Distribution‐free simultaneous tests for location–scale and Lehmann alternative in two‐sample problem

Appl Stoch Models Bus & Ind

Qiu

Marozzi

2021

Self Cite

Distribution-free charting schemes for process monitoring are more robust and reliable than their parametric counterparts when the process distributions are unknown and complicated. Most of the existing control charts are uni-aspect schemes because they can detect a shift in only one aspect of the process distribution, like location or scale. Research on schemes for simultaneous surveillance of location and scale parameters has been very active in recent years, leading to many bi-aspect schemes. These schemes do not explicitly deal with the shape of the process distribution. However, a shift in the shape parameter with or without a location or scale shift can occur in practice, especially in production or time-to-event processes. Note that there are nonparametric schemes for detecting general shifts based on goodness-of-fit test statistics or empirical likelihood ratio statistics, which cannot isolate if there is a shift in location, scale, shape, or combination of these parameters. This article introduces a new distribution-free process monitoring scheme that can detect a shift in the location, scale or shape parameters, or any combinations of them. The new scheme is based on a combined statistic designed via Euclidean distance of the standardized Wilcoxon statistic for location, the standardized Ansari-Bradley statistic for scale, and the standardized Savage-type statistic for shape. We discuss the implementation design using the average run-length as the performance metric and investigate the proposed scheme's in-control performance. It is shown that the new charting scheme is in-control robust irrespective of the underlying process distribution and therefore applicable to monitor any univariate continuous processes. An out-of-control performance comparison study of the new scheme with many existing schemes shows that the new scheme is preferable to the existing schemes. We illustrate the proposed chart with real data to monitor a passenger train's arrival delays in Italy's regional route. The proposed scheme is comprehensive because it integrates the follow-up procedure via integrated sub-charts for classifying the signaling component.

“…observations. Then, as

\min false{m, n false} \to \infty,

the limiting IC distribution of

T_{j}, \forall j,

is approximately equivalent to the distribution of 1.73 Z + 0.27, where Z follows a

χ^{2}

distribution with 1.579 df.Proof See Kössler and Mukherjee 40 …”

Section: Implementation Of the Lvs Chartmentioning

confidence: 98%

“…Suppose that test samples come from a univariate continuous distribution function denoted by

G_{Y}

. Kössler and Mukherjee 40 introduced a versatile shift model, combining the general location‐scale alternative and the Lehmann alternative as

G_{Y} false(x false) = {([], F_{X} ((), \frac{x - θ}{e^{ϑ}}))}^{e^{δ}}, false(θ, ϑ, δ false) \in ℝ^{3},

where

θ, ϑ, δ

are respectively the location, scale and shape parameters. Equivalently, in a simplified notation, one may write

G_{Y} false(x false) = {([], F_{X} ((), \frac{x - θ}{ϑ^{'}}))}^{δ^{'}}, θ \in ℝ, ϑ^{'} \in ℝ^{+}, δ^{'} \in ℝ^{+},

where

θ

is as before,

ϑ^{'}

and

δ^{'}

are the reparameterized scale and shape components, respectively.…”

Section: The Distribution‐free Tri‐aspect Schemementioning

confidence: 99%

“…While the alternatives of types

H_{l}, H_{v}, H_{s,}

H_{italiclv}

Section: The Distribution‐free Tri‐aspect Schemementioning

confidence: 99%

Section: F I G U R E 3 Histogram and Density Plot Of The Phase-i Obse...mentioning

confidence: 99%

Section: The Distribution-free Tri-aspect Schemementioning

confidence: 99%

See 3 more Smart Citations

A comprehensive distribution‐free scheme for tri‐aspect surveillance of complex processes

Appl Stoch Models Bus & Ind

Qiu

Marozzi

2021

Self Cite

An optimally designed distribution‐free CUSUM procedure for tri‐aspect surveillance of continuous processes

Tang

Quality & Reliability Eng

2023

Self Cite

In the past few decades, research on nonparametric process monitoring schemes mainly dealt with the uni‐aspect or bi‐aspect schemes, focusing on monitoring process location or scale separately or jointly. Another critical process characteristic, namely, the process shape, is not explicitly dealt with in‐depth in most existing charting schemes. In classical hypothesis testing, some recent literature clearly showed that the multi‐aspect test statistics, although designed for very restrictive alternatives, often perform as well or better than many statistics for arbitrary distributional shifts. They are often better than Kolmogorov‐Smirnov or Cramér‐von Mises statistics and some empirical likelihood‐based test statistics. The current paper aims to use a tri‐aspect statistic to design a distribution‐free Phase‐II Cumulative Sum (CUSUM) charting scheme for monitoring any arbitrary process changes in the process. The proposal is nonparametric and is equivalent to an unknown standard case. It reflects, in addition, which parameters, among location, scale, or shape, are more responsible for a signal. The construction of the CUSUM scheme from an existing tri‐aspect Shewhart‐type chart is simple, so significant attention is devoted to determining a near‐optimal reference parameter of the chart in keeping an unknown shift type in mind. Comparisons of the optimal performance of various competitors are considered in terms of the median run length (MRL) metric. The proposed charting scheme designed with the tri‐aspect statistic compares highly favorably with many existing SPM schemes. The same is evident from our findings based on the Monte‐Carlo simulation. Finally, the proposed schemes are illustrated with a flow‐width measurement monitoring example.

A distribution‐free procedure for testing versatile alternative in medical multisample comparison studies

Kössler

Marozzi

2022

Statistics in Medicine

Self Cite

We propose a test for multisample comparison studies that can be applied without strict assumptions, especially when the underlying population distributions are far from normal. The new test can detect differences not only in location or scale but also in shape parameters among parent population distributions. We are motivated by numerous medical studies, where the variables are not normally distributed and may present in the various groups more complex differences than simple differences in a particular aspect of underlying distributions, such as location or scale. In these situations, traditional ANOVA and Kruskal-Wallis tests are unreliable since the underlying assumptions are not valid. The proposed procedure also allows the researcher to determine which aspects are more responsible for a significant result. This is an important practical advantage over procedures that test for general differences among the distribution functions but cannot identify which aspects lead to significant results. The asymptotic distribution of the test statistic is analyzed along with its small sample behavior against several competing tests. The practical advantages of the proposed procedure are illustrated with a multisample comparison study of a biomarker for liver damage in patients with hepatitis C.