我們利用Maple數學軟體中的maplet應用程序發展出一套使用者介面mTEXO來檢測指數分配資料中的較大或較小離群值。不同於傳統的計算方法需要花很多時間來做積分與組合代數的運算, 我們則利用Huffer and Lin (2001)的演算法先將文獻知名的指數資料離群值的統計量之分配算出, 再用mTEXO將所算出的結果呈現出來。

We develop a user-interface package mTEXO in the maplet application for testing exponential upper and/or lower outliers. The distributions of some well-known test statistics for exponential outliers in the literature can be easily evaluated using mTEXO, which saves a lot of computational time on the integrations and combinatorial algebra in the traditional approach

1 Introduction 1
2 The Distribution of Test Statistics 4
3 Tests for Exponiential Outliers in Maple using mTEXO 9
3.1 Overview of mTEXO functionality 9
3.2 mTEXO Interface and Functions 9
3.3 Calculating the Tail Probabilities 9
3.4 Application with Real Data 10
4   Conclusion and future development 14
References 14
Appendix   19

Albert, J. (2002). Sums of uniformly distributed variables: a combinatorial approach. College Mathematical Journal 33, 201–206.
Ali, M. M. (1982). Distribution of linear combinations of exponential variates. Communications in Statistics-Theory and Methods 11, 1453–1463.
Amari, S. V. and Misra, R. B. (1997). Closed-form Expressions for distribution of sum of exponential random variables. IEEE Transactions on Reliability 46, 519–522.
Arnold, B. C., Balakrishnan, N. and Nagaraja, H. N. (1992). A First Course in Order Statistics. John Wiley & Sons, New York.
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.
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.
Basu, A. P. and Lochner, R. H. (1971). On the distribution of the ratio of two random variables
having generalized life distributions. Technometrics 13, 281–287.
Chapman, D. G. (1950). Some two-sample tests. Annals of Mathematical Statistics 21, 601–606.
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.
Christopeit, N. and Helmes, K. (1979). A convergence theorem for random linear combinations of independent normal random variables. Annals of Statistics 7, 795–800.
Davies, R. B. (1980). Algorithm AS 155: The distribution of a linear combination of chi-squared random variables. Applied Statistics 29, 323–333.
Dobson, A. J., Kulasmaa, K. and Scherer, J. (1991). Confidence intervals for weighted sums of Poisson parameters. Statistics in Medicine 10, 457–462.
Epstein, B. (1960). Tests for the validity of the assumption that the underlying distribution of life is exponential: Part II. Technometrics 2, 167–183.
Farebrother, R. W. (1984) Algorithm AS 204: The distribution of a positive linear combination
of chi-squared random variables. Applied Statistics 33, 332–339.
Fisher, R. A. (1935). The fiducial argument in statistical inference. Annals of Eugenics 6, 391–398.
Hawkins, D. L. and Han, C.-P. (1986). Bivariate distributions of some ratios of independent noncentral chi-square random variables. Communications in Statistics-Theory and Methods 15, 261–277.
Hitezenko, P. (1998). A note on a distribution of weighted sums of iid Rayleigh random variables. Sankhy¯ a A 60, 171–175.
Hu, C.-Y. and Lin, G. D. (2001). An inequality for the weighted sums of pairwise iid generalized Rayleigh random variables. Journal of Statistical Planning and Inference 92, 1–5.
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), 7866–7875, John Wiley & Sons, Hoboken, New Jersey.
Iyer, S. (1991). The mean time-to-failure of some special series-parallel arrays. IEEE Transactions on Reliability 40, 20–21.
Kabe, D. G. (1970). Testing outliers from an exponential population. Metrika 15, 15–18.
Kamgar-Parsi, B. and Brosh, M. (1995). Distribution and moments of weighted sum of uniform random variables with applications in reducing Monte Carlo simulations. Journal of Statistical Computation and Simulation 52, 399–414.
Kimber, A. C. and Stevens, H. J. (1981). The null istribution of a test for two upper outliers in an exponential sample. Applied Statistics 30, 153–157.
Kimber, A. C. (1982). Tests for many outliers in an exponential sample. Applied Statistics 31, 263–271.
Korhonen, P. J. and Narula, S. C. (1989). The probability distribution of the ratio of the absolute values of two normal variables. Journal of Statistical Computation and Simulation 33, 173–182.
Kordonsky, K. B. and Gertsbakh, I. B. (1995a). System state monitoring and lifetime scales 1. Reliability Engineering & System Safety 47, 1–14.
Kordonsky, K. B. and Gertsbakh, I. B. (1995b). System state monitoring and lifetime scales 2. Reliability Engineering & System Safety 49, 145–154.
Kumar, N. (2015). Testing of suspected observations in an exponential sample with unknown origin. Communications in Statistics – Theory and Methods 44, 3668–3679.
Kumar, N. and Lalitha, S. (2012). Testing for upper outliers in gamma sample. Communications in Statistics – Theory and Methods 41, 820–828.
Lewis, T. and Fieller, N. R. J. (1979). A recursive algorithm for null distribution for outliers: I. Gamma samples. Technometrics 21, 371–376.
Likes, J. (1987). Some tests for k ≥ 2 upper outliers in an exponential sample. Biometrical Journal 29, 313–324.
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. and Wang, S. C. (2015). Discordancy tests for two-parameter exponential samples. Statistical Papers 56, 569–582.
Marsaglia, G. (1965). Ratios of normal variables and ratios of sums of uniform variables. Journal of the American Statistical Association 60, 193–204. Moschopoulos, P. G. (1985). The distribution of the sum of independent gamma random variables. Annals of the Institute of Statistical Mathematics 37, 541–544.
Narayanan, A. and Narayanan, G. (2010). On Maplet development and programming tutorial for science and engineering students. 2010 Annual Conference & Exposition, Engineering Technology.
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.
Pham-Gia, T. (2000). Distributions of the ratios of independent beta variables and applications. Communications in Statistics-Theory and Methods 29, 2693–2715.
Pham-Gia, T. and Turkkan, N. (1993). Bayesian analysis of the difference of two proportions. Communications in Statistics-Theory and Methods 22, 1755–1771.
Pham-Gia, T. and Turkkan, N. (1994). Reliability of a standby system with beta distributed component lives. IEEE Transactions on Reliability 43, 71–75.
Press, S. J. (1969). The t ratio distribution. Journal of the American Statistical Association 64, 242–252.
Provost, S. B. (1989a). On sums of independent gamma random variables. Statistics 20, 583–591.
Provost, S. B. (1989b) On the distribution of the ratio of powers of sums of gamma random variables. Pakistan Journal Statistics 5, 157–174.
Shcolnick, S. M. (1985). On the ratio of independent stable random variables. InStability Problems for Stochastic Models (Uzhgorod, 1984) Lecture Notes in Mathematics 1155, 349–354, Springer, Berlin.
Witkovsky, V. (2001). Computing the distribution of a linear combination of inverted gamma variables. Kybernetika 37, 79–90.
Zerbet, A. and Nikulin, M. (2003). A new statistic for detecting outliers in exponential case. Communications in Statistics – Theory and Methods 32, 573–583.
Zhang, J. (1998). Tests for multiple upper or lower outliers in an exponential sample. Journal of Applied Statistics 25, 245–255.