ON THE EVALUATION OF CONTROL CHART LIMITS BASED ON PREDICTIVE DISTRIBUTIONS

ABSTRACT

We propose a Bayesian approach to obtaining control charts when there is parameter uncertainty. Our approach consists of two stages, (i) construction of the control chart where we use a predictive distribution based on a Bayesian approach to derive the rejection region, and (ii) evaluation of the control chart where we use a sampling theory approach to examine the performance of the control chart under various hypothetical specifications for the data generation model.     

A NOTE ON CHARACTERIZATIONS OF DISTRIBUTIONS BY REGRESSIONS OF NON-ADJACENT GENERALIZED ORDER STATISTICS

We investigate the problem of characterizations of distributions by regressions of generalized order statistics (GOSs) based on a continuous distribution function F . We show that F is uniquely determined if the regressions of a GOS given two other consecutive GOSs are known.     

ASYMPTOTIC DISTRIBUTIONS OF MATCHED-PARIS PERMUTATIO STATISTICS BASED ON DISTANCE MEASURES

A technique for deriving asymptotic distributions of matched-pairs permutation statistics based on distance measures is discussed and applied to four specific examples recently considered by Mielke & Berry (1982).     

CHARACTERISATIONS OF SOME INCOME DISTRIBUTIONS BASED ON MULTIPLICATIVE DAMAGE MODELS

This paper characterises the Pareto and scaled beta distributions within the context of multiplicative damage and generating models. The results obtained allow the destructive or generating mechanism to have more general distributions on (0,1) than the beta(e, l) distribution considered by several authors, thus generalising some recent results. Errors in earlier work are mentioned.     

ON CHARACTERIZING DISTRIBUTIONS VIA LINEARITY OF REGRESSION FOR ORDER STATISTICS

Let X₁X_n be a random sample from an absolutely continuous distribution with the corresponding order statistics X_1:n≤X_2:n≤X_n:n. A complete solution of the problem, posed in 1967 by T. Ferguson, of determining the distribution by linearity of regression of X_k+2:n with respect to X_k:n is given. The only possible distributions are of the exponential, power and Pareto type. A linear regression relation for exponents of order statistics is also considered.     

FINITE MIXTURE OF CERTAIN DISTRIBUTIONS

ABSTRACT

A two parameter extended form of standard gamma function is suggested which provide extra flexibility to the density function over positive range. A finite mixture of beta distribution is defined by using the suggested extended form. The shape of density function of extended gamma distribution and also that of finite mixture of beta distribution for various values of the parameters are shown. Inverted distribution of extended gamma and that of finite mixture of beta distribution are given.     

MODEL‐BASED DIRECT ESTIMATION OF SMALL‐AREA DISTRIBUTIONS

Much of the small‐area estimation literature focuses on population totals and means. However, users of survey data are often interested in the finite‐population distribution of a survey variable and in the measures (e.g. medians, quartiles, percentiles) that characterize the shape of this distribution at the small‐area level. In this paper we propose a model‐based direct estimator (MBDE, Chandra and Chambers) of the small‐area distribution function. The MBDE is defined as a weighted sum of sample data from the area of interest, with weights derived from the calibrated spline‐based estimate of the finite‐population distribution function introduced by Harms and Duchesne, under an appropriately specified regression model with random area effects. We also discuss the mean squared error estimation of the MBDE. Monte Carlo simulations based on both simulated and real data sets show that the proposed MBDE and its associated mean squared error estimator perform well when compared with alternative estimators of the area‐specific finite‐population distribution function.     

A CHARACTERIZATION OF CONTINUOUS DISTRIBUTIONS VIA REGRESSION ON PAIRS OF RECORD VALUES

The exponential type is characterized in terms of the regression of a (possibly non‐linear) function of a record value with its adjacent record values as covariates. Monotone transformations extend this result to more general settings, and these are illustrated with some specific examples.     

A CUSUM SCHEME BASED ON THE EXPONENTIAL DISTRIBUTION FOR SURVEDLLANCE OF RARE CONGENITAL MALFORMATIONS

A method of monitoring the incidence of malformations is described. It is suitable for systems where the number of births between successive malformations is known or can be estimated with reasonable accuracy. The method utilises a cusum technique based on the exponential distribution to detect an increase in the incidence of malformations above a baseline level. Adequate information to enable the implementation of the method is presented. The proposed method compares favourably with others such as the Poisson cusum and the modified sets technique.     

A NEW APPROACH TO MAXIMUM LIKELIHOOD ESTIMATION OF THE THREE-PARAMETER GAMMA AND WEIBULL DISTRIBUTIONS

A new approach, is proposed for maximum likelihood (ML) estimation in continuous univariate distributions. The procedure is used primarily to complement the ML method which can fail in situations such as the gamma and Weibull distributions when the shape parameter is, at most, unity. The new approach provides consistent and efficient estimates for all possible values of the shape parameter. Its performance is examined via simulations. Two other, improved, general methods of ML are reported for comparative purposes. The methods are used to estimate the gamma and Weibull distributions using air pollution data from Melbourne. The new ML method is accurate when the shape parameter is less than unity and is also superior to the maximum product of spacings estimation method for the Weibull distribution.     

ASYMPTOTIC OPTIMALITY OF LIKELIHOOD RATIO TESTS FOR EXPONENTIAL DISTRIBUTIONS UNDER TYPE II CENSORING

Consider the model of k two parameter exponential populations under a type II censoring scheme. In this paper we establish optimality in the sense of Bahadur of the likelihood ratio test for an arbitrary testing problem by requiring only Condition A in Hsieh (1979). This result unifies and generalizes the results in Samanta (1986).     

ON THE INDEPENDENCE OF THE SAMPLE MEAN AND TRANSLATION-INVARIANT STATISTICS FOR MATRIX NORMAL DISTRIBUTIONS

We present an explicit characterization of the joint dependency structure of an n×p matrix normal random matrix such that the p-dimensional sample mean vector is independent of all translation invariant statistics.     

DETERMINATION OF DOMAINS OF ATTRACTION BASED ON A SEQUENCE OF MAXIMA

Suppose that the maximum of a random sample from a distribution F(x) may be obtained in each of k equally spaced observation periods. This paper proposes a test to determine the domain of attraction of F(x), and investigates the properties when the sample size is very large and perhaps unknown and k is fixed and small. The test statistic is a function of the spacings between the order statistics based on the sequence of maxima and is suggested by reference to one studied previously when inference was based on the largest k observations of a random sample. A Monte Carlo study shows that the proposed test is more powerful than its main competitor. The test is illustrated by two examples.     

CONSTRUCTION OF RESOLVABLE ROW-COLUMN DESIGNS BASED ON CONTRACTIONS

This paper defines the contraction of a resolvable row‐column design for more than two replicates. It shows that the (M,S)‐optimality criterion for the row‐column designs can be expressed simply in terms of the elements of the row and column incidence matrices of the contraction. This allows the development of a very fast algorithm to construct optimal or near‐optimal resolvable row‐column designs. The performance of such an algorithm is compared with an existing algorithm.     

ON THE NULL DISTRIBUTIONS OF THE ENTROPY TESTS FOR THE GAUSSIAN AND INVERSE GAUSSIAN MODELS

Vasicek [1] Vasicek, O. 1976. A test for normality based on sample entropy. J. R. Statist. Soc. B, 38: 54–59.  [Google Scholar]used the “convolution of twelve uniforms” for a Monte Carlo tabulation of the 5% critical values for his entropy test for normality. We employ a superior normal generator to construct a corrected and extended tabulation for his test. Interestingly, it is shown that, the same tables can be used for implementing Mudholkar and Tian's [2] Mudholkar, G. S. and Tian, L. 1999. “An entropy characterization of the inverse Gaussian distribution and related goodness-of-fit test”. In Tech. Rep., University of Rochester Rochester, NY Submitted for publication [Google Scholar]entropy test for the composite inverse Gaussian hypothesis. The finding extends the known Gaussian, inverse Gaussian analogies.     

THE EFFECTS OF CORRELATION AMONG OBSERVATIONS ON THE ACCURACY OF APPROXIMATING THE DISTRIBUTION OF SAMPLE MEAN BY ITS ASYMPTOTIC DISTRIBUTION1

In this paper, we investigate the effects of correlation among observations on the accuracy of approximating the distribution of sample mean by its asymptotic distribution. The accuracy is investigated by the Berry-Esseen bound (BEB), which gives an upper bound on the error of approximation of the distribution function of the sample mean from its asymptotic distribution for independent observations. For a given sample size (n₀) the BEB is obtained when the observations are independent. Let this be BEB. We then find the sample size (n_*) required to have BEB below BEB₀, when the observations are dependent. Comparison of n_* with n₀ reveals the effects of correlation among observations on the accuracy of the asymptotic distribution as an approximation. It is shown that the effects of correlation among observations are not appreciable if the correlation is moderate to small but it can be severe for extreme correlations.     

ON QUANTILES OF SUMS

In this paper we investigate the relationship between the quantiles of a sum of independent continuous random variables and those of its components. Results concerning this relationship are given for the special cases of symmetric distributions, gamma distributions, and for the difference of identically distributed random variables.