A Distribution-Free Control Chart Based on Order Statistics

In this article, we introduce a new distribution-free Shewhart-type control chart that takes into account the location of a single order statistic of the test sample (such as the median) as well as the number of observations in that test sample that lie between the control limits. Exact formulae for the alarm rate, the run length distribution, and the average run length (ARL) are all derived. A key advantage of the chart is that, due to its nonparametric nature, the false alarm rate and in-control run length distribution are the same for all continuous process distributions, and so will be naturally robust. Tables are provided for the implementation of the chart for some typical ARL values and false alarm rates. The empirical study carried out reveals that the new chart is preferable from a robustness point of view in comparison to a classical Shewhart-type chart and also the nonparametric chart of Chakraborti et al. (2004 Chakraborti , S. , van der Laan , P. , van de Wiel , M. A. ( 2004 ). A class of distribution-free control charts . J. Roy. Statist. Soc. Ser. C-Appl. Statist. 53 ( 3 ): 443 – 462 .[Web of Science ®] , [Google Scholar]).     

The analysis of some biological data

This paper considers the building of stochastic models and the related analysis of discrete data in two biological problems, The first arises from the reproduction of yeast cells, while the secondis concerned with the aggregation of nucleoli. Galton-Watson and aggregation models are constructed for the respective processes and their goodness of fit to the data tested     

A test for bivariate symmetry based on the empirical distribution function

Hollander (1970) proposed a conditionally distribution-free test of bivariate symmetry based on the empirical distribution function. In this paper Hollander’s test statistic is examined In greater detail: in particular; its conditional asymptotic distribution is derived under the null hypothesis as well as under a sequence of local alternatives. Percentage points of the asymptotic distribution are presented; a power comparison between Hollander’s statistic and the likelihood ratio criterion in testing a variant of the sphericity hypothesis in multivariate analysis is made.     

A time series approach to life table construction

This paper deals with the construction of the life table. A discussion of basic facts about the life table is followed by the proposal of a nonstationary, autoregressive model for the life table. The moment structure of the nonstationary, autoregressive model is developed. Some estimation procedures are proposed followed by several examples.     

Time series analysis of stratospheric ozone

Time series analyses of monthly average total ozone measured at 37 stations throughout the world were used to estimate the extent to which the average ozone trend correlates with the depletion curve hypothesized as due to chlorofluorocarbons (CFCs). Statistical characteristics of stations in the ensemble were used to help define appropriate model and station selection criteria. The maximum likelihood procedure developed herein estimates the weighted average trend, its. variance, and the intra- and inter-station variance components of the trend. Correlations among trends at different stations are also taken into account. The models were subjected to much checking and criticism. Variations in statistical methodology are used to show that the results are insensitive to details of the model selection criteria. The method does not discriminate well between the hypothesized CFC trend and a linear trend. The trend estimates represent the sum of all long-term global effects. The variance includes all effects that differ from station-to-station. The estimated trend and 2α limits for 14 stations with 20-year records (1958-79) is an ozone increase through 1979 of (1.5+1.0) percent. At the 23 stations with shorter records, the trend is (1.0=1.7) percent. It is concluded that no significant depletion in stratospheric ozone has occurred from any cause through the end of 1979.     

Test for Parameter Change in Linear Processes Based on Whittle's Estimator

In this article, we develop a cusum test for testing for parameter changes in linear processes based on Whittle's estimator. It is shown that under regularity conditions, the test statistic converges to the sup of a Brownian bridge. The result is particularly useful in handling the change point test in stationary ARMA processes. A simulation result is provided for illustration.     

Testing for the Equality of Two Autoregressive Functions Using Quasi-Residuals

This article discusses the problem of testing the equality of two nonparametric autoregressive functions against one-sided alternatives. The heteroscedastic errors and stationary densities of the two independent strong mixing strictly stationary time series can be possibly different. The article adapts the idea of using sum of quasi-residuals to construct the test and derives its asymptotic null distributions. The article also shows that the test is consistent for general alternatives and obtains its limiting distributions under a sequence of local alternatives. Then a Monte Carlo simulation is conducted to study the finite sample level and power behavior of these tests at some alternatives. We also compare the test to an existing lag matched test theoretically and by Monte Carlo experiments.     

Locking the Wiener Process to its Level-Crossing Time

We consider the specific transformation of a Wiener process {X(t), t ≥ 0} in the presence of an absorbing barrier a that results when this process is “time-locked” with respect to its first passage time T _a through a criterion level a, and the evolution of X(t) is considered backwards (retrospectively) from T _a . Formally, we study the random variables defined by Y(t) ≡ X(T _a  ? t) and derive explicit results for their density and mean, and also for their asymptotic forms. We discuss how our results can aid interpretations of time series “response-locked” to their times of crossing a criterion level.     

The effects of correlation among observations on the consistency property of sample variance

It is well known that even when the sample observations are correlated and not normal the sample variance, S² converges in probability to E(S²). But the required sample size for S² to be a consistent estimator of E(S²) is an open question. Some light is shed on this question in this paper. In particular the relation between the rate of convergence and the correlation property of the observations is explored. It is shown that the retardation to the rate of convergence is not appreciable if the correlation is moderate but it can be severe for extreme correlations.