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LAND USE LAND COVER IMPACTS US TEMPERATURE TRENDS<br />

is reas<strong>on</strong>able to attach to ˆd, it is comm<strong>on</strong> to provide a<br />

c<strong>on</strong>fidence interval (CI) that c<strong>on</strong>tains d with a certain<br />

<str<strong>on</strong>g>c<strong>over</strong></str<strong>on</strong>g>age probability (0.90 in our study). The unknown<br />

value <str<strong>on</strong>g>of</str<strong>on</strong>g> d may be c<strong>on</strong>sidered positive if its CI c<strong>on</strong>tains<br />

<strong>on</strong>ly positive numbers, as is <strong>the</strong> case for 11 out <str<strong>on</strong>g>of</str<strong>on</strong>g> <strong>the</strong> 14<br />

stati<strong>on</strong>s in our analysis. Note also that it is incorrect to<br />

compare MSD1 and MSD2 by computing CIs for each<br />

and <strong>the</strong>n c<strong>on</strong>sidering MSD1 and MSD2 different if <strong>the</strong>ir<br />

CIs do not <strong>over</strong>lap (see, e.g. Schenker and Gentleman,<br />

2001).<br />

Classical statistical methods for computing CIs are<br />

based <strong>on</strong> assumpti<strong>on</strong>s about <strong>the</strong> data-generating mechanisms<br />

that are rarely met in climatology. One such<br />

assumpti<strong>on</strong> is that observati<strong>on</strong>s follow a Gaussian distributi<strong>on</strong>.<br />

It has been realized, however, that even small<br />

deviati<strong>on</strong>s from <strong>the</strong> assumpti<strong>on</strong>s may result in misleading<br />

inference (e.g. Wilcox, 2003). Fortunately, modern<br />

computer-intensive resampling (bootstrap) techniques<br />

(e.g. Efr<strong>on</strong> and Tibshirani, 1993; Davis<strong>on</strong> and Hinkley,<br />

1997; Lahiri, 2003) permit obtaining reliable inference<br />

without making questi<strong>on</strong>able assumpti<strong>on</strong>s about <strong>the</strong> data.<br />

The CIs in Table I were computed using <strong>the</strong> basic bootstrap.<br />

This implies, however, that <strong>the</strong> observati<strong>on</strong>s are<br />

independent and identically distributed, while climatological<br />

variables are typically serially correlated. It is known<br />

that bootstrap may underestimate <strong>the</strong> width <str<strong>on</strong>g>of</str<strong>on</strong>g> CIs in this<br />

case (e.g. Zwiers, 1990). Thus, our results regarding statistical<br />

significance may need refinement, which could<br />

be accomplished by employing ano<strong>the</strong>r bootstrap technique,<br />

subsampling (Politis et al., 1999), whose practical<br />

implementati<strong>on</strong> is now under active development (e.g.<br />

Gluhovsky et al., 2005; Gluhovsky and Agee, 2007).<br />

The same applies to our results <strong>on</strong> uncertainties in<br />

<strong>trends</strong> that may, in this respect, be c<strong>on</strong>sidered as incremental.<br />

In time series analysis, <strong>the</strong> assumpti<strong>on</strong> is <str<strong>on</strong>g>of</str<strong>on</strong>g>ten<br />

made that <strong>the</strong> trend is linear, while <strong>the</strong> residuals from<br />

<strong>the</strong> trend follow a linear autoregressive model. Bloomfield<br />

(1992) fitted such a model and a linear trend to<br />

an 1861–1989 <strong>temperature</strong> time series and found a linear<br />

trend <str<strong>on</strong>g>of</str<strong>on</strong>g> 0.58 with 95% (classical) CI, (0.37, 0.76).<br />

More recently, Craigmile et al. (2004) and Kallache et al.<br />

(2005) employed wavelets to assess <strong>trends</strong> while modeling<br />

fluctuati<strong>on</strong>s with fracti<strong>on</strong>al ARIMA models that<br />

incorporate l<strong>on</strong>g-range dependence.<br />

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Copyright © 2009 Royal Meteorological Society Int. J. Climatol. (2009)<br />

DOI: 10.1002/joc

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