUTHOR]

For an estimator of quantiles, the efficiency of the sample quantile can be improved by considering linear combinations of order statistics, that is, L estimators. A variety of such methods have appeared in the literature, an important aspect of this article is that asymptotically several of these are shown to be kernel estimators with a Guassian kernel, and the bandwidths are identified. It is seen that some implicit choices of the smoothing parameter are asymptotically suboptimal. In addition, the theory of this article suggests a method for choosing the smoothing parameter. How much reliance should be placed on the theoretical results is investigated through a simulation study. Over a variety of distributions little consistent difference is found between various estimators. An important conclusion, made during the theoretical analysis, is that all of these estimators usually provide only modest improvement over the sample quantile. The results indicate that even if one knew the best estimator for each situation, one can expect an average improvement in efficiency of only 15%. Given the well-known distribution-free reference procedures (e. g., easily constructed confidence intervals) associated with the sample quantile, as well as the ease with which it can be calculated, it will often be a reasonable choice as a quantile estimator. [ABSTRACT FROM AUTHOR]

Plotting order statistics versus some variant of the normal scores is a standard graphical technique for assessing the assumption of normality. To obtain an objective evaluation of the normal assumption, it is customary to calculate the correlation coefficient associated with this plot. The Shapiro-Francia statistic is the square of the correlation between the observed order statistics and the expected values of standard normal order statistics, whereas the Shapiro-Wilk statistic also involves the covariances of the standard normal order statistics. In a wide variety of applications, an investigation of the plausibility of the normal (or lognormal) model is needed when the observations on strength or life length are right-censored. The plotting procedure still applies if the observations are censored at a fixed order statistic or a fixed time. Here, the corresponding distribution theory for some modified versions of the Shapiro-Wilk correlation statistic is investigated. Because the asymptotic theory used in this article shows a surprisingly slow rate of convergence even for complete samples, a table of critical values based on a Monte Carlo study is provided. Results from an empirical power study are also presented. Finally, large-sample critical values are obtained and compared with the Monte Carlo values. [ABSTRACT FROM AUTHOR]

Suppose that the first n order statistics from a random sample of N positive random variables are observed, where N is unknown. This, the general order statistic model, has been applied to the study of market penetration, capture-recapture, burn-in in repairable systems, software reliability growth, the estimation of the number of individuals exposed to radiation, and the estimation of the number of unseen species. Inference is to be made about the unknown parameters, especially N, and future observations are to be predicted. A Bayes empirical Bayes approach to inference is presented. This permits the comparison of competing, perhaps nonnested, models for the distribution of the random variables in a natural way. It also provides easily implemented inference and prediction procedures that avoid the difficulties of non-Bayesian methods. One such difficulty is that the maximum likelihood estimator of N may be infinite. Results are given for the case in which vague prior information about the model parameters is approximated by limiting, improper, prior forms. Applications to three software reliability data sets indicate that the much-used exponential order statistic may give rather optimistic estimates of system reliability and that the not previously considered Weibull order statistic model seems promising for such applications. [ABSTRACT FROM AUTHOR]

Linear combinations of order statistics, or L-estimators, have played an extremely important role in the development of robust methods for the one-sample problem. We suggest analogs of L-estimators for the parameters of the linear model based on the p-dimensional "regression quantiles" proposed by Koenker and Bassett (1978). A uniform, Bahadur-type asymptotic representation of regression quantiles is established, and this yields a general asymptotic theory of L-estimators for the linear model. A leading example of the proposed estimators is an analog of the trimmed mean (TRQ), which is asymptotically equivalent to the trimmed least squares estimator studied by Ruppert and Carroll (1980), but appears to be somewhat less sensitive to influential design observations. This estimator is also asymptotically equivalent to the well-known Huber M-estimator, but offers the significant advantage that it is scale invariant. We illustrate the methods by reconsidering a mid-18th century linear model analyzed by Boscovich. It is apparent that the proposed methods yield estimators that are weighted averages of coefficient vectors determined by certain p-element subsets of the n sample observations. The subset of p-element subsets that generate solutions to the regression quantile optimization problem play the role of order statistics for the linear model and may be useful in other applications. We also investigate two proposals for estimating the covariance matrix for the trimmed regression quantile estimator. One approach employs residuals from the TRQ fit to estimate a winsorized variance, the other employs the empirical quantile function suggested in Bassett and Koenker (1982). Using the Monte Carlo methods of Gross (1977) we find that either approach yields test statistics with critical values close to those of the conventional t test and good expected... [ABSTRACT FROM AUTHOR]

This article suggests nonparametric methods of testing for a patterned alternative. As an example, suppose that we suspect that with increasing age, one's ability to perform a certain job tends to improve, but that after some point advancing age tends to mean diminishing performance. Suppose that N employees are grouped by age into k groups and employees' job performances are ranked from 1 to N. Viewing performance as the dependent variable, the particular pattern of increasing group locations followed by decreasing group locations is called an umbrella (pattern). The tests proposed herein allow the experimenter to test for this or any pattern of his choice. Given k groups, let R[sub j] denote the mean rank of the data in group j. Considering only statistics of the form SIGMA a[sub j]R[sub j], where SIGMA a[sub j] = 0, we find a statistic having maximum asymptotic local power and greatest efficiency when the alternative hypothesis consists of a specific pattern. The case in which the researcher specifies a vague pattern is handled by letting the alternative consist of a collection of appropriate specific patterns. As a special case, the alternative "not all medians are equal" produces a test equivalent to the Kruskal-Wallis rank test. The approach extends to the additive two-way layout with b blocks and k treatments. In the "not all medians are equal" case the standardized test statistic equals the square root of the extended Friedman test statistic. Other special cases also lead to tests equivalent to known tests. The tests are reformulated using ranks within pairs of samples. How to adjust when covariates are present is discussed. Also discussed is the particular case of testing for umbrella alternatives, as are inherent differences with a competing procedure of Mack and Wolfe (1981). [ABSTRACT FROM AUTHOR]

Let pi[sub N] be a finite population of size N whose elements have distinct X values X[sub (1)] < ... < X[sub (N)]. Let chi ([sub 1]) < ... < x(sub (n)] denote the order statistics of a simple random sample of size n taken from pi [sub N] without replacement. An outer confidence interval is formed for the quantile interval [X ([sub t]), X [sub (u)] (1

A method is proposed for testing normality, in the case of general Type II censored data--that is, data for which only a subset of the order statistics are available. Three test statistics are proposed, which are generalizations of the statistics proposed in LaRiccia (1986), and have many of the same properties. Specifically, they are designed to be asymptotically optimal with respect to specific alternatives and are easily adjusted to be asymptotically optimal with respect to many other types of alternatives. Under the null hypothesis, irrespective of the type or amount of censoring, the proposed test statistics are asymptotically distributed as chi-squared random variables. Further, results of a simulation study are presented, indicating that these statistics converge quite rapidly in distribution to the appropriate chi-squared random variables and that the asymptotic critical values provide a useful approximation to the small sample critical values even for n = 25. The results of a simulation study comparing the power of the proposed tests with some standard tests of normality are presented. These results indicate that, for the cases considered, these statistics compare favorably with the standard procedures. [ABSTRACT FROM AUTHOR]

The techniques of exploratory data analysis include a resistant rule for identifying possible outliers in univariate data. Using the lower and upper fourths, F[sub L] and F[sub U] (approximate quartiles), it labels as "outside" any observations below F[sub L] - 1.5(F[sub U] - F[sub L]) or above F[sub U] + 1.5( F[sub U] - F[sub L]). For example, in the ordered sample -5, -2, 0, l, 8, F[sub L] = -2 and F[sub U] = 1, so any observation below -6.5 or above 5.5 is outside. Thus the rule labels 8 as outside. Some related rules also use cutoffs of the form F[sub L] -k(F[sub U] - F[sub L]) and F[sub U] + k(F[sub U] - F[sub L]). This approach avoids the need to specify the number of possible outliers in advance; as long as they are not too numerous, any outliers do not affect the location of the cutoffs. To describe the performance of these rules, we define the some-outside rate per sample as the probability that a sample will contain one or more outside observations. Its complement is the all-inside rate per sample. We also define the outside rate per observation as the average fraction of outside observations. For Gaussian data the population all-inside rate per sample (0) and the population outside rate per observation (.7%) substantially understate the corresponding small-sample values. Simulation studies using Gaussian samples with n between 5 and 300 yield detailed information on the resistant rules. The main resistant rule (k = 1.5) has an all-inside rate per sample between 67% and 86% for 5 less than or equal to n less than or equal to 20, and corresponding estimates of its outside rate per observation range from 8.6% to 1.7%. Both characteristics vary with n in ways that lead to good empirical approximations. Because of the way in which the fourths are defined, the sample sizes separate into four classes, according to whether dividing n by 4 leaves a remainder of 0, 1, 2, or 3. Within these... [ABSTRACT FROM AUTHOR]

Saddlepoint approximations are extended to general statistics. The technique is applied to derive approximations to the density of linear combinations of order statistics, including trimmed means. A comparison with exact results shows the accuracy of these approximations even in very small sample sizes. [ABSTRACT FROM AUTHOR]

The problem of estimating the mean of an exponential distribution is studied, when the data available are a random sample of time-censored rth-order statistics. Examples for such empirical situations are cited. Maximum likelihood estimators (MLE's) and moment-equation estimators (MEE's) are studied. Theoretical derivations are provided for the large sample variances and distributions of these estimators. The efficiency of the MEE compared to the MLE is studied. [ABSTRACT FROM AUTHOR]

This article gives necessary and sufficient conditions for the equivalence of the simple linear unbiased estimator (SLUE) and the best linear unbiased estimator (BLUE) of the parameters when there are k independent sources of Type II censored data from a location scale distribution in which the location parameter is an unknown linear function of certain given independent variables and the scale parameter is an unknown constant. It also shows that the BLUE can be obtained in a two-stage procedure that has the same model-checking features and computational simplicity of the SLUE. [ABSTRACT FROM AUTHOR]

Suppose that y[sub 1] ..... y[sub k] are normal random variables with ordered means u[sub 1] less than or equal to u[sub 2] less than or equal to ... less than or equal to u[sub k].. A confidence bound is found for each u[sub i] that takes advantage of the ordering. The upper and lower bounds are the maximum and minimum values of x such that the hypotheses that x < u[sub i] and u, < x are accepted by their respective likelihood ratio tests. These bounds are often more precise than the usual bounds and can be used to determine the safety of a toxin and to bound the virtual safe dose in a low-dose extrapolation. Simultaneous intervals for u[sub 1],..., u[sub k] are also derived. [ABSTRACT FROM AUTHOR]

A smooth alternative to the conventional sample quantile function as a nonparametric estimator of a population quantile function is proposed. The proposed estimator is essentially a kernel type of estimator and has the same asymptotic distribution as the conventional sample quantile. The mean squared convergence rate of the proposed estimator is also estimated. Monte Carlo studies are conducted to compare the proposed estimator with the sample quantile and the estimator proposed by Kaigh and Lachenbruch. The feasibility of using bootstrap techniques to estimate the optimal window width for the proposed estimator is also considered. [ABSTRACT FROM AUTHOR]

In multiple comparison procedures often the ordering of populations is of more interest than the testing of hypotheses. In such a situation it is important to control the probability of a Type III error (introduced by Hatter 1957 to indicate that one population is concluded to be better than another when actually it is worse). The studentized differences between means may be used to give ordering conclusions on all pairs of populations under consideration. Although this may be done by setting up confidence intervals using Tukey's honest significant difference, a more efficient method (when the confidence intervals themselves are not of interest) is suggested here, and tables are provided for its implementation. [ABSTRACT FROM AUTHOR]

We address the problem of forecasting the future records for an athletic event on the basis of the observed past records in that event. The records are viewed as realizations of a random process. The bivariate distribution and covariance of the record in any two periods is derived by a simple extension of the theory of order statistics. In the special case of a uniform, normal, or extreme-value parent, minimum-variance-linear-unbiased (MVLU) estimates of the parameters and "best" forecasts of future records may be obtained by generalized least squares. As an illustration, forecasts of the world record in six major running events are calculated for a 15-year period. [ABSTRACT FROM AUTHOR]

The problem of choosing the best k objects out of n is considered when, instead of measurements y[sub i] (i = 1 ,..., n) of primary interest, only associated measurements x[sub i] are readily obtainable. Large measurements are regarded as desirable. It is assumed that the n pairs (x[sub i], y[sub l]) are a random sample from a continuous population. A general expression is developed for the probability pi that the s objects with the largest X-values include the k objects (k less than or equal to s) with the largest Y-values. When X and Y are bivariate normal with correlation rho, a table of pi is presented for n less than or equal to 15; this table gives immediately the smallest s for which pi is greater than or equal to P[sup *], where P[sup *] is preassigned. Several applications are treated, especially one to the random effects model for n varieties with r replications. It is shown how reasonable values of r may be found to provide a reduction in the number of varieties from n to s, such that the chosen s contain the k best varieties with at least probability P[sup *]. A method conditional on the observed values of the X's is also developed. [ABSTRACT FROM AUTHOR]

Let pi[sub I], pi[sub 2] ..... pi[sub k] be k is greater than or equal to 2 sources of observations (treatments, populations) and suppose the "goodness" of treatment pi[sub I] is characterized by the size of an unknown real-valued parameter theta[sub I]. Let theta[sub k] = max[sub 1 less than or equal to I less than or equal to k] theta[sub I] If pi[sub I] is preferred to pi[sub j] when theta[sub I] > theta[sub j], the parameters delta[sub l] = theta[sub [k]] - theta[sub I], I = 1, 2 ..... k reflect in an inverse sense the "goodness" of each treatment relative to the "best" treatment. A general technique for obtaining simultaneous confidence intervals on the delta[sub I] is demonstrated with several examples. This technique can be applied in any setting where comparison-with-control intervals can be computed regarding any pi[sub j] as the control. These results have special importance in ranking and selection problems in that the process of generating upper bounds on the delta[sub I] generates traditional confidence statements of both the indifference zone and the subset selection schools, simultaneously, as established by Hsu (1981). [ABSTRACT FROM AUTHOR]

This article explores the use of t statistics based on two- or four-order statistics. The functions of the order statistics, which are used to define the t statistics, are the pivots and bipivots. The pivots are exact order statistics, while the bipivots are the means of two adjacent order statistics. Two t statistics, based on the pivots and bipivots, are examined and compared to other t statistics, including Student's t, under various sampling situations. We conclude that the proposed t statistics are not only simple to use but are robust and perform quite well in terms of the lengths of their confidence intervals. We recommend that these t statistics be used for sample sizes between 4 and 20, because they perform well and are easy to use. Tables of percentage points are provided to facilitate the use of the t statistics. [ABSTRACT FROM AUTHOR]

Tests for symmetry of a distribution function about an unknown value are investigated. The asymptotic distributions under symmetry as well as under near alternatives are given for two kinds of tests: trimmed Wilcoxon tests and tests based on gaps. It turns out that these procedures have a good stability of level under unimodal densities. In a special case (skewness to the left) the efficacies have been calculated. Monte Carlo results are also given. [ABSTRACT FROM AUTHOR]

The behavior of linear combinations of order statistics (L-statistics) under jackknifing is discussed. The asymptotic behavior of a jackknifed L-statistic is described and the pseudovalue-based variance estimate is shown to be consistent under moderate smoothness and a trimming condition on the weight function. The results extend easily to functions of L-statistics and thus answer a question posed in Miller (1974). Monte Carlo results support small-sample applicability of the large-sample results for the construction of approximate confidence intervals. [ABSTRACT FROM AUTHOR]

Let (X[sub I], Y[sub I])(I = 1, 2,..., n) be independent identically distributed as (X, Y). Then the rth ordered X variate is denoted by X[sub r:n] and the associated Y variate, the concomitant of the rth order statistic, by Y[r:.]. This paper considers statistics of the form n[sup -1] Sigma[sup n, sub I = 1] J(I/(n + 1)) Y[sub [I:n]] and more generally of the form n[sup -1] Sigma[sup n, sub I = 1] J(I/(n + 1))H(X[sub I:n], Y[sub [I:n]), where J is a bounded smooth function and may depend on n. Under certain regularity conditions, the asymptotic normality of these statistics is established. These statistics are used to construct consistent estimators of various conditional quantities, for example E(Y|X = x), P(Y is an element of A|X = x) and var(Y|X = x). [ABSTRACT FROM AUTHOR]

A study is performed of McCool's method for constructing linear estimates from order statistics when the coefficients of a linear estimate for a smaller sample size are available. The efficiency of the method is investigated, using the theory of U statistics. Among other things, it is shown that in the case of the logistic distribution, the method produces an asymptotically efficient location parameter estimate and, in the case of the Pareto distribution, an asymptotically efficient scale parameter estimate. [ABSTRACT FROM AUTHOR]

In nomination sampling, the largest values from several independent random samples (nominees) are rank ordered, and an estimate of the population median is formed by interpolating between two of these order statistics. The resulting estimate compares favorably to the sample median of a simple random sample from the same population. When historical data sets retain only extreme values, nomination sampling may offer the only practical way to estimate the population median. The approach may also be useful when potential survey respondents will only participate if they can actively influence the selection of eases for analysis. [ABSTRACT FROM AUTHOR]

When a system consisting of many independently functioning components of the same type is put into service, some action may be appropriate as soon as some proportion of the components has failed. It is, therefore, useful to be able to predict later failure times from earlier ones. We compare the information in the latest available failure time to that contained in all previous failures, for predicting later failures. For many of the well-known failure distributions commonly used to model component life, the relative amount of prediction information in the latest available failure time is surprisingly high. [ABSTRACT FROM AUTHOR]

Based upon Ferguson's Dirichlet process, we introduce an order-statistic process, a technical device which may be viewed as defining a continuum of fractional order statistics for any sample size. The order-statistic process provides an alternative to the quantile function essentially based on stochastic rather than linear interpolation. Its use in large-sample theory is suggested, and a simple proof is given that the order-statistic process, suitably normalized, converges to a Gaussian process. Applications to small-sample theory and to the passage-time distribution of Yule processes are also considered, and a class of censored-data problems is related to mixtures of these processes. [ABSTRACT FROM AUTHOR]