Monitoring process mean level using auxiliary information

In this study, a Shewhart‐type control chart is proposed for the improved monitoring of process mean level (targeting both moderate and large shifts which is the major concern of Shewhart‐type control charts) of a quality characteristic of interest Y. The proposed control chart, namely the M_r chart, is based on the regression estimator of mean using a single auxiliary variable X. Assuming bivariate normality of (Y, X), the design structure of M_r chart is developed for phase I quality control. The comparison of the proposed chart is made with some existing control charts used for the same purpose. Using power curves as a performance measure, better performance of the proposedM_r chart is observed for detecting the shifts in mean level of the characteristic of interest.     

Beware of averages!

Summary Let X₁,.,., X_m, and Y₁, Y_n, be two independent samples from the same distribution and let X and Y be the means of these samples. What is the maximal value of P(X < Y)?     

Two-sample rank tests with truncated populations

LetX ₁,…,X _m andY ₁,…,Y _n be two independent samples from continuous distributionsF andG respectively. Using a Hoeffding (1951) type theorem, we obtain the distributions of the vector S=(S ₍₁₎,…,S _(n)), whereS _(j)=# (X _i ’s≤Y _(j)) andY _(j) is thej-th order statistic ofY sample, under three truncation models: (a)G is a left truncation ofF orG is a right truncation ofF, (b)F is a right truncation ofH andG is a left truncation ofH, whereH is some continuous distribution function, (c)G is a two tail truncation ofF. Exploiting the relation between S and the vectorR of the ranks of the order statistics of theY-sample in the pooled sample, we can obtain exact distributions of many rank tests. We use these to compare powers of the Hajek test (Hajek 1967), the Sidak Vondracek test (1957) and the Mann-Whitney-Wilcoxon test. We derive some order relations between the values of the probagility-functions under each model. Hence find that the tests based onS ₍₁₎ andS _(n) are the UMP rank tests for the alternative (a). We also find LMP rank tests under the alternatives (b) and (c).     

Evaluation of P(Xt>Yt ) when both Xt and Yt are residual lifetimes of two systems

Let the random variables X and Y denote the lifetimes of two systems. In reliability theory to compare between the lifetimes of X and Y there are several approaches. Among the most popular methods of comparing the lifetimes are to compare the survival functions, the failure rates and the mean residual lifetime functions of X and Y. Assume that both systems are operating at time t > 0. Then the residual lifetimes of them are X_t=X?t | X>t and Y_t=Y?t | Y>t, respectively. In this paper, we introduce, by taking into account the age of systems, a time‐dependent criterion to compare the residual lifetimes of them. In other words, we concentrate on function R(t ):=P(X_t>Y_t) which enables one to obtain, at time t, the probability that the residual lifetime X_t is greater than the residual lifetime Y_t. It is mentioned, in Brown and Rutemiller (IEEE Transactions on Reliability, 22 , 1973) that the probability of type R(t) is important for designing as long‐lived a product as possible. Several properties of R(t) and its connection with well‐known reliability measures are investigated. The estimation of R(t) based on samples from X and Y is also discussed.     

Estimation of P[Y < X] for generalized exponential distribution

This paper deals with the estimation of P[Y < X] when X and Y are two independent generalized exponential distributions with different shape parameters but having the same scale parameters. The maximum likelihood estimator and its asymptotic distribution is obtained. The asymptotic distribution is used to construct an asymptotic confidence interval of P[Y < X]. Assuming that the common scale parameter is known, the maximum likelihood estimator, uniformly minimum variance unbiased estimator and Bayes estimator of P[Y < X] are obtained. Different confidence intervals are proposed. Monte Carlo simulations are performed to compare the different proposed methods. Analysis of a simulated data set has also been presented for illustrative purposes.Part of the work was supported by a grant from the Natural Sciences and Engineering Research Council     

Characterization of the multivariate normal distribution by conditional normal distributions

Summary LetX andY be two random vectors with values in ℝ ^k and ℝ∝, respectively. IfZ=(X ^T,Y ^T) ^T is multivariate normal thenX givenY=y andY givenX=x are (multivariate) normal; the converse is wrong. In this paper simple additional conditions are stated such that the converse is true, too. Furthermore, the case is treated that the random vectorZ=(X ₁ ^T , …,X _t ^T ) ^T is splitted intot≥3 partsX ₁, …,X _t.     

One-Sample Bayesian Predictive Interval of Future Ordered Observations for the Pareto Distribution

Nigm et al. (2003, statistics 37: 527–536) proposed Bayesian method to obtain predictive interval of future ordered observation Y _(j) (r < j≤ n ) based on the right type II censored samples Y ₍₁₎ < Y ₍₂₎ < ... < Y _(r) from the Pareto distribution. If some of Y ₍₁₎ < ... < Y _(r-1) are missing or false due to artificial negligence of typist or recorder, then Nigm et al.’s method may not be an appropriate choice. Moreover, the conditional probability density function (p.d.f.) of the ordered observation Y _(j) (r < j ≤ n ) given Y ₍₁₎ <Y ₍₂₎ < ... < Y _(r) is equivalent to the conditional p.d.f. of Y _(j) (r < j ≤ n ) given Y _(r). Therefore, we propose another Bayesian method to obtain predictive interval of future ordered observations based on the only ordered observation Y _(r), then compares the length of the predictive intervals when using the method of Nigm et al. (2003, statistics 37: 527–536) and our proposed method. Numerical examples are provided to illustrate these results.     

Locally minimax test of independence in elliptically symmetrical distributions with additional observations

Summary LetX=(X _ij )=(X ₁, ...,X _n )’,X’ _i =(X _i1, ...,X _ip )’,i=1,2, ...,n be a matrix having a multivariate elliptical distribution depending on a convex functionq with parameters, 0,σ. Let ϱ²=ϱ ₂ ^-2 be the squared multiple correlation coefficient between the first and the remainingp ₂+p ₃=p−1 components of eachX _i . We have considered here the problem of testingH ₀:ϱ²=0 against the alternativesH ₁:ϱ ₁ ^-2 =0, ϱ ₂ ^-2 >0 on the basis ofX andn ₁ additional observationsY ₁ (n ₁×1) on the first component,n ₂ observationsY ₂(n ₂×p ₂) on the followingp ₂ components andn ₃ additional observationsY ₃(n ₃×p ₃) on the lastp ₃ components and we have derived here the locally minimax test ofH ₀ againstH ₁ when ϱ ₂ ^-2 →0 for a givenq. This test, in general, depends on the choice ofq of the familyQ of elliptically symmetrical distributions and it is not optimality robust forQ.     

On Some Nonparametric Tests of Dispersion

Let X and Y be random variables with distribution functions F and G and medians ζand η, respectively. X will be said to be more dispersed about the median than Y if F(x+ζ) – F(–x+ζ) G (x+η) – G (–x+η)for all x > O, with sharp inequality holding for some positive x. Asymptotically nonparametric tests for the hypothesis that F (x+ζ) = G (x+η) against alternatives of the above type (for ζ and η unknown), provided F(x+ζ) (but not necessarily G (x+η)) is symmetric and Fand G have bounded densities, are constructed from a subclass Of SUKHATME'S modified generalized U - statistics [12]. One simple test of this subclass turns out to be the MANN-WHITNEY test as performed on the |X_i–ζ| and | Y_j–η| where ζ and η are the corresponding sample medians. Consistency and PITMAN efficiencies of some of the tests proposed are discussed under two families of alternatives in which the Y's differ from the X's through departure from symmetry as well as by being more (less) dispersed: a model of “heterogeneous response” and the wellknown gross error model. The usual scale alternatives are also studied and the relationship of our tests to those previously given in the literature is pointed out.     

Estimating the conditional mode of a stationary stochastic process from noisy observations

Let {(X _i, Y _i,)}, i≥1, be a strictly stationary process from noisy observations. We examine the effect of the noise in the response Y and the covariates X on the nonparametric estimation of the conditional mode function. To estimate this function we are using deconvoluting kernel estimators. The asymptotic behavior of these estimators depends on the smoothness of the noise distribution, which is classified as either ordinary smooth or super smooth. Uniform convergence with almost sure convergence rates is established for strongly mixing stochastic processes, when the noise distribution is ordinary smooth. Received: April 1998     

The association between two random elements: A complete characterization and odds ratio models

For random elements X and Y (e.g. vectors) a complete characterization of their association is given in terms of an odds ratio function. The main result establishes for any odds ratio function and any pre-specified marginals the unique existence of a corresponding joint distribution (the joint density is obtained as a limit of an iterative procedure of marginal fittings). Restricting only the odds ratio function but not the marginals leads to semi-parmetric association models for which statistical inference is available for samples drawn conditionally on either X or Y. Log-bilinear association models for random vectors X and Y are introduced which generalize standard (regression) models by removing restrictions on the marginals. In particular, the logistic regression model is recognized as a log-bilinear association model. And the joint distribution of X and Y is shown to be multivariate normal if and only if both marginals are normal and the association is log-bilinear.Acknowledgements The author thanks both referees for their helpful comments which improved the first draft of the paper.     

CUMIN charts

Classical control charts are very sensitive to deviations from normality. In this respect, nonparametric charts form an attractive alternative. However, these often require considerably more Phase I observations than are available in practice. This latter problem can be solved by introducing grouping during Phase II. Then each group minimum is compared to a suitable upper limit (in the two-sided case also each group maximum to a lower limit). In the present paper it is demonstrated that such MIN charts allow further improvement by adopting a sequential approach. Once a new observation fails to exceed the upper limit, its group is aborted and a new one starts right away. The resulting CUMIN chart is easy to understand and implement. Moreover, this chart is truly nonparametric and has good detection properties. For example, like the CUSUM chart, it is markedly better than a Shewhart X-chart, unless the shift is really large.     

Regression analysis and dependence

In this paper we study the relationship between regression analysis and a multivariate dependency measure. If the general regression model Y=f() holds for some function f, where 1i₁< i₂<···i_m k, and X₁,...,X_k is a set of possible explanatory random variables for Y. Then there exists a dependency relation between the random variable Y and the random vector (). Using the dependency statistic defined below, we can detect such dependency even if the function f is not linear. We present several examples with real and simulated data to illustrate this assertion. We also present a way to select the appropriate subset among the random variables X₁,X₂,...,X_k, which better explain Y.     

Three dual regression schemes for the Lukacs theorem

In the paper we study regressional versions of Lukacs' characterization of the gamma law. We consider constancy of regression instead of Lukacs' independence condition in three new schemes. Up to now the constancy of regressions of U=X/(X + Y) given V=X + Y for independent X and Y has been considered in the literature. Here we are concerned with constancy of regressions for X and Y while independence of U and V is assumed instead.     

Strong consistency of the MLE under random censoring

LetX ₁, ...,X _n be an i.i.d. sample from some parametric family {_θ :θ (Θ} of densities. In the random censorship model one observesZ _i =min (X _i ,Y _i ) andδ _i =1_{ x _i ≤Y _i}, whereY _i is a censoring variable being independent ofX _i . In this paper we investigate the strong consistency ofθ _n maximizing the modified likelihood function based on (Z _i ,δ _i , 1≤i≤n. The main result constitutes an extension of Wald’s theorem for complete data to censored data. Work partially supported by the “Deutsche Forschungsgemeinschaft”.     

On testing for independence against right tail increasing in bivariate models

A random variableY is right tail increasing (RTI) inX if the failure rate of the conditional distribution ofX givenY>y is uniformly smaller than that of the marginal distribution ofX for everyy0. This concept of positive dependence is not symmetric inX andY and is stronger than the notion of positive quadrant dependence. In this paper we consider the problem of testing for independence against the alternative thatY is RTI inX. We propose two distribution-free tests and obtain their limiting null distributions. The proposed tests are compared to Kendall's and Spearman's tests in terms of Pitman asymptotic relative efficiency. We have also conducted a Monte Carlo study to compare the powers of these tests.Research supported by an NSERC Canada operating grant at the University of Alberta.Part of this research was done while visiting the University of Alberta supported by the NSERC Canada grant of the first author.     

Optimal process parameters under LINEX loss function with general input quality characteristic

In this paper we generalize the quality and cost trade-off problem of Chang and Hung (Qual Quant 41: 291–301, 2007) under the LINEX loss function. We consider the general input characteristic given by the random variable X with moment generating function m _X (t) and output characteristic given by the deterministic transformation Y  =  g(X). The two cases we consider are when g(X) is an affine function of X and X follows (1) the gamma distribution, and (2) the double exponential distribution.     

On the estimation ofPr{Y<X} for the two-parameter exponential distribution

In this paper the Blackwell-Rao and Lehmann-Scheffé theorems are used to derive the minimum variance unbiased estimator ofP=Pr{Y when the independent random variablesX andY follow the two-parameter exponential distribution. Following a Bayesian approach, an estimator ofP is also obtained for this distribution. These results are extended for the case of censored samples.     

On the Ratio of Fréchet Random Variables

The distribution of the ratio X/Y is derived when X and Y are independent Fréchet random variables. Extensive tabulations of the associated percentage points are also given.