Rethinking the truncated normal distribution

Cha, Jinho; Cho, Byung Rae; Sharp, Julia L.

doi:10.1504/ijedpo.2013.059667

Cited by 21 publications

(15 citation statements)

References 32 publications

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“…Detailed proofs for the mean and variance of the DTND can be found in Cha et al . . Table shows that both

φ ((), \frac{x_{l} - μ}{σ})

and

normalΦ ((), \frac{x_{l} - μ}{σ})

converge to zero in the mean and variance of the DTND as the lower truncation point, x l , goes negative infinity.…”

Section: A Review Of the Properties Of A Truncated Normal Distributionmentioning

confidence: 95%

“…The CLT says that the distribution of the mean of a random sample taken from any population with a finite variance converges to the standard normal distribution as the sample size becomes large. As discussed by Cha et al ., the variance of its truncated normal distribution, σ T , becomes finite if the variance of the normal distribution is finite. Section 4.1 provides a proposed theorem, which proves the CLT for a truncated normal distribution by using the moment generating function.…”

Section: Investigation Of the Central Limit Theorem For The Truncatedmentioning

confidence: 99%

“…Applications of a truncated distribution are found in many industry settings. Khasawneh et al 1 and Cha et al 2 illustrated examples in quality and reliability control. Final products are often subject to screening before being sent to the customer.…”

Section: Introductionmentioning

confidence: 99%

“…/ are the probability density function and the cumulative distribution function, respectively, of a standard normal random variable Z. Detailed proofs for the mean and variance of the DTND can be found in Cha et al 2 . Table I shows that both xl andˆ xl converge to zero in the mean and variance of the DTND as the lower truncation point, x l , goes negative infinity.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Classical Statistical Inference Extended to Truncated Populations for Continuous Process Improvement: Test Statistics, P‐values, and Confidence Intervals

Cha

Cho

2014

Quality & Reliability Eng

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Statistical hypothesis testing is useful for controlling and improving processes, products, and services. This most fundamental, yet powerful, continuous improvement tool has a wide range of applications in quality and reliability engineering. Some application areas include statistical process control, process capability analysis, design of experiments, life testing, and reliability analysis. It is well-known that most parametric hypothesis tests on a population mean, such as z-test and t-test, require a random sample from the population under study. However, there are special situations in engineering, where the specification limits, such as the lower and upper specification limits, on the process are implemented externally, and the product is typically reworked or scrapped if the performance of a product does not fall in the range. As such, a random sample needs to be taken from a truncated distribution; however, there has been little work on the theoretical foundation of statistical hypothesis procedures under this special situation. The objective of this paper is twofold. First, we provide the mathematical justifications that the central limit theorem works quite well for a large sample size, given samples taken from a truncated distribution. We also verify this finding using simulation. Second, we then develop the new one-sided and two-sided z-test and t-test procedures, including their test statistics, confidence intervals, and P-values, using appropriate truncated statistics.

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“…Detailed proofs for the mean and variance of the DTND can be found in Cha et al . . Table shows that both

φ ((), \frac{x_{l} - μ}{σ})

and

normalΦ ((), \frac{x_{l} - μ}{σ})

converge to zero in the mean and variance of the DTND as the lower truncation point, x l , goes negative infinity.…”

Section: A Review Of the Properties Of A Truncated Normal Distributionmentioning

confidence: 95%

Section: Investigation Of the Central Limit Theorem For The Truncatedmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Classical Statistical Inference Extended to Truncated Populations for Continuous Process Improvement: Test Statistics, P‐values, and Confidence Intervals

Cha

Cho

2014

Quality & Reliability Eng

Self Cite

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show abstract

“…It is also observed that the variance of the distribution, after implementing a truncation, will no longer be the same as the original variance associated with the untruncated normal distribution f X (x). Applications of the truncated normal distribution are found in Khasawneh et al, 5,6 Hong and Cho, 7 Shin and Cho, 8 Cha et al, 9 and Cha and Cho. 10 Regarding the field of process capability analysis within contemporary literature, this paper offers several contributions that have not been previously explored.…”

Section: Introductionmentioning

confidence: 99%

Integrating Customer Perception into Process Capability Measures

Khamkanya

Cho

Goethals

2015

Qual. Reliab. Engng. Int.

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Process capability indices provide a measure of the output of an in-control process that conforms to a set of specification limits. These measures, which assume that process output is approximately normally distributed, are intended for measuring process capability for manufacturing systems. When the performance of a system results in a product that fails to fall within a given specification range, however, the product is typically scrapped or reworked, and the actual distribution that the customer perceives after inspection is truncated. In this paper, the concept of a truncated measure for three types of quality characteristics is introduced as the key to linking customer perception to process capability. Subsequently, a set of customerperceived process capability indices is presented as an extension of traditional manufacturer-based counterparts. Finally, data transformation-based process capability indices are also discussed. A comparative study and numerical example reveal considerable differences among the traditional and proposed process capability indices. It is believed that the proposed process capability index for various quality characteristics may more aptly lead to process improvement by facilitating a better understanding of the integrated effects found in engineering design problems.

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Comparative Study on Value of a Statistical Life in Road Traffic Based on Mixed Logit Model

Liu

Zhao

2020

Green, Smart and Connected Transportation Systems

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Rethinking the truncated normal distribution

Cited by 21 publications

References 32 publications

Classical Statistical Inference Extended to Truncated Populations for Continuous Process Improvement: Test Statistics, P‐values, and Confidence Intervals

Classical Statistical Inference Extended to Truncated Populations for Continuous Process Improvement: Test Statistics, P‐values, and Confidence Intervals

Integrating Customer Perception into Process Capability Measures

Comparative Study on Value of a Statistical Life in Road Traffic Based on Mixed Logit Model

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