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A model-based multiple imputation approach for analyzing sample data with non-detects is proposed. The imputation approach involves randomly generating observations below the detection limit using the detected sample values and then analyzing the data using complete sample techniques, along with suitable adjustments to account for the imputation. The method is described for the normal case and is illustrated for making inferences for constructing prediction limits, tolerance limits, for setting an upper bound for an exceedance probability and for interval estimation of a log-normal mean. Two imputation approaches are investigated in the paper: one uses approximate maximum likelihood estimates (MLEs) of the parameters and a second approach uses simple ad hoc estimates that were developed for the specific purpose of imputations. The accuracy of the approaches is verified using Monte Carlo simulation. Simulation studies show that both approaches are very satisfactory for small to moderately large sample sizes, but only the MLE-based approach is satisfactory for large sample sizes. The MLE-based approach can be calibrated to perform very well for large samples. Applicability of the method to the log-normal distribution and the gamma distribution (via a cube root transformation) is outlined. Simulation studies also show that the imputation approach works well for constructing tolerance limits and prediction limits for a gamma distribution. The approach is illustrated using a few practical examples.

The problems of interval estimating the parameters and the mean of a two-parameter Rayleigh distribution are considered. We propose pivotal-based methods for constructing confidence intervals for the mean, quantiles, survival probability and for constructing prediction intervals for the mean of a future sample. Pivotal quantities based on the maximum likelihood estimates (MLEs), moment estimates (MEs) and the L-moments estimates (L-MEs) are proposed. Interval estimates based on them are compared via Monte Carlo simulation. Comparison studies indicate that the results based on the MEs and the L-MEs are very similar. The results based on the MLEs are slightly better than those based on the MEs and the L-MEs for small to moderate sample sizes. The methods are illustrated using an example involving lifetime data. [ABSTRACT FROM AUTHOR]

Standardized likelihood ratio test (SLRT) for testing the equality of means of several log-normal distributions is proposed. The properties of the SLRT and an available modified likelihood ratio test (MLRT) and a generalized variable (GV) test are evaluated by Monte Carlo simulation and compared. Evaluation studies indicate that the SLRT is accurate even for small samples, whereas the MLRT could be quite liberal for some parameter values, and the GV test is in general conservative and less powerful than the SLRT. Furthermore, a closed-form approximate confidence interval for the common mean of several log-normal distributions is developed using the method of variance estimate recovery, and compared with the generalized confidence interval with respect to coverage probabilities and precision. Simulation studies indicate that the proposed confidence interval is accurate and better than the generalized confidence interval in terms of coverage probabilities. The methods are illustrated using two examples. [ABSTRACT FROM AUTHOR]

An approximate closed-form one-sided tolerance limit (TL) in a general mixed model is proposed. One-sided TLs for the distribution of observable random variable and for the distribution of unobservable random variable in one-way random model are obtained as special cases from the one for the general mixed model. Applications to a two-way nested random model are also given. The merits of the TLs are evaluated using Monte Carlo simulation and compared with the existing ones. Our comparison studies indicate that the approximate TLs are quite satisfactory for all parameter and sample size configurations, and better than the existing ones in some cases. Approximate confidence intervals for exceedance probabilities in one-way random effects model are also proposed. The procedures are illustrated using three examples. [ABSTRACT FROM PUBLISHER]

Signed-likelihood ratio tests (SLRTs) are derived for testing the shape and scale parameters and the mean of a gamma distribution. The properties of the proposed tests are evaluated by Monte Carlo simulation and compared with the other tests available in the literature. SLRTs are also developed for two-sample problems of comparing two shape parameters, two scale parameters and two means, and their merits are evaluated by Monte Carlo simulation. Evaluation studies indicate that the SLRTs are accurate even for small samples and are comparable with or better than other tests. Furthermore, simple parametric bootstrap (PB) methods based on the maximum likelihood estimates are proposed for interval estimation. The PB confidence intervals are satisfactory even for small samples. The methods are illustrated and compared using two examples. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

The problem of testing the equality of coefficients of variation of independent normal populations is considered. For comparing two coefficients, we consider the signed-likelihood ratio test (SLRT) and propose a modified version of the SLRT, and a generalized test. Monte Carlo studies on the type I error rates of the tests indicate that the modified SLRT and the generalized test work satisfactorily even for very small samples, and they are comparable in terms of power. Generalized confidence intervals for the ratio of (or difference between) two coefficients of variation are also developed. A modified LRT for testing the equality of several coefficients of variation is also proposed and compared with an asymptotic test and a simulation-based small sample test. The proposed modified LRTs seem to be very satisfactory even for samples of size three. The methods are illustrated using two examples. [ABSTRACT FROM AUTHOR]

Abstract: Inferential procedures for estimating and comparing normal correlation coefficients based on incomplete samples with a monotone missing pattern are considered. The procedures are based on the generalized variable (GV) approach. It is shown that the GV methods based on complete or incomplete samples are exact for estimating or testing a simple correlation coefficient. Procedures based on incomplete samples for comparing two overlapping dependent correlation coefficients are also proposed. For both problems, Monte Carlo simulation studies indicate that the inference based on incomplete samples and those based on samples after listwise or pairwise deletion are similar, and the loss of efficiency by ignoring additional data is not appreciable. The proposed GV approach is simple, and it can be readily extended to other problems such as the one of estimating two non-overlapping dependent correlations. The results are illustrated using two examples. [Copyright &y& Elsevier]

Abstract: Exact methods for constructing two-sided tolerance intervals (TIs) and tolerance intervals that control percentages in both tails for a location-scale family of distributions are proposed. The proposed methods are illustrated by constructing TIs for a normal, logistic, and Laplace (double exponential) distributions based on type II singly censored samples. Factors for constructing one-sided and two-sided TIs for a logistic distribution are tabulated for the case of uncensored samples. Factors for constructing TIs based on censored samples for all three distributions are also tabulated. The factors for all cases are estimated by Monte Carlo simulation. An adjustment to the tolerance factors based on type II censored samples is proposed so that they can be used to find approximate TIs based on type I censored samples. Coverage studies of the approximate TIs based on type I censored samples indicate that the approximation is satisfactory as long as the proportion of censored observations is no more than 0.70. The methods are illustrated using some practical examples. [ABSTRACT FROM AUTHOR]

Interval estimation of the mean and quantiles of a lognormal distribution is addressed based on a Type I singly censored sample. A special case of interest is that of a sample containing values below a single detection limit. Generalized inferential procedures which use maximum likelihood estimation based approximate pivotal quantities, and some likelihood based methods, are proposed. The latter include methodology based on the signed log-likelihood ratio test (SLRT) statistic and the modified signed log-likelihood ratio test (MSLRT) statistic. The merits of the methods are evaluated for a left-censored sample using Monte Carlo simulation. For inference concerning the lognormal mean, the SLRT is to be preferred for left-tailed testing, generalized inference for right-tailed testing, and all three approaches provide nearly the same performance for two-tailed testing. These conclusions hold even when the proportion of censored values is as large as 0.70. For inference concerning quantiles, both the generalized inference approach and the MSLRT approach are satisfactory. In view of its simplicity and ease of understanding and implementation, the generalized inference procedure is to be preferred. The results are illustrated with two examples. Technical derivations are given on the Technometrics website as supplementary material. [ABSTRACT FROM AUTHOR]

In this article, we consider the problem of comparing several multivariate normal mean vectors when the covariance matrices are unknown and arbitrary positive definite matrices. We propose a parametric bootstrap (PB) approach and develop an approximation to the distribution of the PB pivotal quantity for comparing two mean vectors. This approximate test is shown to be the same as the invariant test given in [Krishnamoorthy and Yu, Modified Nel and Van der Merwe test for the multivariate Behrens-Fisher problem, Stat. Probab. Lett. 66 (2004), pp. 161-169] for the multivariate Behrens-Fisher problem. Furthermore, we compare the PB test with two existing invariant tests via Monte Carlo simulation. Our simulation studies show that the PB test controls Type I error rates very satisfactorily, whereas other tests are liberal especially when the number of means to be compared is moderate and/or sample sizes are small. The tests are illustrated using an example. [ABSTRACT FROM AUTHOR]

Abstract: The problem of interval estimation of the stress–strength reliability involving two independent Weibull distributions is considered. An interval estimation procedure based on the generalized variable (GV) approach is given when the shape parameters are unknown and arbitrary. The coverage probabilities of the GV approach are evaluated by Monte Carlo simulation. Simulation studies show that the proposed generalized variable approach is very satisfactory even for small samples. For the case of equal shape parameter, it is shown that the generalized confidence limits are exact. Some available asymptotic methods for the case of equal shape parameter are described and their coverage probabilities are evaluated using Monte Carlo simulation. Simulation studies indicate that no asymptotic approach based on the likelihood method is satisfactory even for large samples. Applicability of the GV approach for censored samples is also discussed. The results are illustrated using an example. [Copyright &y& Elsevier]

In this article we propose inferential procedures for a gamma distribution using the Wilson-Hilferty (WH) normal approximation. Specifically, using the result that the cube root of a gamma random variable is approximately normally distributed, we propose normal-based approaches for a gamma distribution for (a) constructing prediction limits, one-sided tolerance limits, and tolerance intervals; (b) for obtaining upper prediction limits for at least I of m observations from a gamma distribution at each of r locations; and (c) assessing the reliability of a stress-strength model involving two independent gamma random variables. For each problem, a normal-based approximate procedure is outlined, and its applicability and validity for a gamma distribution are studied using Monte Carlo simulation. Our investigation shows that the approximate procedures are very satisfactory for all of these problems. For each problem considered, the results are illustrated using practical examples. Our overall conclusion is that the WH normal approximation provides a simple, easy-to-use unified approach for addressing various problems for the gamma distribution. [ABSTRACT FROM AUTHOR]

The problem of comparing several alternate sampling devices to the Occupational Safety and Health Administration (OSHA) standard, or the comparison of several sampling devices among themselves, is considered. A test based on the OSHA criterion that states that "90% of the readings of the sampling device should be within ± 25% of the readings obtained by the standard" is developed. Type I error rates and powers of the test are studied using Monte Carlo simulation. The study indicates that the proposed test is quite satisfactory for applications. The method is illustrated using a simulated data set. [ABSTRACT FROM AUTHOR]

Abstract: This article concerns inference on the correlation coefficients of a multivariate normal distribution. Inferential procedures based on the concepts of generalized variables (GVs) and generalized -values are proposed for elements of a correlation matrix. For the simple correlation coefficient, the merits of the generalized confidence limits and other approximate methods are evaluated using a numerical study. The study indicates that the proposed generalized confidence limits are uniformly most accurate even for samples as small as three. The results are extended for comparing two independent correlations, overlapping and non-overlapping dependent correlations. For each problem, the properties of the GV approach and other asymptotic methods are evaluated using Monte Carlo simulation. The GV approach produces satisfactory results for all the problems considered. The methods are illustrated using a few practical examples. [Copyright &y& Elsevier]

Abstract: Inferential procedures for the difference between two multivariate normal mean vectors based on incomplete data matrices with different monotone patterns are developed. Assuming that the population covariance matrices are equal, a pivotal quantity, similar to the Hotelling statistic, is proposed, and its approximate distribution is derived. Hypothesis testing and confidence estimation of the difference between the mean vectors based on the approximate distribution are outlined. The validity of the approximation is investigated using Monte Carlo simulation. Monte Carlo studies indicate that the approximate method is very satisfactory even for small samples. A multiple comparison procedure is outlined and the proposed methods are illustrated using an example. [Copyright &y& Elsevier]

In this article, we evaluate the merits of some tests for testing about the common mean vector of several multivariate normal populations with different covariance matrices. Five tests, including the Fisher's exact test, are compared with respect to their powers. Powers are estimated using Monte Carlo simulation. Based on the comparison studies, we found that the Fisher's test and one of the test due to Zhou and Mathew [Zhou, L., Mathew, T. (1994). Combining independent tests in multivariate linear models. J. Multivariate Anal. 51:265-276] are in general more powerful than other three tests. The tests are illustrated using simulated data. [ABSTRACT FROM AUTHOR]

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