Evaluation of the bias of the isotonic regression

The systematic error (bias) of the isotonic regression analysis of temporal spacings between failure events is investigated by means of numerical simulation. Spacings that are sampled from an exponential distribution with a constant failure rate (CFR) arc subjected to an isotonic regression search for a declining failure rate (DFR). The results indicate a considerable declining trend (bias) that is imposed upon these CFR-data by isotonic regression analysis. The corresponding results for an increasing trend can be readily obtained through transformation. For practical applications, the results of 100,000 simulations have been approximated by simple analytical expressions. For the evaluation of a trend in a specific set of isotonized spacings (or rates) the results of the latter analysis can be compared with the isotonic bias of a set of CFR data for the same number of events. Alternatively, the specific set of isotonized spacings can be suitably related to the corresponding isotonized CFR data to reduce the bias by largely eliminating the CFR contribution.     

On the quality of reinterview data with application to the Current Population Survey

"The [U.S.] Current Population Survey (CPS) reinterview sample consists of two subsamples: (a) a sample of CPS households is reinterviewed and the discrepancies between the reinterview responses and the original interview responses are reconciled for the purpose of obtaining more accurate responses..., and (b) a sample of CPS households, nonoverlapping with sample (a), is reinterviewed 'independently' of the original interview for the purpose of estimating simple response variance (SRV). In this article a model and estimation procedure are proposed for obtaining estimates of SRV from subsample (a) as well as the customary estimates of SRV from subsample (b).... Data from the CPS reinterview program for both subsamples (a) and (b) are analyzed both (1) to illustrate the methodology and (2) to check the validity of the CPS reinterview data. Our results indicate that data from subsample (a) are not consistent with the data from subsample (b) and provide convincing evidence that errors in subsample (a) are the source of the inconsistency."     

On the minimum expected quantity for the validity of the chi-squared test in 2 2 2 tables

A 2 2 2 contingency table can often be analysed in an exact fashion by using Fisher's exact test and in an approximate fashion by using the chi-squared test with Yates' continuity correction, and it is traditionally held that the approximation is valid when the minimum expected quantity E is E S 5. Unfortunately, little research has been carried out into this belief, other than that it is necessary to establish a bound E>E*, that the condition E S 5 may not be the most appropriate (Martín Andrés et al., 1992) and that E* is not a constant, but usually increasing with the growth of the sample size (Martín Andrés & Herranz Tejedor, 1997). In this paper, the authors conduct a theoretical experimental study from which they ascertain that E* value (which is very variable and frequently quite a lot greater than 5) is strongly related to the magnitude of the skewness of the underlying hypergeometric distribution, and that bounding the skewness is equivalent to bounding E (which is the best control procedure). The study enables estimating the expression for the above-mentioned E* (which in turn depends on the number of tails in the test, the alpha error used, the total sample size, and the minimum marginal imbalance) to be estimated. Also the authors show that E* increases generally with the sample size and with the marginal imbalance, although it does reach a maximum. Some general and very conservative validity conditions are E S 35.53 (one-tailed test) and E S 7.45 (two-tailed test) for alpha nominal errors in 1% h f h 10%. The traditional condition E S 5 is only valid when the samples are small and one of the marginals is very balanced; alternatively, the condition E S 5.5 is valid for small samples or a very balanced marginal. Finally, it is proved that the chi-squared test is always valid in tables where both marginals are balanced, and that the maximum skewness permitted is related to the maximum value of the bound E*, to its value for tables with at least one balanced marginal and to the minimum value that those marginals must have (in non-balanced tables) for the chi-squared test to be valid.     

An approximation for the hotelling-lawley trace in the noncentral case

A procedure for estimating power in conjunction with the Hotelling-Lawley trace is developed. By approximating a non-central Wishart distribution with a central Wishart, and using McKeon's (1974) F-type approximation, a relatively simple procedure for obtaining power estimates is obtained. The accuracy of the approximation is investigated by comparing the approximate results with those for a wide range of conditions given in Olson's (1973) extensive Monte Carlo study. Siotani's (1971) asymptotic expansion is used to provide further comparative assessments. It is demonstrated that the approximation is of sufficient accuracy to be used in practical applications.     

On the strengths of the self-updating process clustering algorithm

The self-updating process (SUP) is a clustering algorithm that stands from the viewpoint of data points and simulates the process how data points move and perform self-clustering. It is an iterative process on the sample space and allows for both time-varying and time-invariant operators. By simulations and comparisons, this paper shows that SUP is particularly competitive in clustering (i) data with noise, (ii) data with a large number of clusters, and (iii) unbalanced data. When noise is present in the data, SUP is able to isolate the noise data points while performing clustering simultaneously. The property of the local updating enables SUP to handle data with a large number of clusters and data of various structures. In this paper, we showed that the blurring mean-shift is a static SUP. Therefore, our discussions on the strengths of SUP also apply to the blurring mean-shift.     

Comments on two papers concerning estimation of the parameters of the Pareto distribution in the presence of outliers

In this paper, we examine and correct various results relating to estimation of a Pareto distribution in the presence of outliers according to a model introduced by Dixit and Jabbari Nooghabi (2011) [1] and further studied by Dixit and Jabbari Nooghabi (2011) [2]. In particular, Dixit and Jabbari Nooghabi (2011) [2] state that the maximum likelihood estimators for the parameters appearing in their model do not exist. We show that these estimators can in fact exist, and we present and illustrate a method for determining them when they do. Two numerical illustrations using actual insurance data are included.     

Threshold Cointegration and the PPP Hypothesis

Self-Exciting Threshold Autoregressive (SETAR) models are a non-linear variant of conventional linear Autoregressive (AR) models. One advantage of SETAR models over conventional AR models lies in its flexible nature in dealing with possible asymmetric behaviour of economic variables. The concept of threshold cointegration implies that the Error Correction Mechanism (ECM) at a particular interval is inactive as a result of adjustment costs, and active when deviations from equilibrium exceed certain thresholds. For instance, the presence of adjustment costs can, in many circumstances, justify the fact that economic agents intervene to recalibrate back to a tolerable limit, as in the case when the benefits of adjustment are superior to its costs. We introduce an approach that accounts for potential asymmetry and we investigate the presence of the relative version of the purchasing power parity (PPP) hypothesis for 14 countries. Based on a threshold cointegration adaptation of the unit root test procedure suggested by Caner & Hansen (2001), we find evidence of an asymmetric adjustment for the relative version of PPP for eight pairs of countries.     

Small sample distribution of the likelihood ratio test in the random effects model

In this paper we examine the small sample distribution of the likelihood ratio test in the random effects model which is often recommended for meta-analyses. We find that this distribution depends strongly on the true value of the heterogeneity parameter (between-study variance) of the model, and that the correct p-value may be quite different from its large sample approximation. We recommend that the dependence of the heterogeneity parameter be examined for the data at hand and suggest a (simulation) method for this. Our setup allows for explanatory variables on the study level (meta-regression) and we discuss other possible applications, too. Two data sets are analyzed and two simulation studies are performed for illustration.     

Choosing Between the Sample-Selection Model and the Multi-Part Model

Hay and Olsen (1984) incorrectly argue that a multi-part model, the two-part model used in Duan et al. (1982,1983), is nested within the sample-selection model. Their proof relies on an unmentioned restrictive assumption that cannot be satisfied. We provide a counterexample to show that the propensity to use medical care and the level of expense can be positively associated in the two-part model, contrary to their assertion. The conditional specification in the multi-part model is preferable to the unconditional specification in the selection model for modeling actual (v. potential) outcomes. The selection model also has poor statistical and numerical properties and relies on untestable assumptions. Empirically the multi-part estimators perform as well as or better than the sample selection estimator for the data set analyzed in Duan et al. (1982, 1983).     

On the use of the scholz-sievers method for dlstribution-free inference about the difference in slopes of two regression lines

A simple adaptation of a distribution-free method due to Scholz (1978) and Sievers (1978) for inference in a single regression setting is proposed for inference about the difference in slopes of two regression lines. We assume that the data are obtained from a designed experiment with common regression constants. A comparison of the proposed method to its competitors-one due to Hollander and the other due to Rao and Gore-indicates superiority of the new method.     

Conditions for the coincidence of the TFR,TRV and CE models

Bhattacharyya and Soejoeti (1989) put forward the tampered failure rate model (TFR Model) for step-stress Accelerated Life Tests(ALT). This paper studies the conditions for the coincidence of the TRV, TFR and CE models, gives the definitions of the coincidence, offers a proof of the necessary and sufficient condition for the coincidence of the TRV and TFR models in [1], and points out a mistake that appeared in the counterexample provided in [3].     

A model of the ring-recovery reporting process for the Cape Griffon Gyps coprotheres

By their very nature, statistical models are constructed on the basis of a number of simplifying assumptions. It is usual to assume that all the individuals in a 'group' or 'cohort' have similar survival, recovery or reporting probabilities. From a number of earlier studies of the Cape Griffon Gyps coprotheres in southern Africa, it is clear that there have been many violations of these assumptions of homogeneity. To get a better understanding of the process whereby a dead ringed bird is found and reported, an analysis was undertaken of 575 recoveries from 7130 individuals ringed as nestlings. From a series of univariate generalized linear models, it was found that the proportion of ringed birds reported dead varied with the following factors: (1) ring prefix (representing different grades and thicknesses of aluminium): there was considerable variation in reporting rate between cohorts fitted with different ring prefix series used; (2) metal type: birds fitted with monel metal rings were reported at a rate twice that of those bearing aluminium rings; (3) colour rings: recoveries of birds with colour rings were much more likely to be reported than birds with only a metal ring; (4) epoch: the reporting rate has increased steadily from the 1950s through to the mid-1980s. All of these factors are confounded and so a number of multivariate generalized linear models were constructed. It was found that the variations in the cohort-specific reporting rate could be described well by a model including factors for metal-ring type and the presence or absence of colour rings. Using the tougher monel metal ring along with a set of colour rings more than doubles the reporting rate and their continued use is strongly recommended for future studies. The year-to-year variations could be accounted for by this model but the colony of ringing did not enter the model. The models used were based on two assumptions: (i) the reporting rate was constant for all individuals within a given cohort and (ii) the recoveries were complete. It is argued that the results are congruent with these assumptions. There is now a clearer model of the manner in which the ring-recovery reporting process proceeds and this has opened the way to building a more realistic statistical model to estimate survival in the Cape Griffon.     

Modified inference about the mean of the exponential distribution using moving extreme ranked set sampling

The maximum likelihood estimator (MLE) and the likelihood ratio test (LRT) will be considered for making inference about the scale parameter of the exponential distribution in case of moving extreme ranked set sampling (MERSS). The MLE and LRT can not be written in closed form. Therefore, a modification of the MLE using the technique suggested by Maharota and Nanda (Biometrika 61:601–606, 1974) will be considered and this modified estimator will be used to modify the LRT to get a test in closed form for testing a simple hypothesis against one sided alternatives. The same idea will be used to modify the most powerful test (MPT) for testing a simple hypothesis versus a simple hypothesis to get a test in closed form for testing a simple hypothesis against one sided alternatives. Then it appears that the modified estimator is a good competitor of the MLE and the modified tests are good competitors of the LRT using MERSS and simple random sampling (SRS).     

Pseudo maximum likelihood estimation for the dirichlet-multinomial distribution

Pseudo maximum likelihood estimation (PML) for the Dirich-let-multinomial distribution is proposed and examined in this pa-per. The procedure is compared to that based on moments (MM) for its asymptotic relative efficiency (ARE) relative to the maximum likelihood estimate (ML). It is found that PML, requiring much less computational effort than ML and possessing considerably higher ARE than MM, constitutes a good compromise between ML and MM. PML is also found to have very high ARE when an estimate for the scale parameter in the Dirichlet-multinomial distribution is all that is needed.     

A model of the ring-recovery reporting process for the Cape Griffon Gyps coprotheres

By their very nature, statistical models are constructed on the basis of a number of simplifying assumptions. It is usual to assume that all the individuals in a 'group' or 'cohort' have similar survival, recovery or reporting probabilities. From a number of earlier studies of the Cape Griffon Gyps coprotheres in southern Africa, it is clear that there have been many violations of these assumptions of homogeneity. To get a better understanding of the process whereby a dead ringed bird is found and reported, an analysis was undertaken of 575 recoveries from 7130 individuals ringed as nestlings. From a series of univariate generalized linear models, it was found that the proportion of ringed birds reported dead varied with the following factors: (1) ring prefix (representing different grades and thicknesses of aluminium): there was considerable variation in reporting rate between cohorts fitted with different ring prefix series used; (2) metal type: birds fitted with monel metal rings were reported at a rate twice that of those bearing aluminium rings; (3) colour rings: recoveries of birds with colour rings were much more likely to be reported than birds with only a metal ring; (4) epoch: the reporting rate has increased steadily from the 1950s through to the mid-1980s. All of these factors are confounded and so a number of multivariate generalized linear models were constructed. It was found that the variations in the cohort-specific reporting rate could be described well by a model including factors for metal-ring type and the presence or absence of colour rings. Using the tougher monel metal ring along with a set of colour rings more than doubles the reporting rate and their continued use is strongly recommended for future studies. The year-to-year variations could be accounted for by this model but the colony of ringing did not enter the model. The models used were based on two assumptions: (i) the reporting rate was constant for all individuals within a given cohort and (ii) the recoveries were complete. It is argued that the results are congruent with these assumptions. There is now a clearer model of the manner in which the ring-recovery reporting process proceeds and this has opened the way to building a more realistic statistical model to estimate survival in the Cape Griffon.     

On the distribution of independence test for the residuals of space-time arma models

The procedure for building space-time autoregressive moving average (STARMA) models depends on the form of the variance-covariance matrix G of the underlying errors (see Pfeifer Deutsch (1980a,c)). In this paper the distribu¬tion of the statistic for testing the hypothesis that G is diagonal is obtained in a very convenient computational form. A table of critical values for the test is given Comparison is made with the approximate values obtained by Pfeifer Deutsch (1980c)     

Efficient Estimation of the PDF and the CDF of the Weibull Extension Model

The Weibull extension model is a useful extension of the Weibull distribution, allowing for bathtub shaped hazard rates among other things. Here, we consider estimation of the PDF and the CDF of the Weibull extension model. The following estimators are considered: uniformly minimum variance unbiased (UMVU) estimator, maximum likelihood (ML) estimator, percentile (PC) estimator, least squares (LS) estimator, and weighted least squares (WLS) estimator. Analytical expressions are derived for the bias and the mean squared error. Simulation studies and real data applications show that the ML estimator performs better than others.     

Transformations for Testing the Fit of the Inverse-Gaussian Distribution

Abstract

Use of the MVUE for the inverse-Gaussian distribution has been recently proposed by Nguyen and Dinh [Nguyen, T. T., Dinh, K. T. (2003). Exact EDF goodnes-of-fit tests for inverse Gaussian distributions. Comm. Statist. (Simulation and Computation) 32(2):505–516] where a sequential application based on Rosenblatt's transformation [Rosenblatt, M. (1952). Remarks on a multivariate transformation. Ann. Math. Statist. 23:470–472] led the authors to solve the composite goodness-of-fit problem by solving the surrogate simple goodness-of-fit problem, of testing uniformity of the independent transformed variables. In this note, we observe first that the proposal is not new since it was proposed in a rather general setting in O'Reilly and Quesenberry [O'Reilly, F., Quesenberry, C. P. (1973). The conditional probability integral transformation and applications to obtain composite chi-square goodness-of-fit tests. Ann. Statist. I:74–83]. It is shown on the other hand that the results in the paper of Nguyen and Dinh (2003) are incorrect in their Sec. 4, specially the Monte Carlo figures reported. Power simulations are provided here comparing these corrected results with two previously reported goodness-of-fit tests for the inverse-Gaussian; the modified Kolmogorov–Smirnov test in Edgeman et al. [Edgeman, R. L., Scott, R. C., Pavur, R. J. (1988). A modified Kolmogorov-Smirnov test for inverse Gaussian distribution with unknown parameters. Comm. Statist. 17(B): 1203–1212] and the A ² based method in O'Reilly and Rueda [O'Reilly, F., Rueda, R. (1992). Goodness of fit for the inverse Gaussian distribution. T Can. J. Statist. 20(4):387–397]. The results show clearly that there is a large loss of power in the method explored in Nguyen and Dinh (2003) due to an implicit exogenous randomization.     

On the estimation of the mean vector of a multivariate normal distribution under symmetry

The problem of estimation of the mean vector of a multivariate normal distribution with unknown covariance matrix, under uncertain prior information (UPI) that the component mean vectors are equal, is considered. The shrinkage preliminary test maximum likelihood estimator (SPTMLE) for the parameter vector is proposed. The risk and covariance matrix of the proposed estimato are derived and parameter range in which SPTMLE dominates the usual preliminary test maximum likelihood estimator (PTMLE) is investigated. It is shown that the proposed estimator provides a wider range than the usual premilinary test estimator in which it dominates the classical estimator. Further, the SPTMLE has more appropriate size for the preliminary test than the PTMLE.     

A simple condition for the multivariate CLT and the attraction to the Gaussian of Lévy processes at long and short times

We show that a necessary and sufficient condition for the sum of iid random vectors to converge (under appropriate centering and scaling) to a multivariate Gaussian distribution is that the truncated second moment matrix is slowly varying at infinity. This is more natural than the standard conditions, and allows for the possibility that the limiting Gaussian distribution is degenerate (so long as it is not concentrated at a point). We also give necessary and sufficient conditions for a d-dimensional Lévy process to converge (under appropriate centering and scaling) to a multivariate Gaussian distribution as time approaches zero or infinity.