Development of statistical convolutions of truncated normal and truncated skew normal distributions with applications

Convolutions of normal random variables often arise in engineering problems, and the probability densities of the sums of these random variables are known in the literature. There are practical situations where specification limits on a process are imposed externally, and the product is typically scrapped if its performance does not fall in the specification range. As such, the actual distribution after inspection is truncated. Despite the practical importance of the role of truncated distributions, there has been little work on the theoretical foundation of convolutions associated with truncated random variables. This is paramount, since convolutions are often used as an important standard in statistical tolerance analysis. In this article, the convolutions of the combinations of truncated normal and truncated skew normal random variables on double and triple truncations are developed. The successful completion of this research task on convolution could help obtain a better understanding of integrated effects of statistical tolerance analysis in engineering design, leading to process and quality improvement.

[1]  Wichai Chattinnawat.  Statistical tolerance design to minimize dual-responses of APFA height deviations with tolerance cost-quality loss model , 2015 .

[2]  Fritz Scholz.  Tolerance Stack Analysis Methods , 1995 .

[3]  Byung Rae Cho,et al.  Joint optimization of process target mean and tolerance limits with measurement errors under multi-decision alternatives , 2007, Eur. J. Oper. Res..

[4]  J. Jawitz.  Moments of truncated continuous univariate distributions , 2004 .

[5]  C. Roberts.  A Correlation Model Useful in the Study of Twins , 1966 .

[6]  G. Henzold.  Handbook of Geometrical Tolerancing: Design, Manufacturing and Inspection , 1995 .

[7]  Jian Ma,et al.  Operational Impacts of Wind Generation on California Power Systems , 2009, IEEE Transactions on Power Systems.

[8]  A. Azzalini,et al.  Statistical applications of the multivariate skew normal distribution , 2009, 0911.2093.

[9]  Øyvind Bjørke,et al.  Computer Aided Tolerancing , 1989 .

[10]  R. Arellano-Valle,et al.  Definition and probabilistic properties of skew-distributions , 2002 .

[11]  Paul L. Goethals,et al.  Solving the optimal process target problem using response surface designs in heteroscedastic conditions , 2011 .

[12]  Samuel Kotz,et al.  Skew Distributions Generated from Different Families , 2006 .

[13]  E. M. Mansoor.  The Application of Probability to Tolerances Used in Engineering Designs , 1963 .

[14]  W. H. Greenwood,et al.  A New Tolerance Analysis Method for Designers and Manufacturers , 1987 .

[15]  Günter Kirschling.  Qualitätssicherung und Toleranzen , 1988 .

[16]  Byung Rae Cho,et al.  Rethinking the truncated normal distribution , 2013 .

[17]  Arjun K. Gupta,et al.  A class of multivariate skew-normal models , 2004 .

[18]  Tomotsuka Takayama,et al.  Computational Improvement for Expected Sliding Distance of a Caisson-Type Breakwater by Introduction of a Doubly-Truncated Normal Distribution , 2003 .

[19]  A. Azzalini,et al.  The multivariate skew-normal distribution , 1996 .

[20]  Chang-Hsin Kuo,et al.  A novel statistical tolerance analysis method for assembled parts , 2012 .

[21]  Byung Rae Cho,et al.  Classical Statistical Inference Extended to Truncated Populations for Continuous Process Improvement: Test Statistics, P-values, and Confidence Intervals , 2015, Qual. Reliab. Eng. Int..

[22]  Rajarshi Das,et al.  Characteristics of the design surface of damage tolerance parameters and their relation to shape optimisation , 2015 .

[23]  Query 2: The Sum of Values from a Normal and a Truncated Normal Distribution (Continued) , 1964 .

[24]  A two-parameter generalized skew-normal distribution , 2008 .

[25]  Oliver R. Wade,et al.  Tolerance control in design and manufacturing , 1967 .

[26]  D. Barr,et al.  Mean and Variance of Truncated Normal Distributions , 1999 .

[27]  Michael Yu Wang,et al.  Simultaneous parameter and tolerance optimization of structures via probability-interval mixed reliability model , 2015 .

[28]  Earlwood T. Fortini.  Dimensioning for interchangeable manufacture , 1967 .

[29]  V. J. Francis.  On the Distribution of the Sum of N Sample Values Drawn from a Truncated Normal Population , 1946 .

[30]  Joshua U. Turner,et al.  Review of statistical approaches to tolerance analysis , 1995, Comput. Aided Des..

[31]  A. Azzalini.  The Skew‐normal Distribution and Related Multivariate Families * , 2005 .

[32]  Byung Rae Cho,et al.  Studies on a biobjective robust design optimization problem , 2009 .

[33]  Byung Rae Cho,et al.  Tables of a Truncated Standard Normal Distribution: A Doubly Truncated Case , 2004 .

[34]  John Gilson.  A new approach to engineering tolerances : a critical presentation of the considerations necessary for the allocation and maintenance of realistic tolerances in modern economic production , 1951 .

[35]  A. Azzalini.  A class of distributions which includes the normal ones , 1985 .

[36]  Raluca Vernic.  Multivariate skew-normal distributions with applications in insurance , 2006 .

[37]  D. E. Coleman.  Volume 11: How to Perform Statistical Tolerance Analysis , 1988 .

[38]  M. Weinstein.  Query 2: The Sum of Values from a Normal and a Truncated Normal Distribution , 1964 .

[39]  D. Allard,et al.  Truncated skew-normal distributions: moments, estimation by weighted moments and application to climatic data , 2010 .

[40]  David H. Evans,et al.  Statistical Tolerancing: The State of the Art, Part III. Shifts and Drifts , 1975 .