Page 275 - Contributed Paper Session (CPS) - Volume 7

CPS2099 Takatsugu Yoshioka et al.

and

Anderson and Olkin (1985) developed an approach to derive the maximum
likelihood estimators (MLEs) of the mean vector and the covariance matrix with
several missing patterns. Kanda and Fujikoshi (1998) proposed the distribution
of the MLEs in the cases of two-step, three-step, and general monotone
missing data. For a two-step monotone missing data, Seko, Yamazaki and Seo
(2012) derived Hotelling’s T type statistic and an accurate simple approach to
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give the upper percentiles in one-sample problem, and Seko, Kawsaki and Seo
(2011) provided Hotelling’s T type statistic of testing for two normal mean
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vectors and its approximate upper percentile. Kawasaki and Seo (2016) derived
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the asymptotic expansion of the Hotelling’s T type test statistics for large
sample and proposed the Bartlett corrected statistics with one-sample
problem. Their results are theoretical results; however, the equation is slightly
complicated.
The aim of this study is to propose simple and convenient approximations
with two-step monotone missing data by adjusting the degrees of freedom.
For adjusting degrees of freedom, Yanagihara and Yuan (2005) provided some
approximate solutions to the multivariate Behrens-Fisher problem that are two
F approximations with approximate degrees of freedom for complete data.
Kawasaki and Seo (2015) proposed some new approximate solutions by
deriving the asymptotic expansions up to the term of order N for the
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moments of test statistic under the multivariate Behrens-Fisher problem with
complete data. Note that the asymptotic expansions up to the term of order
N for the moments of test statistic are obtained by Yanagihara and Yuan
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(2005). Krishnamoorthy and Pannala (1999) derived an approximate
distribution of the Hotelling’s type test statistic by a constant time an F
distribution using the decompositions of the statistics.
In the following section, we propose approximate solutions by adjusting
the degrees of freedom of F distribution. We perform Monte Carlo simulations
in Section 3.

2. Methodology
In this section, we will consider approximate solutions with two-step
monotone missing data. In this study, we employ the asymptotic expansion of
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the Hotelling’s T statistic by Kawasaki and Seo (2016) in a situation when

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