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SUMMARY:Optimal design to test for heteroschedasticity in a regression mod
el
DTSTART:20220627T115000Z
DTEND:20220627T121000Z
DTSTAMP:20240301T040400Z
UID:indico-contribution-252@conferences.enbis.org
DESCRIPTION:Speakers: Alessandro Lanteri (University of Milan)\, chiara to
mmasi (University of Milan)\, Samantha Leorato (University of Milan)\n\nTh
e goal of this study is to design an experiment to detect a specific kind
of heteroscedasticity in a non-linear regression model\, i.e.\n$$\ny_i=\\e
ta(x_i\;\\beta)+\\varepsilon_i\,\\\; \\varepsilon_i\\sim N(0\;\\sigma^2 h(
x_i\;\\gamma))\,\\quad i=1\,\\ldots\,n\, \n$$\nwhere $\\eta(x_i\;\\beta)$
is a non-linear mean function\, depending on a vector of regression coef
ficients $\\beta\\in {\\rm I\\!R}^m$\, and $\\sigma^2 h(x_i\;\\gamma)$ is
the error variance depending on an unknown constant $\\sigma^2$ and on
a continuous positive function $h(\\cdot\;\\cdot)$\, completely known ex
cept for a parameter vector $\\gamma\\in {\\rm I\\!R}^s$. \n\nIn many pra
ctical problems\, it may be meaningful to test for the heteroscedasticity\
, that is to consider the null hypothesis $H_0\\!:\\\, \\gamma=\\gamma_0$\
, where $\\gamma_0$ is a specific value leading to the homoscedastic model
\, i.e. $h(x_i\;\\gamma_0)=1$\, and a local alternative $H_1\\!:\\\, \\gam
ma=\\gamma_0+\\lambda/\\sqrt{n}\,$ with $ \\lambda\\neq 0$.\nThe applicati
on of a likelihood-based test (such as log-likelihood ratio\, score or Wal
d statistics) is a common approach to tackle this problem\, since its a
symptotic distribution is known. \n \nThe aim of this study consists in d
esigning an experiment with the goal of maximizing (in some sense) the asy
mptotic power of a likelihood-based test. \nThe majority of the literature
in optimal design of experiments concerns the inferential issue of precis
e parameter estimation. Few papers are related to hypothesis testing. See
for instance\, Stigler (1971)\, Spruill (1990)\, Dette and Titoff (2009)
and the references therein\, which essentially concern designing to check
an adequate fit to the true mean function. \nIn this study\, instead\, we
justify the use of the $\\mathbf{{\\rm D}_s}$-criterion and the KL-optimal
ity (Lopez-Fidalgo\, Tommasi\, Trandafir\, 2007) to design an experiment w
ith the inferential goal of checking for heteroscedasticity. Both $\\mathb
f{{\\rm D}_s}$- and KL-criteria are proved to be related to the noncentral
ity parameter of the asymptotic chi-squared distribution of a likelihood t
est.\n\nhttps://conferences.enbis.org/event/18/contributions/252/
LOCATION:EL5
RELATED-TO:indico-event-18@conferences.enbis.org
URL:https://conferences.enbis.org/event/18/contributions/252/
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