我們利用R統計軟體中的 Shiny 套件開發出一套網路介面 OTL (Outlier Testing in Lifetime Data) 來檢測指數、Weibull、對數常態或對數Laplace分配的資料是否有較大或較小離群值。在許多文獻中，虛無假設下區塊或連續檢定程序之檢定統計量的機率分配計算通常是需要用到非常複雜的積分，而且論文所提供的表格數據都是有限的。因此，我們利用Huffer and Lin (2001)的演算法和Iliopoulos and Balakrishnan (2009)的定理讓OTL 介面可以精確地計算出指數與對數 Laplace 模式之檢定統計量的機率分配。而對於 Weibull 與對數常態模式之檢定統計量的機率分配，我們則用 Monte Carlo 模擬方法取得其機率分配之逼近值，並且將所得數據儲存於資料庫，再與 OTL 介面連結以供使用者可以快速取得並做出判斷。使用OTL 介面不但可以節省搜尋表格的時間，也能擴大使用的數據範圍，更可以直接從介面上看到多個檢定統計量離群值檢測之结果，所以OTL 介面對於研究學者、決策者和企業使用者是一個相當有效且方便判斷檢測壽命時間離群值的線上工具。

A web interface, named OTL, is developed to conduct outlier analysis in exponential, Weibull, Lognormal, and Log-Laplace samples online over the Internet through Shiny and R. It is a tool to evaluate the exact or approximate distributions of block and consecutive test statistics on outliers which have been discussed over the years in the literature. Since it allows for identifying outliers without using the classical extensive tables, OTL can be very useful to academics, policy-makers and businesses to compare and understand the results of outlier testing procedures more efficiently.

1 Introduction 1
2 Main Tools for Evaluating the Null Distributions of Test Statistics from Exponential and Laplace Samples 7
2.1 Conditional Independence of Blocked Order Statistics  8
2.2 Recursions 8
3 The OTL User Interface 10
4 Demonstration with Known Data Sets 13
Example 1  13
Block Tests 13
Consecutive Tests 15
Example 2  17
Example 3  18
Example 4  19
5 Concluding Remarks and Possible Future Directions 21
References 22
Appendix A 27
Appendix B 29

Andrews, D. F., Bickel, P. J., Hampel, F. R., Huber, P. J., Rogers, W. H. and Tukey, J. W.
(1972). Robust Estimates of Location. Princeton, NJ: Princeton University Press.
Bain, L. J. and Engelhardt, M. (1973). Interval estimation for the two-parameter double exponential
distribution. Technometrics 15, 875–887.
Balakrishnan, N. and Zhu, X. (2016). Exact likelihood-based point and interval estimation
for Laplace distribution based on Type-II right censored samples. Journal of Statistical
Computation and Simulation 86, 29–54.
Balasooriya, U. and Gadag, V. (1994). Tests for upper outliers in the two-parameter exponential
distribution. Journal of Statistical Computation and Simulation 50, 249–259.
Barnett, V. and Lewis, T. (1994). Outliers in Statistical Data. Third Edition. John Wiley &
Sons, Chichester, England.
Bartholomew, D. J. (1963). The sampling distribution of an estimate arising in life testing.
Technometrics 5, 361–374.
Basu, A. P. (1965). On some tests of hypotheses relating to the exponential distribution when
some outliers are present. Journal of the American Statistical Association 60, 548–559.
Chikkagoudar, M. S. and Kunchur, S. H. (1983). Distributions of test statistics for multiple
outliers in exponential samples. Communications in Statistics – Theory and Methods 12,
2127–2142.
Chikkagoudar, M. S. and Kunchur, S. H. (1987). Comparison of many outlier procedures for
exponential samples. Communications in Statistics – Theory and Methods 16, 627–645.
Childs, A. (1996). Advances in Statistical Inference and Outliers Related Issues. Ph.D. Thesis,
McMaster University, Hamilton, Canada.
Dey, A. K. and Kundu, D. (2009). Discriminating between the lognormal and log-logistic distributions.
Communications in Statistics – Theory and Methods 39, 280–292.
Dixon, W. J. (1950). Analysis of extreme values. Annals of Mathematical Statistics 21, 488–506.
Fung, K. Y. and Paul, S. R. (1985). Comparison of outlier detection procedures in Weibull or
extreme-value distribution. Communications in Statistics – Simulation and Computation 14,
895–917.
Gasparini, M. (1995). Exact multivariate Bayesian bootstrap distributions of moments. Annals
of Statistics 23, 762–768.
Glaz, J. and Balakrishnan, N. (1999). Scan Statistics and Applications. Birkhäuser, Boston.
Glaz, J. and Koutras, M. V. (2019). Handbook of Scan Statistics. Springer, New York.
Grubbs, F. E. (1950). Sample criteria for testing outlying observations. Annals of Mathematical
Statistics 21, 27–58.
Hoaglin, D. C., Mosteller, F. and Tukey, J. W. (eds.) (1983). Understanding Robust and Exploratory
Data Analysis. New York: Wiley.
Huffer, F. (1988). Divided differences and the joint distribution of linear combinations of spacings.
Journal of Applied Probability 25, 346–354.
Huffer, F. W. and Lin, C. T. (2001). Computing the joint distribution of general linear combinations
of spacings or exponential variates. Statistica Sinica 11, 1141–1157.
Huffer, F. W. and Lin, C. T. (2006). Linear combinations of spacings. In Encyclopedia of
Statistical Sciences, Volume 12, Second Edition (Eds., S. Kotz, N. Balakrishnan, C. B. Read
and B. Vidakovic), pp. 7866–7875. John Wiley & Sons, Hoboken, New Jersey.
Iliopoulos, G. and Balakrishnan, N. (2009). Conditional independence of blocked ordered data.
Statistics & Probability Letters 79, 1008–1015.
Kimber, A. C. (1982). Tests for many outliers in an exponential sample. Applied Statistics 31,
263–271.
Kimber, A. C. and Stevens, H. J. (1981). The null distribution of a test for two upper outliers
in an exponential sample. Applied Statistics 30, 153–157.
Kornacki, A. and Bochniak, A. (2015). The use of outlier detection methods in the log-normal
distribution for the identification of atypical varietal experiments. Biometrical Letters 52,
75–84.
Kumar, N. and Lin, C. T. (2017). Testing for multiple upper and lower outliers in an exponential
sample. Journal of Statistical Computation and Simulation 87, 870–881.
Lawless, J. F. (1982). Statistical Models and Methods for Lifetime Data. John Wiley & Sons,
New York.
Lin, C. T. and Balakrishnan, N. (2009). Exact computation of the null distribution of a test for
multiple outliers in an exponential sample. Computational Statistics & Data Analysis 53,
3281–3290.
Lin, C. T. and Balakrishnan, N. (2014). Tests for multiple outliers from an exponential population.
Communications in Statistics – Simulation and Computation 43, 706–722.
Lin, C. T., Balakrishnan, N. and Ling, M. H. (2020). Exact tests for outliers in Laplace samples.
Communications in Statistics – Simulation and Computation (accepted).
Lin, C. T., Lee, B. Y. and Balakrishnan, N. (2019a). On the computation of distributions of
linear combinations of Laplace order statistics and their applications. Communications in
Statistics – Simulation and Computation 48, 3031–3055.
Lin, C. T., Lee, Y. C. and Balakrishnan, N. (2019b). Package mTEXO for testing the presence
of outliers in exponential samples. Computational Statistics 34, 803–818.
Lin, C. T. and Wang, S. C. (2015). Discordancy tests for two-parameter exponential samples.
Statistical Papers 56, 569–582.
Linhart, H. and Zucchini, W. (1986). Model Selection. John Wiley & Sons, New York.
Mann, N. R. (1982). Optimal outlier tests for a Weibull model – to identify process changes or
predict failure times. Studies in the Management Sciences, 19 261–270.
Mann, N. R. and Fertig, K. W. (1973). Tables for obtaining Weibull confidence bounds and
tolerance bounds based on best linear invariant estimates of parameters of the extreme-value
distribution. Technometrics 15, 87–101
Patil, S. A., Kovner, J. L. and King, R. M. (1977). Table of percentage points of ratios of linear
combinations of order statistics of samples from exponential distributions. Communications
in Statistics – Simulation and Computation 6, 115–136.
Paul, S. R. and Fung, K. Y. (1986). Critical values for Dixon type test statistics for testing
outliers in Weibull or extreme value distributions. Communications in Statistics – Simulation
and Computation 15, 277–283.
Pike, M. C. (1966). A method of analysis of a certain class of experiments in carcinogenesis.
Biometrics 22, 142–161.
Proschan, F. (1963). Theoretical explanation of observed decreasing failure rate. Technometrics
5, 375–383.
Rosner, B. (1975). On the detection of many outliers. Technometrics 17, 221–227.
Sen, T., Singh, S. and Tripathi, Y. M. (2018). Statistical inference for lognormal distribution
with Type-I progressive hybrid censored data. American Journal of Mathematical and Management
Sciences 38, 70–95.
Tiku, M. L., Tan, W. Y. and Balakrishnan, N. (1986). Robust Inference. Marcel Dekker, New
York.
Zerbet, A. and Nikulin, M. (2003). A new statistic for detecting outliers in exponential case.
Communications in Statistics – Theory and Methods 32, 573–583.
Zhu, X. and Balakrishnan, N. (2016). Exact inference for Laplace quantile, reliability, and cumulative
hazard functions based on Type-II censored data. IEEE Transactions on Reliability,
65 164–178.