### Some Inferences on Tests for Inliers

#### Abstract

The present paper studies the problem of inliers in samples and how to test their statistical significance. Inliers are instantaneous or early failures that are natural occurrences of a life test. In them, some of the items may fail immediately or within a short time of the life test due to mechanical failure, inferior quality, or faulty construction of objects and components. We provide procedures for testing hypotheses consists of single and multiple inliers coming from an exponential distribution. It further studies the masking effect on Dixon type tests and the Cochran type test for the case of single inliers. The critical values are theoretically considered and numerically computed. The power of the tests and the error probabilities for the effects of masking and swamping under outward sequential criteria are tabulated for the number of inliers = 2 and 3.

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A. C. Kimber; H. J. Stevens; The null distribution of a test for two upper outliers in an exponential sample, Applied Statistics, 30, No. 2: 153–157 (1981).

A. C. Kimber; Tests for many outliers in an exponential sample, Applied Statistics, 31: 263–271 (1982).

B. K. Kale; Trimmed means and the method of maximum likelihood when spurious observations are present, Applied Statistics, R. P. Gupta, Ed., North Holland, Amsterdam, 177–185 (1975).

B. K. Kale; Modified failure time distributions to accommodate instantaneous and early failures, Industrial Mathematics and Statistics, Ed. J. C. Misra, Narosa Publishing House, New Delhi, 623–648 (2003).

B. K. Kale; K. Muralidharan; Optimal estimating equations in mixture distributions accommodating instantaneous or early failures, Journal of the Indian Statistical Association, 38: 317–329 (2000).

B. K. Kale; K. Muralidharan; Masking effect of inliers, Journal of the Indian Statistical Association, 45(1): 33–49 (2007).

B. K. Kale; K. Muralidharan; Maximum Likelihood estimation in presence of inliers, Journal of Indian Society for Probability and Statistics, 10: 65–80 (2008).

B. Rosner; On the detection of many outliers, Technometrics, 17(2): 221–227 (1975).

C. Lin; N. Balakrishnan; Exact computation of the null distribution of a test for multiple outliers in an exponential sample, Computational Statistics & Data Analysis, 53(9): 3281–3290 (2009).

C. Lin; N. Balakrishnan; Tests for Multiple Outliers in an Exponential Sample, Communications in Statistics–Simulation and Computation, 43(4): 706–722 (2014).

D. M. Hawkins; Identification of outliers, Chapman and Hall, London (1980).

F. Louzada; P. L. Ramos; P. H. Ferreira; Exponential-Poisson distribution: estimation and applications to rainfall and aircraft data with zero occurrence, Communication in Statistics–Simulation and Computation, 46(10), 8118-8139 (2017).

G. Bipin; D. Mintu; Detection of Multiple Upper Outliers in Exponential Sample under Slippage Alternative, International Advanced Research Journal in Science, Engineering and Technology, 2(8): 63–69 (2015).

http://hydro.imd.gov.in/hydrometweb/(S(puzbfiyiec4n0p45dmrgzv45))/PRODUCTS/Publications/Rainfall%20Statistics%20of%20India%20%202016/Rainfall%20Statistics%20of%20India%20-%202016.pdf

https://rbi.org.in/Scripts/NEFTView.aspx

I. Guttman; Care and handling of univariate multivariate outliers in detecting spuriousity- a Bayesian approach, Technometrics, 15: 723–738 (1973).

J. Aitchison; On the distribution of a positive random variable having a discrete probability mass at the origin, Journal of American Statistical Association, 50: 901–908 (1955).

J. R. Veale; Improved estimation of expected life when one identified spurious observation may be present, Journal of American Statistical Association, 70: 398–401 (1975).

J. W. Tukey; A survey of sampling from contaminated distributions. In Contributions to Probability and Statistics: Essays in Honor of Harold Hotelling (I. Olkin, S. G. Ghurye, W. Hoeffding, W. G. Madow and H. B. Mann, eds.), Stanford Univ. Press, 448–485 (1960).

J. Zhang; Tests for multiple upper or lower outliers in an exponential sample, Journal of Applied Statistics, 1998, 25(2): 245–255 (1998).

K. Muralidharan; Inlier prone models: A review, ProbStat Forum, 3: 38–51 (2010).

K. Muralidharan; M. Arti; Analysis of instantaneous and early failures in Pareto distribution, Journal of Statistical Theory and Applications, 7:187–204 (2008).

K. Muralidharan; P. Bavagosai; Some inferential studies on inliers in Gompertz distribution, Journal of Indian Society for Probability and Statistics, 17: 35–55 (2016a).

K. Muralidharan; P. Bavagosai; A revisit to early failure analysis in life testing. Journal of the Indian Statistical Association, 54(1 & 2): 43–69 (2016b).

K. Muralidharan; P. Bavagosai; Analysis of lifetime model with discrete mass at zero and one, Journal of Statistical Theory Practice, 11(4), 670–692 (2017).

K. Muralidharan; P. Bavagosai; A new Weibull model with inliers at zero and one based on type-II censored samples, Journal of Indian Society of Probability and Statistics, 19: 121–151 (2018).

K. Muralidharan; P. Lathika; (2004) The concept of inliers, Proceeding of first Sino-International Symposium on Probability, Statistics and Quantitative Management, Taiwan, October, 77–92.

K. Muralidharan; P. Lathika; Analysis of instantaneous and early failures in Weibull distribution, Metrika, 64(3): 305–316 (2006).

K. Muralidharan; P. Lathika; Statistical modeling of rainfall data using modified Weibull distribution, Mausam, Indian Journal of Meteorology, Hydrology and Geophysics, 56(4): 765–770 (2005).

K. Vannman; Comparing samples from nonstandard mixtures of distributions with applications to quality comparison of wood; Research report: 2, submitted to the division of quality technology, Lulea University, Lulea, Sweden (1991).

K. Vannman; (1995). On the distribution of the estimated mean from the nonstandard mixtures of distribution, Communication in Statistics–Theory and Methods, 24(6): 1569–1584 (1995).

M. S. Chikkagoudar; S. H. Kunchur; Distribution of test statistics for multiple outliers in exponential samples, Communication in Statistics–Theory and Methods, 12: 2127–2142 (1983).

M. S. Chikkagoudar; S. H. Kunchur; Comparison of many outlier procedures for exponential samples, Communication in Statistics–Theory and Methods, 16: 627–645 (1987).

P. Bavagosai; K. Muralidharan; Some inferential study on inliers in Lindley distribution, International Journal of Statistics and Reliability Engineering, 3(2): 108–129 (2016).

R. L. Shinde; A. Shanubhogue; Estimation of parameters and the mean life of a mixed failure time distribution, Communication in Statistics–Theory and Methods, 29(1): 2621–2642 (2000).

S. M. Bendre; B. K. Kale; Masking effect on test for outliers in exponential models, Journal of the American Statistical Association, 80 (392): 1020–1025 (1985).

T. Lewis; N. R. J. Feller; A recursive algorithm for null distributions for outliers: I. Gamma sample, Technometrics, 21: 371–376 (1979).

T. S. Ferguson; Rules of rejection of outliers, Revue Institute Internationale de Statistica, 29(3): 29–43 (1961).

U. Balasooriya; V. Gadag; Test for upper outliers in the two-parameter exponential distribution, Journal of Statistical Computation and Simulation, 50, 3–4: 249–259 (1994).

V. Barnett; T. Lewis; Outliers in Statistical Data, 3rd ed., John Wiley& Sons, New York (1994).

V. P. Jayade; M. S. Prasad; Estimation of parameters of mixed failure time distribution, Communication in Statistics–Theory and Methods, 19(12): 4667–4677 (1990).

W. G. Cochran; The distribution of the largest of a set of estimated variances as a fraction of their total, Annals of Eugenics, 11, 47–52 (1941).

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