Multiple Window Discrete Scan Statistics

In this article, multiple scan statistics of variable window sizes are derived for independent and identically distributed 0-1 Bernoulli trials. Both one and two dimensional, as well as, conditional and unconditional cases are treated. The advantage in using multiple scan statistics, as opposed to single fixed window scan statistics, is that they are more sensitive in detecting a change in the underlying distribution of the observed data. We show how to derive simple approximations for the significance level of these testing procedures and present numerical results to evaluate their performance.     

A fractional order statistic towards defining a smooth quantile function for discrete data

This work is motivated in part by a recent publication by Ma et al. (2011) who resolved the asymptotic non-normality problem of the classical sample quantiles for discrete data through defining a new mid-distribution based quantile function. This work is the motivation for defining a new and improved smooth population quantile function given discrete data. Our definition is based on the theory of fractional order statistics. The main advantage of our definition as compared to its competitors is the capability to distinguish the uth quantile across different discrete distributions over the whole interval, u∈(0,1). In addition, we define the corresponding estimator of the smooth population quantiles and demonstrate the convergence and asymptotic normal distribution of the corresponding sample quantiles. We verify our theoretical results through a Monte Carlo simulation, and illustrate the utilization of our quantile function in a Q-Q plot for discrete data.     

A spatial scan statistic for survival data based on generalized life distribution

ABSTRACT

For many years, detection of clusters has been of great public health interest and widely studied. Several methods have been developed to detect clusters and their performance has been evaluated in various contexts. Spatial scan statistics are widely used for geographical cluster detection and inference. Different types of discrete or continuous data can be analyzed using spatial scan statistics for Bernoulli, Poisson, ordinal, exponential, and normal models. In this paper, we propose a scan statistic for survival data which is based on generalized life distribution model that provides three important life distributions, viz. Weibull, exponential, and Rayleigh. The proposed method is applied to the survival data of tuberculosis patients in Nainital district of Uttarakhand, India, for the year 2004–05. The Monte Carlo simulation studies reveal that the proposed method performs well for different survival distributions.     

An unconditional space–time scan statistic for ZIP‐distributed data

A scan statistic is proposed for the prospective monitoring of spatiotemporal count data with an excess of zeros. The method that is based on an outbreak model for the zero‐inflated Poisson distribution is shown to be superior to traditional scan statistics based on the Poisson distribution in the presence of structural zeros. The spatial accuracy and the detection timeliness of the proposed scan statistic are investigated by means of simulation, and an application on the weekly cases of Campylobacteriosis in Germany illustrates how the scan statistic could be used to detect emerging disease outbreaks. An implementation of the method is provided in the open‐source R package scanstatistics available on the Comprehensive R Archive Network.     

Prospective time periodic geographical disease surveillance using a scan statistic

Most disease registries are updated at least yearly. If a geographically localized health hazard suddenly occurs, we would like to have a surveillance system in place that can pick up a new geographical disease cluster as quickly as possible, irrespective of its location and size. At the same time, we want to minimize the number of false alarms. By using a space–time scan statistic, we propose and illustrate a system for regular time periodic disease surveillance to detect any currently 'active' geographical clusters of disease and which tests the statistical significance of such clusters adjusting for the multitude of possible geographical locations and sizes, time intervals and time periodic analyses. The method is illustrated on thyroid cancer among men in New Mexico 1973–1992.     

A Bayesian analysis for the Wilcoxon signed-rank statistic

A Bayesian analysis is provided for the Wilcoxon signed-rank statistic (T⁺). The Bayesian analysis is based on a sign-bias parameter φ on the (0, 1) interval. For the case of a uniform prior probability distribution for φ and for small sample sizes (i.e., 6 ? n ? 25), values for the statistic T⁺ are computed that enable probabilistic statements about φ. For larger sample sizes, approximations are provided for the asymptotic likelihood function P(T⁺|φ) as well as for the posterior distribution P(φ|T⁺). Power analyses are examined both for properly specified Gaussian sampling and for misspecified non Gaussian models. The new Bayesian metric has high power efficiency in the range of 0.9–1 relative to a standard t test when there is Gaussian sampling. But if the sampling is from an unknown and misspecified distribution, then the new statistic still has high power; in some cases, the power can be higher than the t test (especially for probability mixtures and heavy-tailed distributions). The new Bayesian analysis is thus a useful and robust method for applications where the usual parametric assumptions are questionable. These properties further enable a way to do a generic Bayesian analysis for many non Gaussian distributions that currently lack a formal Bayesian model.     

Estimating the distribution of one-dimensional discrete scan statistics viewed as extremes of 1-dependent stationary sequences

Discrete one-dimensional scan statistics can be viewed as extremes of 1-dependent stationary sequences. A result of Haiman [1999. First passage time for some stationary processes. Stochastic Process. Appl. 80, 231–248] provides approximations of the distributions of extremes of 1-dependent stationary sequences together with sharp bounds for the corresponding errors. We apply this result to scan statistics generated by Bernoulli r.v.'s and to the charge problem.     

Prediction intervals for ordinary and dual generalized order statistics from two independent sequences

ABSTRACT

Based on the observed dual generalized order statistics drawn from an arbitrary unknown distribution, nonparametric two-sided prediction intervals as well as prediction upper and lower bounds for an ordinary and a dual generalized order statistic from another iid sequence with the same distribution are developed. The prediction intervals for dual generalized order statistics based on the observed ordinary generalized order statistics are also developed. The coverage probabilities of these prediction intervals are exact and free of the parent distribution, F. Finally, numerical computations and real examples of the coverage probabilities are presented for choosing the appropriate limits of the prediction.     

Cramér-von Mises statistics for discrete distributions

Cramér-von Mises statistics are developed for use in testing for discrete distributions, and tables are given for tests for the discrete uniform distribution.     

Estimation for Type II domain of attraction based on the W statistic

The paper presents an estimating equation approach to the estimation of high quantiles of a distribution in the Type II domain of attraction based on the k largest order statistics. The estimators are shown to be consistent. The method fits neatly into a general scheme for estimating high quantiles irrespective of the domain of attraction, which includes Wang's approach to optimally choosing k .     

Coup de Grâce for a Tough Old Bull: “Statistically Significant” Expires

ABSTRACT

Many controversies in statistics are due primarily or solely to poor quality control in journals, bad statistical textbooks, bad teaching, unclear writing, and lack of knowledge of the historical literature. One way to improve the practice of statistics and resolve these issues is to do what initiators of the 2016 ASA statement did: take one issue at a time, have extensive discussions about the issue among statisticians of diverse backgrounds and perspectives and eventually develop and publish a broadly supported consensus on that issue. Upon completion of this task, we then move on to deal with another core issue in the same way. We propose as the next project a process that might lead quickly to a strong consensus that the term “statistically significant” and all its cognates and symbolic adjuncts be disallowed in the scientific literature except where focus is on the history of statistics and its philosophies and methodologies. Calculation and presentation of accurate p-values will often remain highly desirable though not obligatory. Supplementary materials for this article are available online in the form of an appendix listing the names and institutions of 48 other statisticians and scientists who endorse the principal propositions put forward here.     

Comparisons of the asymptotic efficiencies of two sample tests for discrete distributions

The asymptotic efficiencies are computed for several popular two sample rank tests when the underlying distributions are Poisson, binomial, discrete uniform, and negative binomial The rank tests examined include the Mann-Whitney test, the van der Waerden test, and the median test. Three methods for handling ties are discussed and compared. The computed asymptotic efficiencies apply also to the k-sample extensions of the above tests, such as the Kruskal-Wallis test, etc.     

A monte carlo investigation of a statistic for a bivariate missing data problem

Testing the equal means hypothesis of a bivariate normal distribution with homoscedastic varlates when the data are incomplete is considered. If the correlational parameter, ρ, is known, the well-known theory of the general linear model is easily employed to construct the likelihood ratio test for the two sided alternative. A statistic, T, for the case of ρ unknown is proposed by direct analogy to the likelihood ratio statistic when ρ is known. The null and nonnull distribution of T is investigated by Monte Carlo techniques. It is concluded that T may be compared to the conventional t distribution for testing the null hypothesis and that this procedure results in a substantial increase in power-efficiency over the procedure based on the paired t test which ignores the incomplete data. A Monte Carlo comparison to two statistics proposed by Lin and Stivers (1974) suggests that the test based on T is more conservative than either of their statistics.     

On a cramér-von mises type statistic for testing bivariate independence

A Cramér-von Mises type statistic for testing bivariate independence, proposed by Hoeffding (1948) and by Blum, Kiefer, and Rosenblatt (1961), is examined in greater detail. The statistic is decomposed into components in the manner of Durbin and Knott (1972), and the components are shown to be related to linear rank statistics. Asymptotic power properties of the Hoeffding statistic and its components in testing for independence with bivariate normal random observations are described; a Monte Carlo study comparing these statistics with other nonparametric statistics for bivariate independence is also reported.     

Computation of the estimated parameters and wald statistic for the generalized growth curve model

Multivariate analysis is difficult when there are missing observations in the response vectors. Kleinbaum (1973) proposed a Wald statistic useful in the analysis of incomplete multivariate data. SUBROUTINE C0EF calculates the estimated parameter matrix g in the generalization of the Potthoff-Roy (1964) growth curve model proposed by Kleinbaum (1973). SUBROUTINE WALD calculates the Wald statistic for hypotheses of the form Hn: H 5 D = 0 as proposed by Kleinbaum (1973).     

A mutual information-based k-sample test for discrete distributions

The two-sample problem and its extension to the k-sample problem are well known in the statistical literature. However, the discrete version of the k-sample problem is relatively less explored. Here in this work we suggest a k-sample non-parametric test procedure for discrete distributions based on mutual information. A detailed power study with comparison with other alternatives is provided. Finally, a comparison of some English soccer league teams based on their goal-scoring pattern is discussed.     

A corrected likelihood ratio statistic for the multivariate regression model with antedependent errors

This paper addresses the problem of inference for the antedependence model (Gabriel, 1961, 1962). Antedependence can be formulated as an autoregressive process of general order which is non-stationary in time. Its primary application is in the analysis of repeated measurements data, that is, data consisting of independent replicates of relatively short time series. Our focus is on testing a general linear hypothesis in the context of a multivariate regression model with multivariate normal antedependent errors. Although the relevant likelihood ratio statistic was first presented by Gabriel (1961), the distribution of this statistics has not yet been derived. We present this derivation and show how this result leads to a simple correction factor to improve the x² approximation of the likelihood ratio statistic.     

Multiple nonlinear regression of the Markovian arrival process for estimating the daily global solar radiation

Abstract

Solar radiation is a global ecological phenomenon that affects life everywhere. In this study, a new statistical method, called the Quartiles-Moment's method, is proposed to estimate the scale and shape parameters of the exponentiated Gumbel maximum distribution (EGMD). The Kolomogorov–Smirnov test and the percentiles of the dataset are thus used to fit the dataset of the daily global solar radiation and the corresponding daily maximum temperature with EGMD. Thence, multiple nonlinear regression of the daily global solar radiation and the corresponding daily maximum temperature are produced and compared with the real dataset accordingly.     

Study of the quality of several asymptotic and near-exact approximations based on moments for the distribution of the Wilks Lambda statistic

In this paper a measure of proximity of distributions, when moments are known, is proposed. Based on cases where the exact distribution is known, evidence is given that the proposed measure is accurate to evaluate the proximity of quantiles (exact vs. approximated). The measure may be applied to compare asymptotic and near-exact approximations to distributions, in situations where although being known the exact moments, the exact distribution is not known or the expression for its probability density function is not known or too complicated to handle. In this paper the measure is applied to compare newly proposed asymptotic and near-exact approximations to the distribution of the Wilks Lambda statistic when both groups of variables have an odd number of variables. This measure is also applied to the study of several cases of telescopic near-exact approximations to the exact distribution of the Wilks Lambda statistic based on mixtures of generalized near-integer gamma distributions.