| Title:
|
D-optimal and highly D-efficient designs with non-negatively correlated observations (English) |
| Author:
|
Katulska, Krystyna |
| Author:
|
Smaga, Łukasz |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
52 |
| Issue:
|
4 |
| Year:
|
2016 |
| Pages:
|
575-588 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we consider D-optimal and highly D-efficient chemical balance weighing designs. The errors are assumed to be equally non-negatively correlated and to have equal variances. Some necessary and sufficient conditions under which a design is D*-optimal design (regular D-optimal design) are proved. It is also shown that in many cases D*-optimal design does not exist. In many of those cases the designs constructed by Masaro and Wong (2008) and some new designs are shown to be highly D-efficient. Theoretical results are accompanied by numerical search, suggesting D-optimality of designs under consideration. (English) |
| Keyword:
|
correlation |
| Keyword:
|
D-efficiency |
| Keyword:
|
D-optimal chemical balance weighing design |
| Keyword:
|
Hadamard matrix |
| Keyword:
|
simulated annealing algorithm |
| Keyword:
|
tabu search |
| MSC:
|
15A18 |
| MSC:
|
62K05 |
| idZBL:
|
Zbl 06644311 |
| idMR:
|
MR3565770 |
| DOI:
|
10.14736/kyb-2016-4-0575 |
| . |
| Date available:
|
2016-10-20T08:11:30Z |
| Last updated:
|
2018-01-10 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145906 |
| . |
| Reference:
|
[1] Angelis, L., Bora-Senta, E., Moyssiadis, C.: Optimal exact experimental designs with correlated errors through a simulated annealing algorithm..Comput. Statist. Data Anal. 37 (2001), 275-296. Zbl 0990.62061, MR 1862514, 10.1016/s0167-9473(01)00011-1 |
| Reference:
|
[2] Banerjee, K. S.: Weighing Designs for Chemistry, Medicine, Economics, Operations Research, Statistics..Marcel Dekker Inc., New York 1975. Zbl 0334.62030, MR 0458751 |
| Reference:
|
[3] Bora-Senta, E., Moyssiadis, C.: An algorithm for finding exact D- and A-optimal designs with $n$ observations and $k$ two-level factors in the presence of autocorrelated errors..J. Combinat. Math. Combinat. Comput. 30 (1999), 149-170. Zbl 0937.62074, MR 1705339 |
| Reference:
|
[4] Bulutoglu, D. A., Ryan, K. J.: D-optimal and near D-optimal $2^k$ fractional factorial designs of resolution $V$..J. Statist. Plann. Inference 139 (2009), 16-22. Zbl 1284.62473, MR 2460547, 10.1016/j.jspi.2008.05.012 |
| Reference:
|
[5] Ceranka, B., Graczyk, M.: Optimal chemical balance weighing designs for $v+1$ objects..Kybernetika 39 (2003), 333-340. Zbl 1248.62128, MR 1995737 |
| Reference:
|
[6] Ceranka, B., Graczyk, M.: Robustness optimal spring balance weighing designs for estimation total weight..Kybernetika 47 (2011), 902-908. Zbl 1274.62492, MR 2907850 |
| Reference:
|
[7] Ceranka, B., Graczyk, M., Katulska, K.: A-optimal chemical balance weighing design with nonhomogeneity of variances of errors..Statist. Probab. Lett. 76 (2006), 653-665. Zbl 1090.62074, MR 2234783, 10.1016/j.spl.2005.09.012 |
| Reference:
|
[8] Ceranka, B., Graczyk, M., Katulska, K.: On certain A-optimal chemical balance weighing design..Comput. Statist. Data Analysis 51 (2007), 5821-5827. MR 2407680, 10.1016/j.csda.2006.10.021 |
| Reference:
|
[9] Cheng, C. S.: Optimal biased weighing designs and two-level main-effect plans..J. Statist. Theory Practice 8 (2014), 83-99. MR 3196641, 10.1080/15598608.2014.840520 |
| Reference:
|
[10] Domijan, K.: tabuSearch: R based tabu search algorithm. R package version 1.1..\url{http://CRAN.R-project.org/package=tabuSearch} (2012) |
| Reference:
|
[11] Ehlich, H.: Determinantenabschätzungen für binäre Matrizen..Math. Zeitschrift 83 (1964), 123-132. Zbl 0115.24704, MR 0160792, 10.1007/bf01111249 |
| Reference:
|
[12] Ehlich, H.: Determinantenabschätzungen für binäre Matrizen mit $n\equiv 3 \mathrm{mod} 4$..Math. Zeitschrift 84 (1964), 438-447. MR 0168573, 10.1007/bf01109911 |
| Reference:
|
[13] Galil, Z., Kiefer, J.: D-optimum weighing designs..Ann. Statist. 8 (1980), 1293-1306. Zbl 0598.62087, MR 0594646, 10.1214/aos/1176345202 |
| Reference:
|
[14] Graczyk, M.: A-optimal biased spring balance weighing design..Kybernetika 47 (2011), 893-901. Zbl 1274.62495, MR 2907849 |
| Reference:
|
[15] Graczyk, M.: Some applications of weighing designs..Biometr. Lett. 50 (2013), 15-26. 10.2478/bile-2013-0014 |
| Reference:
|
[16] Harman, R., Bachratá, A., Filová, L.: Construction of efficient experimental designs under multiple resource constraints..Appl. Stochast. Models in Business and Industry 32 (2015), 1, 3-17. MR 3460885, 10.1002/asmb.2117 |
| Reference:
|
[17] Jacroux, M., Wong, C.S., Masaro, J.C.: On the optimality of chemical balance weighing designs..J. Statist. Planning Inference 8 (1983), 231-240. Zbl 0531.62072, MR 0720154, 10.1016/0378-3758(83)90041-1 |
| Reference:
|
[18] Jenkins, G. M., Chanmugam, J.: The estimation of slope when the errors are autocorrelated..J. Royal Statist. Soc., Ser. B (Statistical Methodology) 24 (1962), 199-214. Zbl 0116.11401, MR 0138154 |
| Reference:
|
[19] Jung, J. S., Yum, B. J.: Construction of exact D-optimal designs by tabu search..Comput. Statist. Data Analysis 21 (1996), 181-191. Zbl 0900.62403, MR 1394535, 10.1016/0167-9473(95)00014-3 |
| Reference:
|
[20] Katulska, K., Smaga, Ł.: D-optimal chemical balance weighing designs with $n\equiv 0 (\text{mod} 4)$ and $3$ objects..Comm. Statist. - Theory and Methods 41 (2012), 2445-2455. Zbl 1271.62175, MR 3003795, 10.1080/03610926.2011.608587 |
| Reference:
|
[21] Katulska, K., Smaga, Ł.: D-optimal chemical balance weighing designs with autoregressive errors..Metrika 76 (2013), 393-407. MR 3041462, 10.1007/s00184-012-0394-8 |
| Reference:
|
[22] Katulska, K., Smaga, Ł.: A note on D-optimal chemical balance weighing designs and their applications..Colloquium Biometricum 43 (2013), 37-45. |
| Reference:
|
[23] Katulska, K., Smaga, Ł.: On highly D-efficient designs with non-negatively correlated observations..REVSTAT - Statist. J. (accepted). |
| Reference:
|
[24] Li, C. H., Yang, S. Y.: On a conjecture in D-optimal designs with $n\equiv 0 (\mathrm{mod} 4)$..Linear Algebra Appl. 400 (2005), 279-290. MR 2132491, 10.1016/j.laa.2004.11.020 |
| Reference:
|
[25] Masaro, J., Wong, C. S.: D-optimal designs for correlated random vectors..J. Statist. Planning Inference 138 (2008), 4093-4106. Zbl 1147.62062, MR 2455990, 10.1016/j.jspi.2008.03.012 |
| Reference:
|
[26] Neubauer, M. G., Pace, R. G.: D-optimal $(0,1)$-weighing designs for eight objects..Linear Algebra Appl. 432 (2010), 2634-2657. Zbl 1185.62134, MR 2608182, 10.1016/j.laa.2009.12.007 |
| Reference:
|
[27] Pukelsheim, F.: Optimal Design of Experiments..John Wiley and Sons Inc., New York 1993. Zbl 1101.62063, MR 1211416 |
| Reference:
|
[28] Team, R Core: R: A language and environment for statistical computing..R Foundation for Statistical Computing, Vienna, Austria. URL http://www.R-project.org/ (2015). |
| Reference:
|
[29] Smaga, Ł.: D-optimal Chemical Balance Weighing Designs with Various Forms of the Covariance Matrix of Random Errors..Ph.D. Thesis, Adam Mickiewicz University, 2013 (in polish). |
| Reference:
|
[30] Smaga, Ł.: Necessary and sufficient conditions in the problem of D-optimal weighing designs with autocorrelated errors..Statist. Probab. Lett. 92 (2014), 12-16. MR 3230466, 10.1016/j.spl.2014.04.027 |
| Reference:
|
[31] Smaga, Ł.: Uniquely E-optimal designs with $n\equiv 2 (\mathrm{mod} 4)$ correlated observations..Linear Algebra Appl. 473 (2015), 297-315. MR 3338337, 10.1016/j.laa.2014.08.022 |
| Reference:
|
[32] Wojtas, M.: On Hadamard's inequality for the determinants of order non-divisible by $4$..Colloquium Mathematicum 12 (1964), 73-83. Zbl 0126.02604, MR 0168574, 10.4064/cm-12-1-73-83 |
| Reference:
|
[33] Yang, C. H.: On designs of maximal $(+1,-1)$-matrices of order $n\equiv 2 (\text{mod} 4)$..Math. Computat. 22 (1968), 174-180. Zbl 0167.01703, MR 0225476, 10.1090/s0025-5718-1968-0225476-4 |
| Reference:
|
[34] Yeh, H. G., Huang, M. N. Lo: On exact D-optimal designs with $2$ two-level factors and $n$ autocorrelated observations..Metrika 61 (2005), 261-275. Zbl 1079.62078, MR 2230375, 10.1007/s001840400336 |
| . |
