Properties and performance of the c-chart for attributes data

The effects of parameter estimation are examined for the well-known c-chart for attributes data. The exact run length distribution is obtained for Phase II applications, when the true average number of non-conformities, c, is unknown, by conditioning on the observed number of non-conformities in a set of reference data (from Phase I). Expressions for various chart performance characteristics, such as the average run length (ARL), the standard deviation of the run length (SDRL) and the median run length (MDRL) are also obtained. Examples show that the actual performance of the chart, both in terms of the false alarm rate (FAR) and the in-control ARL, can be substantially different from what might be expected when c is known, in that an exceedingly large number of false alarms are observed, unless the number of inspection units (the size of the reference dataset) used to estimate c is very large, much larger than is commonly used or recommended in practice. In addition, the actual FAR and the in-control ARL values can be very different from the nominally expected values such as 0.0027 (or ARL₀=370), particularly when c is small, even with large amounts of reference data. A summary and conclusions are offered.     

Lp-norm spherical distribution

In this paper, the spherical distribution is generalized to the L_p-norm spherical distribution, S(n, p), through the stochastic representation. Then the form of its p.d.f. characterizing the S(n, p) distribution is derived. Marginal, conditional distributions and the moments of S(n, p) distribution are obtained. The order statistics and the invariance property of S(n,p) distribution are also investigated.     

Some simulations and applications of forecasting long-memory time-series models

In this paper, we show some results of forecasting based on the ARFIMA(p,d,q) and ARIMA(p,d,q) models. We show, by simulation, that the technique of forecasting of the ARIMA(p,d,q) model can also be used when d is fractional, i.e., for the ARFIMA(p,d,q) model. We also conduct a simulation study to compare the two estimators of d obtained through regression methods. They are used in the hypothesis test to decide whether or not the series has long memory property and are compared on the basis of their k-step ahead forecast errors. The properties of long-memory models are also investigated using an actual set of data.     

Setting the clock back to zero property of a class of bivariate life distributions

In a previous paper. B. R. Rao and Talwalker (1993) considered absolutely continuous life distributions and extended the Lack of Memory Property (L.M.P.) of the exponential distribution and showed that several classes of life distributions have this property, which was called the 'setting the clock back to zero' property. ¶Its analog is discussed in the present paper for hivariate and multivariate classes of life distributions. As a simple application of this analog, it is proved that the Life expectancy and the Percentile Residual Life vectors of a population of individuals under the influence of multiple competing risks have simple expressions if the class of their joint life distributions has the setting the clock back to zero property,     

Bounded integrated processes and unit root tests

In the framework of integrated processes, the problem of testing the presence of unknown boundaries which constrain the process to move within a closed interval is considered. To analyze this problem, the concept of bounded integrated process is introduced, thus allowing to formally define boundary conditions for I(1) processes. A new class of tests, which are based on the rescaled range of the process, is introduced in order to test the null hypothesis of no boundary conditions. The limit distribution of the test statistics involved can be expressed in terms of the distribution of the range of Brownian functionals, while the power properties are obtained by deriving some asymptotic results for I(1) processes with boundary conditions. Both theoretical and simulation investigations show that range-based tests outperform standard unit root tests significantly when used to detect the presence of boundary conditions. A previous draft of the paper (Cavaliere, 2000) was presented at the 8th World Congress of the Econometric Society, Seattle, 11–16 August 2000. I wish sincerely to thank: Martin Jacobsen for his patience in discussing weak convergence to regulated Brownian motions and his valuable suggestions; the Department of Theoretical Statistics of the University of Copenhagen whose hospitality is gratefully acknowledged; Tommaso Proietti for important suggestions; Silvano Bordignon and partecipants at the CIdE seminar, University of Padua, June 2000; two anonymous referees. Partial financial support from 60% M.U.R.S.T. research grants is acknowledged.     

Second order efficiency for a class of estimators within the multinomial family

All the estimators considered by Rao (1961; 1963) belong to a certain class of minimum discrepancy estimators. A new representation of Rao s second measure of second order efficiency is given for estimators belonging to this class.     

Pairwise orthogonal F-rectangle designs

The concept of pairwise orthogonal Latin square design is applied to r row by c column experiment designs which are called pairwise orthogonal F-rectangle designs. These designs are useful in designing successive and/or simulataneous experiments on the same set of rc experimental units, in constructing codes, and in constructing orthogonal arrays. A pair of orthogonal F-rectangle designs exists for any set of v treatment (symbols), whereas no pair of orthogonal Latin square designs of order two and six exists; one of the two construction methods presented does not rely on any previous knowledge about the existence of a pair of orthogonal Latin square designs, whereas the second one does. It is shown how to extend the methods to r=pv row by c=qv column designs and how to obtain t pairwise orthogonal F-rectangle design. When the maximum possible number of pairwise orthogonal F-rectangle designs is attained the set is said to be complete. Complete sets are obtained for all v for which v is a prime power. The construction method makes use of the existence of a complete set of pairwise orthogonal Latin square designs and of an orthogonal array with vⁿ columns, (vⁿ−1)/(v−1) rows, v symbols, and of strength two.     

Log-efficient search for significant inputs of linear model

This paper is concerned with the search for an unknown s-tuple A of significant inputs in a linear model with t random discrete carriers and finitely supported i.i.d. noise. We study the decision based on s maximal values of Shannon information between various input and the output N-sequences. It was studied in Malyutov and Sadaka (Random Oper. Stochastic Equations 6 (4) (1998) 339) (abbreviated as MS98) as one of two nonparametric decisions introduced there inspired by the related study in Csiszár and Körner, Information Theory: Coding Theorems for Discrete Memoryless Systems, Academic Press, New York, NY, 1981. It was shown in MS98 that both decisions have the best asymptotic rates as t→∞ uniformly over arbitrary noise distributions and the set of significant parameters in the corresponding classes of tests. We show here that also provides the best rate of the mean error probability exponential decay (defined in Section 2) among the tests based on separate testing influence of each input to the output sequence for a natural class of random designs. The results of this paper (and MS98 for t→∞) justify the universal optimality of the introduced decisions, when used under random balance designs.     

Estimation of the limit variance for sums under a new weak dependence condition

We prove a self-normalized central limit theorem for a mixing class of processes introduced in Kacem M, Loisel S, Maume-Deschamps V. [Some mixing properties of conditionally independent processes. Commun Statist Theory Methods. 2016;45:1241–1259]. This class is larger than more classical strongly mixing processes and thus our result is more general than [Peligrad M, Shao QM. Estimation of the variance of partial sums for ρ-mixing random variables. J Multivar Anal. 1995;52:140–157; Shi S. Estimation of the variance for strongly mixing sequences. Appl Math J Chinese Univ. 2000;15(1):45–54] ones. The fact that some conditionally independent processes satisfy this kind of mixing properties motivated our study. We investigate the weak consistency as well as the asymptotic normality of the estimator of the variance that we propose.     

Autoregressive moving-average convolution processes on the unit circle

ARMA convolution models for processes in continuous space (in this case the unit circle) and discrete time are derived as a natural extension of the usual Box-Jenkins models. Both weakly time-stationary and nonstationary processes are considered. Sufficient conditions for the existence of weakly time-stationary AR_cMA_c processes are derived, and the covariance functions for some processes are computed. It is demonstrated that the usual scalar and multivariate ARMA processes can be embedded within the larger class of AR_CMA_c models. A possible application of these models to sea-surface temperature prediction is discussed.     

Optimal partially balanced nested row-column designs

A partially balanced nested row-column design, referred to as PBNRC, is defined as an arrangement of v treatments in b p × q blocks for which, with the convention that p q, the information matrix for the estimation of treatment parameters is equal to that of the column component design which is itself a partially balanced incomplete block design. In this paper, previously known optimal incomplete block designs, and row-column and nested row-column designs are utilized to develop some methods of constructing optimal PBNRC designs. In particular, it is shown that an optimal group divisible PBNRC design for v = mn kn treatments in p × q blocks can be constructed whenever a balanced incomplete block design for m treatments in blocks of size k each and a group divisible PBNRC design for kn treatments in p × q blocks exist. A simple sufficient condition is given under which a group divisible PBNRC is Ψ_f-better for all f> 0 than the corresponding balanced nested row-column designs having binary blocks. It is also shown that the construction techniques developed particularly for group divisible designs can be generalized to obtain PBNRC designs based on rectangular association schemes.     

Filtration of ASTA: a weak convergence approach

Consider a stochastic process (X,A), where X represents the evolution of a system over time, and A is an associated point process that has stationary independent increments. Suppose we are interested in estimating the time average frequency of the process X being in a set of states. Often it is more convenient to have a sampling procedure for estimating the time average based on averaging the observed values of X(T_n) (T_n being a point of A) over a long period of time: the event average of the process. In this paper we examine the situation when the two procedures—event averaging and time averaging—produce the same estimate (the ASTA property: Arrivals See Time Averages). We prove a result stronger than ASTA. Under a lack-of-anticipation assumption we prove that the point process, A, restricted to any set of states, has the same probabilistic structure as the original point process. In particular, if the original point process is Poisson the new point process is still Poisson with the same parameter as the original point process. We develop our results in the more general setting of a stochastic process (X,A), that is, a process with an imbedded cumulative process, A={A(t),t0}, which is assumed to be a Levy process with non-decreasing sample paths. This framework allows for modeling fluid processes, as well as compound Poisson processes with non-integer increments. First, we state the result in discrete time; the discrete-time result is then extended to the continuous-time case using limiting arguments and weak-convergence theory. As a corollary we give a proof of ASTA under weak conditions and a simple, intuitive proof of (Poisson Arrivals See Time Averages) under the standard conditions. The results are useful in queueing and statistical sampling theory.     

Bounds on posterior expectations for density bounded class with constant bandwidth

We consider a certain (simplified) version of the density bounded class; see Lavine (J. Amer. Statist. Assoc. 86 (1991) 400–403). We find that this class has certain advantages over many other similar classes in terms of elicitation, ease of calculation and other characteristics of interest. Bounds on posterior expectations are considered. In particular, we show how certain bounds may be found in the (multidimensional) normal linear model problem.     

Forward Moving Average Representations for MA Processes of Finite Order: Multivariate Stationary and Periodically Correlated

Soltani and Mohammadpour (2006 Soltani , A. R. , Mohammadpour , M. (2006). Moving average representations for multivariate stationary processes. J. Time Ser. Anal. 27(6):831–841.[Crossref], [Web of Science ®] , [Google Scholar]) observed that in general the backward and forward moving average coefficients, correspondingly, for the multivariate stationary processes, unlike the univariate processes, are different. This has stimulated researches concerning derivations of forward moving average coefficients in terms of the backward moving average coefficients. In this article we develop a practical procedure whenever the underlying process is a multivariate moving average (or univariate periodically correlated) process of finite order. Our procedure is based on two key observations: order reduction (Li, 2005 Li , L. M. ( 2005 ). Factorization of moving average spectral densities by state space representations and stacking . J. Multivariate Anal. 96 : 425 – 438 .[Crossref], [Web of Science ®] , [Google Scholar]) and first-order analysis (Mohammadpour and Soltani, 2010 Mohammadpour , M. , Soltani , A. R. ( 2010 ). Forward moving average representation for multivariate MA(1) processes . Commun. Statist. Theory Meth. 39 : 729 – 737 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]).     

Recent and classical goodness-of-fit tests for the Poisson distribution

We give a critical synopsis of classical and recent tests for Poissonity, our emphasis being on procedures which are consistent against general alternatives. Two classes of weighted Cramér–von Mises type test statistics, based on the empirical probability generating function process, are studied in more detail. Both of them generalize already known test statistics by introducing a weighting parameter, thus providing more flexibility with regard to power against specific alternatives. In both cases, we prove convergence in distribution of the statistics under the null hypothesis in the setting of a triangular array of rowwise independent and identically distributed random variables as well as consistency of the corresponding test against general alternatives. Therefore, a sound theoretical basis is provided for the parametric bootstrap procedure, which is applied to obtain critical values in a large-scale simulation study. Each of the tests considered in this study, when implemented via the parametric bootstrap method, maintains a nominal level of significance very closely, even for small sample sizes. The procedures are applied to four well-known data sets.     

Modified Shewhart Charts for High Yield Processes

The conventional Shewhart p or np chart is not effective for monitoring a high yield process, a process in which the defect level is close to zero. An improved Shewhart np chart for monitoring high yield processes is proposed. A review of control charts for monitoring high yield processes is first given. The run length performance of the proposed Shewhart chart is then compared with other high yield control charts. A simple procedure for designing the chart for processes subjected to sampling or 100% continuous inspection is provided and this allows the chart to be implemented easily on the factory floor. The practical aspects of implementation of the Shewhart chart are discussed. An application of the Shewhart chart based on a real data set is demonstrated.     

Some new classes of multivariate survival distribution functions

The concept of the univariate mean remaining life (m.r.l.) function is generalized to the multivariate case. The multivariate mean remaining life (m.m.r.l.) function is utilized to introduce four new classes of multivariate survival distribution functions (s.d.f.'s). Each of these classes is a new generalization of the univariate decreasing mean remaining life (DMRL) class of s.d.f.'s. The duals of these classes are introduced. Some properties, physical interpretation, and relationships among these classes are investigated. Also for each case, the class of s.d.f.'s common in a class and its dual is characterized.     

Invariant methods for estimating variance components in mixed linear models 2

The subject of this contribution is to present a survey on new methods for variance component estimation, which appeared in the literature in recent years. Starting from mixed models treated in analysis of variance research work on this field turned over to a more general approach in which the covariance matrix of the vector of observations is assumed to be a unknown linear combination of known symmetric matrices. Much interest has been shown in developing some kinds op optimal estimators for the unknown parameters and most results were obtained for estimators being invariant with respect to a certain group of translations. Therefore we restrict attention to this class of estimates. We will deal with minimum variance unbiased estimators, least squared errors estimators, maximum likelihood estimators. Bayes quadratic estimators and show some relations to the mimimum norm quadratic unbiased estimation principle (MINQUE) introduced by C. R. Rao [20]. We do not mention the original motivation of MINQUE since the otion of minimum norm depends on a measure that is not accepted by all statisticians. Also we do‘nt deal with other approaches like the BAYEsian and fiducial methods which were successfully applied by S. Portnoy [18], P. Rusolph [22], G. C. Tiao, W. Y. Tan [28], M. J. K. Healy [9] and others, although in very special situations, only. Additionally we add some new results and also new insight in the properties of known estimators. We give a new characterization of MINQUE in the class of all estimators, extend explicite expressions for locally optimal quadratic estimators given by C. R. Rao [22] to a slightly more general situation and prove complete class theorems useful for the computation of BAYES quadratic estimators. We also investigate situations in which BAYES quadratic unbiased estimators do'nt change if the distribution of the error terms differ from the normal distribution.     

Seemingly unrelated regression on the autoregressive (Ar(p)) singular equation system

This paper studies the estimation of seemingly unrelated regressions (SUR) of singular equation systems with an autoregressive error process (AR(p)) for each equation.Parameter estimates of the autoregressive singular equation system are not generally invariant to the equation deleted. Under the model specification restriction on the autoregressive parameters, the invariance property is preserved, and this paper shows that a single equation generalized least squares (GLS) estimation for a general autoregressive error process is equivalent to the SURGLS estimation of the AR(p) singular equation system.