Search

A new method is proposed for identifying clusters in continuous data indexed by time or by space. The scan statistic we introduce is derived from the well-known Mann–Whitney statistic. It is completely non parametric as it relies only on the ranks of the marks. This scan test seems to be very powerful against any clustering alternative. These results have applications in various fields, such as the study of climate data or socioeconomic data.

This article reexamines ANOVA problems for autocorrelated data. Using linear prediction techniques for stationary time series, a new test statistic that assesses a null hypothesis of equal means is proposed and investigated. Our test statistic mimics the classical F-type ratio form used with independent data, but substitutes estimated prediction residuals in for the errors. This simple tactic departs from past studies that adjust the quadratic forms in the numerator and denominator in the F ratio for autocorrelation. One of the advantages is that our statistic retains the classical null hypothesis F distribution (now as a limit) with the customary degrees of freedom. The statistic is shown to perform well in simulations. Asymptotic proofs are given in the case of autoregressive random errors; a sports application is supplied.[Received December 2014. Revised August 2015.]

In this paper, we consider the consistency of Cp-type statistics for selecting variables in a normality-assumed linear regression with multiple responses when the dimension of the vector of the response variables may be large. We propose a new consistent Cp-type statistic for which consistency can be achieved whenever the dimension of the response variables vector is fixed or goes to infinity. A high probability of selecting the true subset of explanatory variables can be expected under a moderate sample size when the proposed Cp-type statistic is used to select variables, even when there is a high-dimensional response variables vector.

The C statistic is a commonly reported measure of screening test performance. Optimistic estimation of the C statistic is a frequent problem because of overfitting of statistical models in small data sets, and methods exist to correct for this issue. However, many studies do not use such methods, and those that do correct for optimism use diverse methods, some of which are known to be biased. We used clinical data sets (United Kingdom Down syndrome screening data from Glasgow (1991-2003), Edinburgh (1999-2003), and Cambridge (1990-2006), as well as Scottish national pregnancy discharge data (2004-2007)) to evaluate different approaches to adjustment for optimism. We found that sample splitting, cross-validation without replication, and leave-1-out cross-validation produced optimism-adjusted estimates of the C statistic that were biased and/or associated with greater absolute error than other available methods. Cross-validation with replication, bootstrapping, and a new method (leave-pair-out cross-validation) all generated unbiased optimism-adjusted estimates of the C statistic and had similar absolute errors in the clinical data set. Larger simulation studies confirmed that all 3 methods performed similarly with 10 or more events per variable, or when the C statistic was 0.9 or greater. However, with lower events per variable or lower C statistics, bootstrapping tended to be optimistic but with lower absolute and mean squared errors than both methods of cross-validation.

Barndorff-Nielsen’s celebrated p*-formula and variations thereof have amongst their various attractions the ability to approximate bimodal distributions. In this paper we show that in general this requires a crucial adjustment to the basic formula. The adjustment is based on a simple idea and straightforward to implement, yet delivers important improvements. It is based on recognizing that certain outcomes are theoretically impossible and the density of the MLE should then equal zero, rather than the positive density that a straight application of p* would suggest. This has implications for inference and we show how to use the new p**-formula to construct improved confidence regions. These can be disjoint as a consequence of the bimodality. The degree of bimodality depends heavily on the value of an approximate ancillary statistic and conditioning on the observed value of this statistic is therefore desirable. The p**-formula naturally delivers the relevant conditional distribution. We illustrate these results in small and large samples using a simple nonlinear regression model and errors in variables model where the measurement errors in dependent and explanatory variables are correlated and allow for weak proxies.

Now, we are ready to discuss the relationship between probability and statistical inference. The two key facts to statistical inference are (a) the population parameters are fixed numbers that are usually unknown and (b) sample statistics are known for any sample. For different samples, we get different values of the statistics and hence this variability is accounted for identifying distributions called sampling distributions. In this chapter, we discuss certain distributions that arise in sampling from normal distribution. The other topics covered in this chapter are as follows: unbiasedness, mean square error, consistency, relative efficiency, sufficiency, minimum variance, Fisher information for a function of a parameter, Cramer–Rao lower bound, efficiency, method of moments, maximum likelihood estimation to find estimators analytically and numerically, and asymptotic distributions of maximum likelihood estimators.

Tests of goodness of fit of sparse multinomial models with non-canonical links is proposed by using approximations to the first three moments of the conditional distribution of a modified Pearson Chi-square statistic. The modified Pearson statistic is obtained using a supplementary estimating equation approach. Approximations to the first three conditional moments of the modified Pearson statistic are derived. A simulation study is conducted to compare, in terms of empirical size and power, the usual Pearson Chi-square statistic, the standardized modified Pearson Chi-square statistic using the first two conditional moments, a method using Edgeworth approximation of the p-values based on the first three conditional moments and a score test statistic. There does not seems to be any qualitative difference in size of the four methods. However, the standardized modified Pearson Chi-square statistic and the Edgeworth approximation method of obtaining p-values using the first three conditional moments show power advantages compared to the usual Pearson Chi-square statistic, and the score test statistic. In some situations, for example, for small nominal level, the standardized modified Pearson Chi-square statistic shows some power advantage over the method using Edgeworth approximation of the p-values using the first three conditional moments. Also, the former is easier to use and so is preferable. Two data sets are analyzed and a discussion is given.

In this article, several scan statistics are discussed for detecting a local change in variance for one dimensional normal data. When the length of the scanning window is known, a fixed window scan statistic based on moving sum of squares is proposed. Two approximations for the distribution of this scan statistic are investigated. When the length of the scanning window is unknown, a variable window scan statistic based on a generalized likelihood ratio test and a multiple window minimum P-value scan statistic are proposed for detecting the local change in variance. For a moderate or large shift in variance, numerical results indicate that both the variable and multiple window scan statistics perform well. For large data sets, considering the detection power and computing efficiency, the multiple window scan statistic is recommended.

Pearson's Chi-square statistic for frequency tables depends on what is hypothesized as the expected frequencies. Its partitions also depend on the hypothesis. Lancaster (J R Stat Soc B 13:242---249, 1951) proposed ANOVA-like partitions of Pearson's statistic under several representative hypotheses about the expected frequencies. His expositions were, however, not entirely clear. In this paper, we clarify his method of derivations, and extend it to more general situations. A comparison is made with analogous decompositions of the log likelihood ratio statistic associated with log-linear analysis of contingency tables.

This paper proposes a new simple panel unit-root test by extending the cross-sectionally augmented panel unit-root test (CIPS) developed by Pesaran et al. (2013) to allow for smoothing structural changes in deterministic terms, approximated by a Fourier series. The proposed statistic is the simple average of the individual statistics constructed from the breaks and cross-sectional dependence augmented Dickey-Fuller (BCADF) regression and is called the BCIPS statistic. We initially develop the tests by assuming that the number of factors in the model is known and show that the limiting distribution of the BCADF statistic is free of nuisance parameters. The nonstandard limiting distribution of the (truncated) BCIPS statistic is also shown to exist and its critical values are tabulated. Monte-Carlo experiments point out that the sizes and powers of the BCIPS statistic are generally satisfactory as long as T is greater than or equal to fifty and a hundred, respectively. By using two different methods to determine the number of factors, both the BCIPS and CIPS tests are applied to examine the validity of long-run purchasing power parity. The proposed test complements the panel unit-root tests with breaks using dummy variables.

A new characterization of the Pareto distribution is proposed, and new goodness-of-fit tests based on it are constructed. Test statistics are functionals of U-empirical processes. The first of these statistics is of integral type, it is similar to the classical statistics $$\omega _n^1$$ . The second one is a Kolmogorov type statistic. We show that the kernels of our statistics are non-degenerate. The limiting distribution and large deviations asymptotics of the new statistics under null hypothesis are described. Their local Bahadur efficiency for parametric alternatives is calculated. This type of efficiency is mostly appropriate for the solution of our problem since the Kolmogorov type statistic is not asymptotically normal, and the Pitman approach is not applicable to this statistic. For the second statistic we evaluate the critical values by using Monte-Carlo methods. Also conditions of local optimality of new statistics in the sense of Bahadur are discussed and examples of such special alternatives are given. For small sample size we compare the power of those tests with some common goodness-of-fit tests.

As an important extension of the regular kappa statistic, the weighted kappa statistic has been widely used to assess the agreement between two procedures for independent matched-pair ordinal data. For clustered matched-pair ordinal data, based on the delta method and sampling techniques, a non-parametric variance estimator for the weighted kappa statistic is proposed without within-cluster correlation structure or distributional assumptions. The results of an extensive Monte Carlo simulation study demonstrate that the proposed weighted kappa statistic provides consistent estimation, and the proposed variance estimator behaves reasonably well for at least a moderately large number of clusters (e.g., K ≥ 50 ). Compared with the variance estimator ignoring dependence within a cluster, the proposed variance estimator performs better in maintaining the nominal coverage probability when the intra-cluster correlation is fair ( ρ ≥ 0.3 ), with more pronounced improvement when ρ is further increased. Moreover, under the general analysis of variance setting with systematic variability between procedures and clusters being included as a component of total variation, the equivalence between weighted kappa statistic and intraclass correlation coefficient is established. To illustrate the practical application of the proposed estimator, two real medical research data examples of clustered matched-pair ordinal data are analyzed, including an agreement study to compare two methods for assessing cervical ectopy, and a physician–patients data example from the Enhancing Communication and HIV Outcomes study.

In normal theory exploratory factor analysis, likelihood ratio (LR) statistic plays an important role in evaluating the goodness-of-fit of the model. In this paper, we derive an approximation of the LR statistic. The approximation is then used to show explicitly that the expectation of the LR statistic agrees with the degrees of freedom of the asymptotic chi-square distribution.

A U-statistic for the tail index of a multivariate stable random vector is given as an extension of the univariate case introduced by Fan (2006). Asymptotic normality and consistency of the proposed U-statistic for the tail index are proved theoretically. The proposed estimator is used to estimate the spectral measure. The performance of both introduced tail index and spectral measure estimators is compared with the known estimators by comprehensive simulations and real datasets.

The score statistic S2 is commonly used for general likelihood-based inference. Pearson’s Chi-squared statistic X2 = ∑(O − E)2/E is ubiquitous in contingency table inference. Because tests and confidence intervals based on S2 have been shown to work well in practice and theory and because X2 has such a simple and intuitively appealing form, it is of interest to know when S2 is identical to X2 and when X2 has an approximate Chi-squared distribution. Toward these ends, this paper gives a simple proof that S2 = X2 for the broad class of multinomial-Poisson distributions when the alternative hypothesis is unrestricted in a certain sense. This paper also gives a sufficient condition under which the null distribution of the Pearson score statistic is approximately Chi-squared. Several examples illustrate the utility of the results and counter-examples highlight the importance of the sufficient conditions of the results.

Comparing two suppliers for linear profiles is a very important task for supplier management. The difference test statistic based on the process-yield index is proposed to tackle the better process selection for linear profiles. A simple form of the sampling distribution of the process-yield index is derived by a simulation study. Therefore, the asymptotic normal distribution of the difference test statistic is established. The results provide useful information to practitioners. An example from the leather industry is presented to illustrate the applicability of the proposed method. Copyright © 2014 John Wiley & Sons, Ltd.

Standard nonparametric prediction intervals for a future order statistic are obtained by taking the interval between two order statistics of the initial sample. We obtain improved prediction intervals by taking the shortest of two or more standard intervals.

A 2-stage robust procedure as well as an R package, rsem, were recently developed for structural equation modeling with nonnormal missing data by Yuan and Zhang (2012). Several test statistics that have been used for complete data analysis are employed to evaluate model fit in the 2-stage robust method. However, properties of these statistics under robust procedures for incomplete nonnormal data analysis have never been studied. This study aims to systematically evaluate and compare 5 test statistics, including a test statistic derived from normal-distribution-based maximum likelihood, a rescaled chi-square statistic, an adjusted chi-square statistic, a corrected residual-based asymptotical distribution-free chi-square statistic, and a residual-based F statistic. These statistics are evaluated under a linear growth curve model by varying 8 factors: population distribution, missing data mechanism, missing data rate, sample size, number of measurement occasions, covariance between the latent intercept and s...

Single moments of order statistics from the modified Makeham distribution (MMD) are derived, an identity about the single moments of order statistics is given, and the specific expected value and variance of the single moments of order statistics from the MMD are calculated. In this study, the order statistic from the MMD was applied to the rank sum test in a two-sample problem. The exact critical values of the designated statistics were evaluated. Simulations were used to investigate the power of these statistics for the two-sided alternative with several population distributions. The powers of the statistics were compared with the Wilcoxon rank sum statistic, the Lepage statistic, the modified Baumgartner statistic, the Savage test and the normal score test. The Edgeworth expansion was used to evaluate the upper tail probability for the preferred statistic, given finite sample sizes.

This paper proposes an overlapping-based test statistic for testing the equality of two exponential distributions with different scale and location parameters. The test statistic is defined as the maximum likelihood estimate of the Weitzman's overlapping coefficient, which estimates the agreement of two densities. The proposed test statistic is derived in closed form. Simulated critical points are generated for the proposed test statistic for various sample sizes and significance levels via Monte Carlo Simulations. Statistical powers of the proposed test are computed via simulation studies and compared to those of the existing Log likelihood ratio test.

Hotelling's T2 statistic is the default control statistic for continuous multivariate data, but there are dangers in applying this statistic without the appropriate level of checks and balances. This paper discusses the potential issues with using the Hotelling's T2 statistic when the quality variable measures are highly correlated and provides some solutions that will help mitigate the risks with applying the Hotelling's T2 control charts in such practical examples. Copyright © 2014 John Wiley & Sons, Ltd.

Hotelling's T2 statistic has many applications in multivariate analysis. In particular, it can be used to measure the influence that a particular observation vector has on parameter estimation. For example, in the bivariate case, there exists a direct relationship between the ellipse generated using a T2 statistic for individual observations and the hyperbolae generated using Hampel's influence function for the corresponding correlation coefficient. In this paper, we jointly use the components of an orthogonal decomposition of the T2 statistic and some influence functions to identify outliers or influential observations. Since the conditional components in the T2 statistic are related to the possible changes in the correlation between a variable and a group of other variables, we consider the theoretical influence functions of the correlations and multiple correlation coefficients. Finite-sample versions of these influence functions are used to find the estimated influence function values.

This paper examines the general third-order theory to the log-normal regression model. The interest parameter is its conditional mean. For inference, traditional first-order approximations need large sample sizes and normal-like distributions. Some specific third-order methods need the explicit forms of the nuisance parameter and ancillary statistic, which are quite complicated. Note that this general third-order theory can be applied to any continuous models with standard asymptotic properties. It only needs the log-likelihood function. With small sample settings, the simulation studies for confidence intervals of the conditional mean illustrate that the general third-order theory is much superior to the traditional first-order methods.

The scan statistic is widely used in spatial cluster detection applications of inhomogeneous Poisson processes. However, real data may present substantial departure from the underlying Poisson process. One of the possible departures has to do with zero excess. Some studies point out that when applied to data with excess zeros, the spatial scan statistic may produce biased inferences. In this work, we develop a closed-form scan statistic for cluster detection of spatial zero-inflated count data. We apply our methodology to simulated and real data. Our simulations revealed that the Scan-Poisson statistic steadily deteriorates as the number of zeros increases, producing biased inferences. On the other hand, our proposed Scan-ZIP and Scan-ZIP+EM statistics are, most of the time, either superior or comparable to the Scan-Poisson statistic.

The discrete scan statistic is used in many areas of applied probability and statistics to study local clumping of patterns. Testing based on the statistic requires tail probabilities. Whereas the distribution has been studied extensively, most of the results are approximations, due to the difficulties associated with the computation. Results for exact p-values for the statistic have been given for a binary sequence that is independent or first-order Markovian. We give an algorithm to obtain probabilities for the statistic over multi-state trials that are Markovian of a general order of dependence, and explore the algorithm's usefulness.

The theory of the hypotheses test is well developed when the minimal sufficient statistic is uni-dimensional. In this paper, we discuss the problem of finding a uniformly most powerful test of size α for one-sided hypothesis testing of the parameter of the uniform family for which the minimal sufficient statistic is two-dimensional.

Our new method uses order statistics to judge the fit of a distribution to data. A test-statistic based on quantiles of order-statistics compares favorably with the Kolmogorov-Smirnov (K-S) and Anderson-Darling (A-D) test statistics. The performance of this new goodness-of-fit test statistic is examined with simulation experiments. For certain hypothesis tests, the test statistic is more powerful than the K-S and A-D test statistics. The new test statistic is calculated using a computer algebra system because of the need to compute exact distributions of order statistics.

D. Basu gave a striking bivariate normal example, N 2(0, 0, 1, 1, ρ) with an unknown correlation coefficient ρ, − 1

The PARSCALE G2 statistic is arguably the most popular item fit statistic in operational testing. For long tests, the Type I error rates of the statistic have often been found to be satisfactory. However, the Type I error rates of the statistic have only been studied for sample sizes of up to several thousands. The authors examined the Type I error rates of the PARSCALE G2 statistic in a simulation study using sample sizes much larger than those considered in the literature. For any fixed test length, the Type I error rate of the PARSCALE G2 statistic is found to increase to 1 as the sample size increases. The findings contradict the claim in the PARSCALE software manual that the PARSCALE G2 statistic leads to a large-sample test and also contradict the common belief that the statistic has reasonable Type I error rates for long tests. Thus, this simulation study conveys the important practical message that the use of the PARSCALE G2 statistic cannot always be recommended even for long tests. The Type I error rates of the item fit statistics of Orlando and Thissen were found to be close to the nominal level for all simulation conditions considered here.

This paper considers the methods of combining two independent test statistics to test the homogeneity of several gamma populations. The methods are based on Fisher's method of combining two likelihood ratio statistics, and combining two score statistics. Some simulation results provided here indicate that the Fishers method of combining two independent score statistics performs well relative to the other method. An example is presented.

A wide literature is available on the asymptotic behavior of the Durbin-Watson statistic for autoregressive models. However, it is impossible to find results on the Durbin-Watson statistic for autoregressive models with adaptive control. Our purpose is to fill the gap by establishing the asymptotic behavior of the Durbin Watson statistic for ARX models in adaptive tracking. On the one hand, we show the almost sure convergence as well as the asymptotic normality of the least squares estimators of the unknown parameters of the ARX models. On the other hand, we establish the almost sure convergence of the Durbin-Watson statistic and its asymptotic normality. Finally, we propose a bilateral statistical test for residual autocorrelation in adaptive tracking.

Formal inference in randomized clinical trials is based on controlling the type I error rate associated with a single pre-specified statistic. The deficiency of using just one method of analysis is that it depends on assumptions that may not be met. For robust inference, we propose pre-specifying multiple test statistics and relying on the minimum p-value for testing the null hypothesis of no treatment effect. The null hypothesis associated with the various test statistics is that the treatment groups are indistinguishable. The critical value for hypothesis testing comes from permutation distributions. Rejection of the null hypothesis when the smallest p-value is less than the critical value controls the type I error rate at its designated value. Even if one of the candidate test statistics has low power, the adverse effect on the power of the minimum p-value statistic is not much. Its use is illustrated with examples. We conclude that it is better to rely on the minimum p-value rather than a single statistic particularly when that single statistic is the logrank test, because of the cost and complexity of many survival trials.

This study deals with the small sample likelihood based inference for the ratio of twonormal variances. The small sample likelihood inference is an approximation method.The signed log-likelihood ratio statistic and the modi ed signed log-likelihood ratiostatistic, which converge to standard normal distribution, are proposed for the normalvariance ratio. Through the simulation study, the coverage probabilities of con denceinterval and power of the exact, the signed log-likelihood and the modi ed signed log-likelihood ratio statistic will be compared. A real data example will be provided.Keywords: Likelihood based inference, modi ed signed log-likelihood ratio statistic,normal variance ratio, signed log-likelihood ratio statistic. 1. Introduction The normal distribution plays an important role in statistical inference, and there are somany studies about this distribution. The statistical inference for the ratio of two normalvariances arises in the areas for comparing the precision of two independent normal popula-tions. About the ratio of two normal variances, there also exists an exact statistical inferencewith Fstatistic. This problem is simple, obvious and important.When comparing the dispersion of two normal population, one can use the distributiontable of F statistic. Since F distribution depends on degrees of freedom, the distributiontable of Fdistribution is more than several pages. For the degrees of freedom exceed 30, thetable does not have the value. In this case, one must use an interpolation to obtaining thequantile.An approximation of a statistic to standard normal distribution has been developed inmany statistical models. Since the percentile of standard normal distribution is well known,a statistic, which distributes as standard normal distribution asymptotically, may be veryuseful. But these statistics have the error rate depending on sample size. When the samplesize is small, these approximation to standard normal distribution is quite inaccurate. Forexample, a signed log-likelihood ratio statistic converges to standard normal distributionwith an error of O(n

In this paper, multivariate two-sample testing problems were examined based on the Jureckova–Kalina's ranks of distances. The multivariate two-sample rank test based on the modified Baumgartner statistic for the two-sided alternative was proposed. The proposed statistic was a randomized statistic. Simulations were used to investigate the power of the suggested statistic for various population distributions.

A great deal of inference in statistics is based on making the approximation that a statistic is normally distributed. The error in doing so is generally O(n−1/2), where n is the sample size and can be considered when the distribution of the statistic is heavily biased or skewed. This note shows how one may reduce the error to O(n−(j+1)/2), where j is a given integer. The case considered is when the statistic is the mean of the sample values of a continuous distribution with a scale or location change after the sample has undergone an initial transformation, which may depend on an unknown parameter. The transformation corresponding to Fisher's score function yields an asymptotically efficient procedure.

The asymptotic behavior of unbiased probability estimators for lattice distributions is investigated in the case where the mean is a sufficient statistic.

In this paper, a new approximation of the marginal posterior distribution function is obtained. Moreover, for the location-scale model, by applying the shrinkage argument, a new approximation of the conditional distribution function of the signed likelihood ratio statistic given an ancillary statistic is derived from the approximated marginal posterior distribution.