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Communications in Statistics - Theory and Methods Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/lsta20
Properties of doubly-truncated gamma variables a
Christopher S. Coffey & Keith E. Muller
b
a
Dept of Preventive Medicine , Vanderbilt Unive.School of Med , A-1124 Medical Center North, Nashville, Tennessee, 37232-2637 b
Dept. of Biostatictics CB#7400 , University of North Carolina , 3105C McGavranGreenberg, Chapel Hill, North Carolina, 27599 Published online: 27 Jun 2007.
To cite this article: Christopher S. Coffey & Keith E. Muller (2000) Properties of doubly-truncated gamma variables, Communications in Statistics - Theory and Methods, 29:4, 851-857, DOI: 10.1080/03610920008832519 To link to this article: http://dx.doi.org/10.1080/03610920008832519
PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions
COMMtiN. STATIST.-THEORY
METE., 29(4), 851-857 (2000)
Chiistopher S. Coffey
Keith E. Mullc:
A- i 124 ~ e d i c aCenter i North
3 105C McGavran-Greenberg Dept. of Riostatistics, CR#7400 University uf North Czolina Chapel Hill, North Carolina 27599
-n y t nf' Preventive M ~ d i e i n ~ Vanderbi!: Univ. School of Mcd.
Downloaded by [Stony Brook University] at 01:41 30 October 2014
Nashville, Tennessee 37232-2637
-.
- .- irllnca~ed ..-- gamma distribution has been wideiy studied, primai-iiy in Hetesting and reiiabiiiiy settings, Most work has assumed an upper bomd on the i
>
support of the random variabie, i. e , t'ne space of the distribution is (0, uj. We consider. a doubiy-truncated gamma rartdom 9-ariable restricted by both a lowcr (1) and upper (u) truncation point, both of which are considered known. We provide sirnp!e f o m s for the densityi cumulative distribution function (CDF), moment . . . generating fimctioii, cumulant generating ~ n c t i o n ,charac:enst:c h n c l i m , and
-
moments.
extend the resuiis to describe tiie dsnsi~y,CDF, and momen& 3of a
doubly-truncated noncentral chi-square variable.
1. lNTRGDtTCTION
The truncated garnna distrib~tionhas heer! wide!y stndied, primarily in lifetesting and reliability settings. Most work has assumed an upper bound on the s~upport of the random variabie. i. e . the space of the distributioii is ( 0 , u j (Johnson; Kotz, and Salakrishnan, !994, $8.!1. We consider a dot?b!=trllncated J SLY-
Copyright O 2000 by Marcel Dekker, Inc.
www.dekker.com
COFFEY .4ND MULLER
552
In study pianning igr linear modeis with Gaussian er~:.ors. detenrrin~ngthe szrno!e size reauired to detect a specified effect w t h scme target p v e r requires
-
err;~be aPn!ied ~ ~ piiot stud\,/ i ~ in ~ nrdpr tn p:r_rt~~.t rgaimt m i ~ - ~ p e ~ ~ f inf ~ act2&. The x fir:^! s a q ! sizei ~ !V+ !her? -2i becomes a FGnc;ioii of 'h-.--: ------+ L--.1' :..+--.-' -:I-.& -a -.-, 1L I I G v a ~ I a l w iL ~ L I ~ I I C I I L ~ U ~ IL I~ I L I I I L L I I I ~ I p i LIL ti pi^, v1. .. LoItey and Miiiier 11 k 4s" ,, , rnmwi:c - i . d ; . d
AJL.~.L#
. .
- .~- . ,. js!:nscr., 5 . L., Kow, S , and Bai&r~sriiidli, 3. j ; ~ y y j j , Con:int;:3L;s ;I;:t:iya:~i;::~ Disrri&rianb-2, Ne\i; i'ork: lyi;j:,;,
Downloaded by [Stony Brook University] at 01:41 30 October 2014
+~
"
T.ee. Y . S. 2nd Liz, '1.. K. (1992). ';High Order co,iiish-Fisher E.xp~i;~jio~,"
-
~
.
'4-n!13.2 .+ ' S U ,
a
......,*.-
C?,:?;?:;' "
Lee, Y . S, znd Li.;
0,
.ra --7??_249 -.
-rx,
T . K. (1993;). "(zorrection to Aigoriihm AS 26.9: High Order
Expansion," Applied Siilii~fics,42, 268-263. -1 aylor,Comisb-Fisher D. J. and Muller, K. E. (i996). "Bias in Linear iviodei Power and Sample Size Caicuiarions Due to Es-iirnaiing Noncentralitj;," C ~ m ; x u t : i c d o n sir: Statistics: Theor?, and Methods, 25, i595-1610. Wities, J. and Erittain, F. (1990). "The Role oEInternai Pilot Studies in increasing the Efficency of Cilnical Trials," Sf&Sljc.y it: )nne&cine, 9,65-72, Zheng, Q. (i998). "computing Relations Getwee: S:a:istica! M=mer.ts 2nd
Cumulants," 1998 Proceedings of the ASA Statistical Conzputing Section, 165-170. Received August, 1999; Revised December, 1999
Communications in Statistics - Theory and Methods Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/lsta20
Properties of doubly-truncated gamma variables a
Christopher S. Coffey & Keith E. Muller
b
a
Dept of Preventive Medicine , Vanderbilt Unive.School of Med , A-1124 Medical Center North, Nashville, Tennessee, 37232-2637 b
Dept. of Biostatictics CB#7400 , University of North Carolina , 3105C McGavranGreenberg, Chapel Hill, North Carolina, 27599 Published online: 27 Jun 2007.
To cite this article: Christopher S. Coffey & Keith E. Muller (2000) Properties of doubly-truncated gamma variables, Communications in Statistics - Theory and Methods, 29:4, 851-857, DOI: 10.1080/03610920008832519 To link to this article: http://dx.doi.org/10.1080/03610920008832519
PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions
COMMtiN. STATIST.-THEORY
METE., 29(4), 851-857 (2000)
Chiistopher S. Coffey
Keith E. Mullc:
A- i 124 ~ e d i c aCenter i North
3 105C McGavran-Greenberg Dept. of Riostatistics, CR#7400 University uf North Czolina Chapel Hill, North Carolina 27599
-n y t nf' Preventive M ~ d i e i n ~ Vanderbi!: Univ. School of Mcd.
Downloaded by [Stony Brook University] at 01:41 30 October 2014
Nashville, Tennessee 37232-2637
-.
- .- irllnca~ed ..-- gamma distribution has been wideiy studied, primai-iiy in Hetesting and reiiabiiiiy settings, Most work has assumed an upper bomd on the i
>
support of the random variabie, i. e , t'ne space of the distribution is (0, uj. We consider. a doubiy-truncated gamma rartdom 9-ariable restricted by both a lowcr (1) and upper (u) truncation point, both of which are considered known. We provide sirnp!e f o m s for the densityi cumulative distribution function (CDF), moment . . . generating fimctioii, cumulant generating ~ n c t i o n ,charac:enst:c h n c l i m , and
-
moments.
extend the resuiis to describe tiie dsnsi~y,CDF, and momen& 3of a
doubly-truncated noncentral chi-square variable.
1. lNTRGDtTCTION
The truncated garnna distrib~tionhas heer! wide!y stndied, primarily in lifetesting and reliability settings. Most work has assumed an upper bound on the s~upport of the random variabie. i. e . the space of the distributioii is ( 0 , u j (Johnson; Kotz, and Salakrishnan, !994, $8.!1. We consider a dot?b!=trllncated J SLY-
Copyright O 2000 by Marcel Dekker, Inc.
www.dekker.com
COFFEY .4ND MULLER
552
In study pianning igr linear modeis with Gaussian er~:.ors. detenrrin~ngthe szrno!e size reauired to detect a specified effect w t h scme target p v e r requires
-
err;~be aPn!ied ~ ~ piiot stud\,/ i ~ in ~ nrdpr tn p:r_rt~~.t rgaimt m i ~ - ~ p e ~ ~ f inf ~ act2&. The x fir:^! s a q ! sizei ~ !V+ !her? -2i becomes a FGnc;ioii of 'h-.--: ------+ L--.1' :..+--.-' -:I-.& -a -.-, 1L I I G v a ~ I a l w iL ~ L I ~ I I C I I L ~ U ~ IL I~ I L I I I L L I I I ~ I p i LIL ti pi^, v1. .. LoItey and Miiiier 11 k 4s" ,, , rnmwi:c - i . d ; . d
AJL.~.L#
. .
- .~- . ,. js!:nscr., 5 . L., Kow, S , and Bai&r~sriiidli, 3. j ; ~ y y j j , Con:int;:3L;s ;I;:t:iya:~i;::~ Disrri&rianb-2, Ne\i; i'ork: lyi;j:,;,
Downloaded by [Stony Brook University] at 01:41 30 October 2014
+~
"
T.ee. Y . S. 2nd Liz, '1.. K. (1992). ';High Order co,iiish-Fisher E.xp~i;~jio~,"
-
~
.
'4-n!13.2 .+ ' S U ,
a
......,*.-
C?,:?;?:;' "
Lee, Y . S, znd Li.;
0,
.ra --7??_249 -.
-rx,
T . K. (1993;). "(zorrection to Aigoriihm AS 26.9: High Order
Expansion," Applied Siilii~fics,42, 268-263. -1 aylor,Comisb-Fisher D. J. and Muller, K. E. (i996). "Bias in Linear iviodei Power and Sample Size Caicuiarions Due to Es-iirnaiing Noncentralitj;," C ~ m ; x u t : i c d o n sir: Statistics: Theor?, and Methods, 25, i595-1610. Wities, J. and Erittain, F. (1990). "The Role oEInternai Pilot Studies in increasing the Efficency of Cilnical Trials," Sf&Sljc.y it: )nne&cine, 9,65-72, Zheng, Q. (i998). "computing Relations Getwee: S:a:istica! M=mer.ts 2nd
Cumulants," 1998 Proceedings of the ASA Statistical Conzputing Section, 165-170. Received August, 1999; Revised December, 1999
