Normalization of the origin-shifted exponential distribution for control chart construction

Abstract

This study demonstrates that a location parameter of an exponential distribution significantly influences normalization of the exponential. The Kullback-Leibler information number is shown to be an appropriate index for measuring data normality using a location parameter. Control charts based on probability limits and transformation are compared for known and estimated location parameters. The probabilities of type II error (β-risks) and average run length (ARL) without a location parameter indicate an ability to detect an out-of-control signal of an individual chart using a power transformation similar to using probability limits. The β-risks and ARL of control charts with an estimated location parameter deviate significantly from their theoretical values when a small sample size of n≤50 is used. Therefore, without taking into account of the existence of a location parameter, the control charts result in inaccurate detection of an out-of-control signal regardless of whether a power or natural logarithmic transformation is used. The effects of a location parameter should be eliminated before transformation. Two examples are presented to illustrate these findings.location parameter, exponential distribution, power transformation, natural logarithmic transformation, Kullback-Leibler information number,

Improved Phase I Control Charts for Monitoring Times Between Events

“…compare some of these charts including the cumulative sum and exponentially‐weighted moving average charts. Some authors suggest applying the regular Shewhart charts after a transformation of the data . Jones and Champ emphasize the early stage process improvement activities when failures occur according a homogeneous Poisson process .…”

Exact simultaneous location-scale tests for two shifted exponential samples

“…Singh and Abebe [20] compared multiple exponential populations with more than one control. Kao [10] showed that the amount of current needed to break an insulator of the P-type high-voltage metal oxide semiconductor (MOS) transistor (HPM) on a non-volatile drive memory wafer follows the two-parameter exponential distribution and this article nicely demonstrates the importance of the two-parameter exponential distribution in the engineering sciences. In the same year, Wu [27] proposed an interval estimation technique for the scale parameter of a two-parameter exponential distribution based on Type-II progressive censoring.…”

“…For example, in the engineering sciences it is used to model the lifespan of electronic components, transmission, engine and mechanical equipment. Nevertheless, many researchers, including Huang, Mukherjee and Yang [8], Kao [10], Krishnamoorthy and Xia [11], Roy and Mathew [17], and Wu [27], have found that in various practical applications, such as measuring the reliability of a product, monitoring the high-voltage of current in certain metal oxide semiconductor transistor on a flash memory wafer, modelling the consumer lifetime, among others, the two-parameter (shifted) exponential distribution should be preferred to the one-parameter exponential distribution. The probability density function (pdf) of the two-parameter exponential distribution is f (x; θ, λ) = 1 λ exp −(x−θ)/λ ; x > θ > 0, λ > 0.…”

Simultaneous monitoring of origin and scale of a shifted exponential process with unknown and estimated parameters

“…One may see, among others, Wright et al, 1 Mitra and Patankar, 2 Baker et al, 3 Perry and Pignatiello, 4 Baklizi, 5 Doostparast and Balakrishnan, 6 Han et al, 7 Ganguly et al, 8 Cheng et al, 9 and Krishnamoorthy and Xia 10 . Kao 11 emphasized the importance of monitoring‐shifted exponential processes in the context of monitoring high voltage of current in a typical semiconductor transistor on a flash memory wafer. He also illustrated its use in health care monitoring.…”