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and testing variance components. This is accomplished by allowing test variables to depend on observable random vectors as well as their observed values, as in the
Bayesian treatment of the problem, but without having to treat constant parameters as random variables.
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Krishnamoorthy, K. and Tian, L. (2007), “Inferences on the ratio of means of two inverse
Gaussian distributions: the generalized variable approach”, Journal of Statistical Planning and Inferences, Volume 138, Issue 7, 1, Pages
119:
that are valid only when the sample size is large. With small samples, such methods often have poor performance. Use of approximate and asymptotic methods may lead to misleading conclusions or may fail to detect truly
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135:-values are exact statistical methods in that they are based on exact probability statements. While conventional statistical methods do not provide exact solutions to such problems as testing
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834:
893:
Mathew, T. and Webb, D. W. (2005). Generalized p-values and confidence intervals for variance components: Applications to Army test and evaluation, Technometrics, 47, 312-322.
611:{\displaystyle R={\frac {{\overline {x}}S}{s\sigma }}-{\frac {{\overline {X}}-\mu }{\sigma }}={\frac {\overline {x}}{s}}{\frac {\sqrt {U}}{\sqrt {n}}}~-~{\frac {Z}{\sqrt {n}}},}
475:
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Tian, L. and Wu, Jianrong (2006) “Inferences on the Common Mean of
Several Log-normal Populations: The Generalized Variable Approach”, Biometrical Journal.
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Gamage J, Mathew T, and
Weerahandi S. (2013). Generalized prediction intervals for BLUPs in mixed models, Journal of Multivariate Analysis}, 220, 226-233.
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Li, X., Wang J., Liang H. (2011). Comparison of several means: a fiducial based approach. Computational
Statistics and Data Analysis, 55, 1993-2002.
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Hamada, M., and
Weerahandi, S. (2000). Measurement System Assessment via Generalized Inference. Journal of Quality Technology, 32, 241-253.
46:
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and its observed value are both free of nuisance parameters. Therefore, a test of a hypothesis with a one-sided alternative such as
68:
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154:-values, Tsui and Weerahandi extended the classical definition so that one can obtain exact solutions for such problems as the
116:
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Zhou, L., and Mathew, T. (1994). Some Tests for
Variance Components Using Generalized p-Values, Technometrics, 36, 394-421.
272:
be the sample mean and the sample variance. Inferences on all unknown parameters can be based on the distributional results
103:
Conventional statistical methods do not provide exact solutions to many statistical problems, such as those arising in
155:
39:
33:
50:
121:
896:
Wu, J. and Hamada, M. S. (2009) Experiments: Planning, Analysis, and
Optimization. Wiley, Hoboken, New Jersey.
836:, a quantity that can be easily evaluated via Monte Carlo simulation or using the non-central t-distribution.
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171:-values in a simple example, consider a situation of sampling from a normal population with the mean
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under unequal variances, exact tests for such problems can be obtained based on generalized
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100:, which except in a limited number of applications, provides only approximate solutions.
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911:-values in significance testing of hypotheses in the presence of nuisance parameters"
481:-values, the task can be easily accomplished based on the generalized test variable
115:. As a result, practitioners often resort to approximate statistical methods or
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349:{\displaystyle Z={\sqrt {n}}({\overline {X}}-\mu )/\sigma \sim N(0,1)}
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XPro, Free software package for exact parametric statistics
443:
Now suppose we need to test the coefficient of variation,
433:{\displaystyle U=nS^{2}/\sigma ^{2}\sim \chi _{n-1}^{2}.}
150:
In order to overcome the shortcomings of the classical
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477:. While the problem is not trivial with conventional
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69:Learn how and when to remove this message
774:{\displaystyle H_{A}:\rho <\rho _{0}}
93:is an extended version of the classical
32:This article includes a list of general
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829:{\displaystyle p=Pr(R\geq \rho _{0})}
7:
905:Tsui, K. and Weerahandi, S. (1989):
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167:To describe the idea of generalized
470:{\displaystyle \rho =\mu /\sigma }
38:it lacks sufficient corresponding
14:
781:can be based on the generalized
715:. Note that the distribution of
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668:{\displaystyle {\overline {X}}}
641:{\displaystyle {\overline {x}}}
238:{\displaystyle {\overline {X}}}
961:Statistical hypothesis testing
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117:asymptotic statistical methods
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867:Tsui & Weerahandi (1989)
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156:Behrens–Fisher problem
928:Springer-Verlag, New York.
211:{\displaystyle \sigma ^{2}}
131:Tests based on generalized
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648:is the observed value of
53:more precise citations.
922:Weerahandi, S. (1995)
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265:{\displaystyle S^{2}}
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184:{\displaystyle \mu }
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855:Weerahandi (1995)
728:{\displaystyle R}
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688:{\displaystyle s}
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59:January 2017
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126:experiments
122:significant
51:introducing
887:2082-2089.
875:References
83:statistics
34:references
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