Testing for multiple upper and lower outliers in an exponential sample

Due to wide applicability and simplicity, the exponential distribution is the most commonly used distribution in reliability engineering and other life testing experiments. In this paper a test statistic for testing upper and lower outliers simultaneously in an exponential sample is proposed. However, the distribution of test statistic under the alternative is rather intricate, the null distribution is derived and critical values are obtained. A simulation study is also carried out to compare the performance of test and is found that the test based on this statistic is more powerful than the other two selected tests.     

Test for multiple upper outliers in an exponential sample irrespective of origin

An approach for testing multiple upper outliers with slippage alternative in an exponential sample, irrespective of origin, is discussed. The outlier detection procedure is based on a ratio of two estimates, obtained by the maximization of the two log-likelihood functions. One is the complete data log-likelihood and the other is its conditional expectation, given the regular observations. The exact null distribution of the test statistic is derived and no new table for critical values is required. A simulation study is also carried out to compare the performance of the test with the earlier work.     

Multiple outlier test for upper outliers in an exponential sample

In this paper, a test statistic for testing upper outliers with a slippage alternative, in an exponential sample is proposed. No tables for critical values are required as they can be calculated easily for any sample size. A simulation study is also carried out to compare the performance of the test with the maximum likelihood ratio test and other existing tests.     

Unmasking test for multiple upper or lower outliers in normal samples

SUMMARY The discordancy test for multiple outliers is complicated by problems of masking and swamping. The key to the settlement of the question lies in the determination of k , i.e. the number of 'contaminants' in a sample. Great efforts have been made to solve this problem in recent years, but no effective method has been developed. In this paper, we present two ways of determining k , free from the effects of masking and swamping, when testing upper (lower) outliers in normal samples. Examples are given to illustrate the methods.     

Distributions of test statistics for multiple outliers in exponential samples

Null as well as alternative distributions of two types of statistics used to test for multiple outliers in exponential samples are obtained. Of these two types one is based on the ratio of sum of the observations suspected to be outliers to sum of the sample observations and the other is Dixon's type. Powers of the tests based on these statistics are compared.

     

A note on determining the number of outliers in an exponential sample by least squares procedure

In this paper, we suggest a least squares procedure for the determination of the number of upper outliers in an exponential sample by minimizing sample mean squared error. Moreover, the method can reduce the masking or “swamping” effects. In addition, we have also found that the least squares procedure is easy and simple to compute than test test procedure T _k suggested by Zhang (1998) for determining the number of upper outliers, since Zhang (1998) need to use the complicated null distribution of T _k . Moreover, we give three practical examples and a simulated example to illustrate the procedures. Further, simulation studies are given to show the advantages of the proposed method. Finally, the proposed least squares procedure can also determine the number of upper outliers in other continuous univariate distributions (for example, Pareto, Gumbel, Weibull, etc.). Received: May 10, 1999; revised version: June 5, 2000     

Masking and swamping effects on tests for multiple outliers in normal sample

A study of some commonly used multiple outlier tests in case of normal samples is presented. When the number of outliers in the sample is unknown, two phenomena, namely, the masking and the swamping effect can occur. The performance of the tests is studied using the measures of masking and swamping effects proposed by Bendre and Kale (1985) and Bendre (1985). The effects are illustrated in case of the Murphy test, Tietjen—Moore test and Dixon test. A small simulation study is carried out to indicate these effects.     

A small sample test for an absolutely continuous bivariate exponential model

The finite sample distribution of the likelihood ratio sta-tistic is obtained for testing independence, given marginal homo-geneity, in the absolutely continuous bivariate exponential distri-bution of Block and Basu (1974). This test is discussed in light of the analysis of Gross and Lam (1981) on times to relief of head-aches for standard and new treatments on ten subjects.     

Detection of outliers in the exponential distribution based on prediction

In this paper, we present a test procedure to detect outliers in the one-parameter exponential distribution based on prediction. The distribution of the test statistic is obtained. The proposed test can be used to detect more than one outlier and the required percentage points can be easily determined. Furthermore, the test provides a simple procedure to detect whether a given set of data is free from outliers or spurious observations.     

On determining the number of outliers in exponential and Pareto samples

In the present paper we suggest a procedure for the determination of the number of outliers in exponential and Pareto samples, using the predictive interval approach.     

Asymptotic results for lower exponential spacings

Abstract

In this paper, we investigate distributional and asymptotic properties of lower exponential spacings. We find conditions under which the sequence of logarithmic lower spacings converges in probability and with probability one.     

Tests of hypotheses for reliability functions in two-parameter exponential models

This paper derives a test procedure for testing hypotheses about the reliability function of the two-parameter exponential model. The exact distribution of the test statistic is obtained and it is shown that the test is UMP invariant. Applications to problems in quality control are also considered.     

A statistical-test-complemented graphical method for detecting multiple outliers in two-way tables

A statistic Rk which has a simple relationship with Qk is proposed for the analysis of outliers in two-way tables and the rationale is discussed. The critical values of the test statistic, minimum Rk, can be well approximated by existing values based on univariate Grubbs-type outlier test statistics. The test statistics are complemented with plots of the largest Wk values, which have a simple monotonic inverse relationship with the values of Rk, against their expected quantiles which are approximated using a conditional independence argument. Two examples are analysed with satisfactory results.     

A modified multiple comparisons with a control for exponential location parameters based on doubly censored sample under heteroscedasticity

ABSTRACT

In this paper, a modified one-stage multiple comparison procedures with a control for exponential location parameters based on the doubly censored sample under heteroscedasticity is proposed. A simulation study is done and the results show that the proposed procedures have shorter confidence length with coverage probabilities closer to the nominal ones compared with the one proposed in Wu (2017 Wu, S. F. 2017. Multiple comparisons of exponential location parameters with a control based on doubly censored sample under heteroscedasticity. Communications in Statistics: Simulation and Computation 46 (3):1858–1870. doi: 10.1080/03610918.2015.1017582.[Taylor & Francis Online] , [Google Scholar]). At last, an example of comparing the duration of remission for four drugs as the treatment of leukemia is given to demonstrate the proposed procedures.     

Estimation of generalized exponential distribution based on an adaptive progressively type-II censored sample

In this paper, based on an adaptive Type-II progressively censored sample from the generalized exponential distribution, the maximum likelihood and Bayesian estimators are derived for the unknown parameters as well as the reliability and hazard functions. Also, the approximate confidence intervals of the unknown parameters, and the reliability and hazard functions are calculated. Markov chain Monte Carlo method is applied to carry out a Bayesian estimation procedure and in turn calculate the credible intervals. Moreover, results from simulation studies assessing the performance of our proposed method are included. Finally, an illustrative example using real data set is presented for illustrating all the inferential procedures developed here.     

Tests for generalized exponential laws based on the empirical Mellin transform

Recently, many standard families of distributions have been generalized by exponentiating their cumulative distribution function (CDF). In this paper, test statistics are constructed based on CDF–transformed observations and the corresponding moments of arbitrary positive order. Simulation results for generalized exponential distributions show that the proposed test compares well with standard methods based on the empirical distribution function.     

Tests for outliers in the inverse Gaussian distribution,with application to first hitting time models

The inverse Gaussian (IG) distribution is often applied in statistical modelling, especially with lifetime data. We present tests for outlying values of the parameters (μ, λ) of this distribution when data are available from a sample of independent units and possibly with more than one event per unit. Outlier tests are constructed from likelihood ratio tests for equality of parameters. The test for an outlying value of λ is based on an F-distributed statistic that is transformed to an approximate normal statistic when there are unequal numbers of events per unit. Simulation studies are used to confirm that Bonferroni tests have accurate size and to examine the powers of the tests. The application to first hitting time models, where the IG distribution is derived from an underlying Wiener process, is described. The tests are illustrated on data concerning the strength of different lots of insulating material.     

Jacobi and Laguerre polynomial approximations for the distributions of statistics useful in testing for outliers in exponential and gamma samples

Recently, Sanjel and Balakrishnan [A Laguerre Polynomial Approximation for a goodness-of-fit test for exponential distribution based on progressively censored data, J. Stat. Comput. Simul. 78 (2008), pp. 503–513] proposed the use of Laguerre orthogonal polynomials for a goodness-of-fit test for the exponential distribution based on progressively censored data. In this paper, we use Jacobi and Laguerre orthogonal polynomials in order to obtain density approximants for some test statistics useful in testing for outliers in gamma and exponential samples. We first obtain the exact moments of the statistics and then the density approximants, based on these moments, are expressed in terms of Jacobi and Laguerre polynomials. A comparative study is carried out of the critical values obtained by using the proposed methods to the corresponding results given by Barnett and Lewis [Outliers in Statistical Data, 3rd ed., John Wiley & Sons, New York, 1993]. This reveals that the proposed techniques provide very accurate approximations to the distributions. Finally, we present some numerical examples to illustrate the proposed approximations. Monte Carlo simulations suggest that the proposed approximate densities are very accurate.     

Exact likelihood inference based on an unified hybrid censored sample from the exponential distribution

Recently, exact confidence bounds and exact likelihood inference have been developed based on hybrid censored samples by Chen and Bhattacharyya [Chen, S. and Bhattacharyya, G.K. (1998). Exact confidence bounds for an exponential parameter under hybrid censoring. Communications in Statistics—Theory and Methods, 17, 1857–1870.], Childs et al. [Childs, A., Chandrasekar, B., Balakrishnan, N. and Kundu, D. (2003). Exact likelihood inference based on Type-I and Type-II hybrid censored samples from the exponential distribution. Annals of the Institute of Statistical Mathematics, 55, 319–330.], and Chandrasekar et al. [Chandrasekar, B., Childs, A. and Balakrishnan, N. (2004). Exact likelihood inference for the exponential distribution under generalized Type-I and Type-II hybrid censoring. Naval Research Logistics, 51, 994–1004.] for the case of the exponential distribution. In this article, we propose an unified hybrid censoring scheme (HCS) which includes many cases considered earlier as special cases. We then derive the exact distribution of the maximum likelihood estimator as well as exact confidence intervals for the mean of the exponential distribution under this general unified HCS. Finally, we present some examples to illustrate all the methods of inference developed here.