Dealing with small samples and dimensionality issues in data envelopment analysis

Transcription

1 MPRA Munch Personal RePEc Archve Dealng wth small samples and dmensonalty ssues n data envelopment analyss Panagots Zervopoulos Unversty of Western Greece 5. February 2012 Onlne at MPRA Paper No , posted 5. June :25 UTC

2 Dealng wth small samples and dmensonalty ssues n Data Envelopment Analyss Panagots D. Zervopoulos Department of Busness Admnstraton of Food and Agrcultural Enterprses Unversty of Western Greece, 2 Georgou Sefer St, Agrno, Greece pzervopoulos@uo.gr Abstract Data Envelopment Analyss (DEA) s a wdely appled nonparametrc method for comparatve evaluaton of frms effcency. A defcency of DEA s that the effcency scores assgned to each frm are senstve to samplng varatons, partcularly when small samples are used. In addton, an upward bas s present due to dmensonalty ssues when the sample sze s lmted compared to the number of nputs and output. As a result, n case of small samples, DEA effcency scores cannot be consdered as relable measures. The DEA Bootstrap addresses ths lmtaton of the DEA method as t provdes the effcency scores wth stochastc propertes. However, the DEA Bootstrap s stll napproprate n the presence of small samples. In ths context, we ntroduce a new method that draws on random data generaton procedures, unlke Bootstrap whch s based on resamplng, and Monte Carlo smulatons. Keywords: Data envelopment analyss; Data generaton process; Random data; Bootstrap; Bas correcton; Effcency 1. Introducton Data Envelopment Analyss (DEA) s a wdely appled nonparametrc method for assessng operatonal effcency of homogeneous unts. The unts or, decson makng unts (DMUs) nvolved n the effcency evaluaton process are predomnantly a sample of a broader populaton. Populaton data are ether dffcult to collect or unknown. Consderng the nonparametrc property of DEA, or even ts lmted statstcal underpnnng, the yelded effcency scores are senstve to samplng varatons (Smar and Wlson 1998). Hence, the effcency scores assgned to the sample DMUs should not be consdered as global relatve assessment measures, but rather solely as local. Another ssue rased n the DEA lterature s assocated wth the dmensonalty curse that plagues DEA effcency scores. A plethora of scholars hghlght the upward bas of the DEA effcency scores when the sample sze s nadequate for the number of nput and output 1

3 varables (Perelman and Santn 2009; Cooper et al. 2007; Smar 2007; Sherman and Zhu 2006; Coell et al. 2005; Staat 2001; Smth 1997; Banker 1993). Cooper et al. (2007), Zhang and Bartels (1998), and Smth (1997) have defned an approprate sample sze for bas-free estmatons of up to 160 unts, or a sample adjusted accordngly to the number of utlzed nput and output varables. Bootstrap, and partcularly the DEA Bootstrap put forth by Smar and Wlson (1998) tackles the problem of relablty of the DEA effcency scores when sample data are utlzed n the evaluaton process. The DEA Bootstrap, or smoothed Bootstrap, s a combnaton of the orgnal Bootstrap (Efron 1979) modfed wth a smoothng parameter (Slverman 1986) and DEA (Charnes et al. 1978). To be more precse, Smar and Wlson manage to estmate bas n the DEA effcency scores that s due to samplng varatons. They apply a smoothed Bootstrap for generatng randomly sampled effcency scores that are then used for estmatng bootstrapped nputs (nput-orented approach) or outputs (output-orented approach). Subsequently, the bootstrapped nputs or outputs are ntroduced to the DEA lnear programmng models for bas-corrected effcency scores. The DEA Bootstrap nherts the vrtues of the orgnal Bootstrap wthout avodng though ts lmtatons. A major lmtaton of the Bootstrap method when t s appled to nonparametrc settngs s the mnmum requred sample data for estmatng the varablty of the populaton data (Chernck 2008). Ths weakness s also mpled by Efron and Tbshran (1998). In ths context, Chernck (2008) proposed a mnmum sample sze of 50 observatons for estmatng relable scores consstent wth the populaton dstrbuton. The proposed method overcomes the lmtaton of Bootstrap, partcularly of the DEA Bootstrap, as t yelds effcency scores to DMUs that resemble, more so than those obtaned by the DEA Bootstrap, the true effcency scores when small samples of observatons are avalable. The new method also cures the dmensonalty problem of DEA as the adaptablty of the estmated sample effcency scores to the true populaton scores ncreases aganst the DEA Bootstrap results when more nput and output varables are ncorporated n the producton process. 2. Breakdown of the new bas-correcton method The ntroduced method s not a resamplng as Bootstrap, rather t draws on truncated random data generaton processes to estmate the unknown populaton dstrbuton F from the emprcal dstrbuton ˆF. The scope of the new method s to estmate the populaton effcency scores p, p 1,2,..., m by producng an estmator ˆF of the populaton dstrbuton 2

4 from the effcency scores ˆ ˆ, 1, 2,..., F n defned by DEA. Bas-corrected effcency scores, 1,2,..., ˆ n are generated by F n the pursut of and. p Let a DMU n y 1 u x, where denotes the k-number nputs ( ) and stands k x k x k y for the -number outputs ( y Varable Returns to Scale (VRS) model (Banker et al., 1984) mn st.. X x Y y o 1 o ). By applyng DEA, for nstance, the nput-orented 0 (1) we obtan effcency scores ˆ ˆ 0 ˆ 1 n 1 for every DMU. Accordngly, n the case of the output-orented VRS DEA model, we defne ˆ ˆ ˆ effcency scores 1 n 1 for every DMU. In the followng analyss we presume nput orentaton s appled. Based on the effcency scores (.e., 1,2,..., n) assgned to the sample DMUs, a ˆ truncated random data generaton process T s utlzed to produce a sequence of pseudo-numbers x for every DMU. Every sequence of pseudo-numbers orgnates 1 from every sngle effcency score or from a combnaton of a targeted effcency score and the average scores of the sample. ˆ ˆ to produce o 1, 2, T x..., or n ˆ ˆ 1 ˆ T z (1 z) n to produce xo 1,2,..., n; 1,2,..., (2) 1 where x mn, ˆ o x 3

5 In addton, T( x ) N( ˆ, se 2 ) and T( x ) N( ˆ ( ), ˆ ( ) cv ) (3) n ˆ ˆ 1 () (1 ) ˆ z z n 1 where z s a user-defned credblty score that denotes the magntude of a sngle effcency score, and complementary of the sample mean effcency scores, on the generaton of a truncated random sequence of data (scores). In fact, there s nherent dependency between the effcency scores of the sample DMUs that s due to the comparatve assessment procedure appled through DEA. Moreover, x represents the randomly generated data, the x o expresses selected randomly generated replcas of the effcency score for the -number elements of the sequence, and cv stands for the coeffcent of varaton. The bas-corrected effcency score for every DMU s defned as follows sx ( ) 1,2,..., n; 1,2,..., (4) o where s s a statstc (.e., mean) It s straghtforward that the bas s expressed as ˆ bas TRDG where [0,1) (5) The standard error of the proposed truncated random data generaton (TRDG) process s TRDG 1 2 se [ xo s ( )] (6) 1 1/2 where s () 1 x o 1 Takng nto account equatons (4) and (6), the confdence nterval of the bas-corrected effcency scores are formed as follows TRDG (1 a/2) (1 a/2) t( 1) se, t( 1) se TRDG (7) 4

6 where denotes the level of sgnfcance, we prove that ˆ ub Pr ob, 1,2,..., n 0 (8) and 1 L ˆ ub l l1 L Pr ob, 1,2,..., n e (9) where ub stands for the upper bound of the confdence nterval of the bas-corrected effcency scores. Acknowledgng the nhert randomness n the proposed method, all the provded proofs or statements result from teratve procedures. In formulaton (9), the probablty, that s the average of L=1000 teratons, s equal to an nfntesmal value. The cases n whch ths nfntesmal probablty s present are dentfed and presented n order to be avoded by the user of the proposed method. The nhert randomness n the proposed method s regarded as a drawback because t s a source of nstablty for the obtaned results when the method s appled repeatedly. To overcome ths drawback, a stablzaton parameter s ntroduced n the procedure that elmnates up to 99% the varaton of the bas-corrected scores. The parameter expresses the number of teratons for the formulatons (2)-(7). The reported results are average scores. The proposed method for dealng wth samplng varatons and dmensonalty ssues n DEA s expressed by the followng functon f ˆ(, cv, z,,, n,var ) ex ex TRDG (10) ex In formulaton (10), two exogenous parameters n and var ex are ncluded whch denote the number of DMUs n the orgnal sample and the number of nput and output varables, respectvely, that are utlzed for defnng the effcency scores through DEA. These two parameters mplctly nfluence the bas-correcton procedure. Based on a numercal example and on the results that are tested through Monte Carlo so that to elmnate randomness, the proposed method yelds better estmators ( populaton effcency scores ( ) than the DEA Bootstrap ( boot TRDG ) for the ) when the orgnal sample conssts of less than 50 DMUs. In addton, the adaptve power of TRDG s to ncreases aganst boot s when the number of nput and output varables ncreases. 5

7 3. Concluson In ths paper, a new method for correctng bas n DEA effcency scores s presented. Commonly, DEA yelds overestmated effcency scores when sample data rather than populaton data are used, and the number of DMUs s lmted compared to the number of varables. In some studes, adequate sample szes have been determned for obtanng unbased effcency scores. However, n many cases the requred sample sze cannot be collected (e.g., automoble ndustry, power companes, water companes). In ths paper s presented a new method for correctng bas n DEA effcency scores when small samples are avalable (.e., n<50 DMUs). The new method enhances the applcablty of DEA when the DEA Bootstrap fals due to the lmted number of DMUs under evaluaton, or the nadequate sample sze compared to the number of nput and output varables. The new approach does not draw on resamplng but on an teratve truncated random number generaton procedure. Despte the nhert randomness of the new method, the results are robust and the proposed procedure does not suffer from nstablty. In addton, t s proved that the results obtaned by the proposed method are more adaptve to realty than those estmated by the DEA Bootstrap when small samples are avalable. References Banker RD (1993) Maxmum-Lkelhood, Consstency and Data Envelopment Analyss - a Statstcal Foundaton. Manage Sc 39 (10): Charnes A, Cooper WW, Rhodes E (1978) Measurng the effcency of decson makng unts. Eur J Oper Res 2 (6): Chernck MR (2008) Bootstrap Methods: A Gude for Practtoners and Researchers. John Wley & Sons, New Jersey Coell T, Rao P, O'Donnell CJ, Battese G (2005) An Introducton to Effcency and Productvty Analyss. Sprnger, New York Cooper WW, Seford LM, Tone K (2007) Data envelopment analyss: a comprehensve text wth models, applcatons, references and DEA-Solver software. 2nd edn. Sprnger Scence + Busness Meda, New York Efron B (1979) Bootstrap methods; another look at the jacknfe. Annals of Statstcs 7:1-26 Efron B, Tbshran RJ (1998) An Introducton to the Bootstrap. Chapman & Hall/CRC, Boca Raton Perelman S, Santn D (2009) How to generate regularly behaved producton data? A Monte Carlo expermentaton of DEA scale effcency measurement. Eur J Oper Res 19: Sherman HD, Zhu J (2006) Benchmarkng wth qualty-adjusted DEA (Q-DEA) to seek lower-cost hgh-qualty servce: Evdence from a US bank applcaton. Ann Oper Res 145: do:doi /s Slverman BW (1986) Densty Estmaton for Statstcs and Data Analyss. Chapman and Hall, London 6

8 Smar L (2007) How to mprove the performances of DEA/FDH estmators n the presence of nose? J Prod Anal 28: Smar L, Wlson PW (1998) Senstvty analyss of effcency scores: How to bootstrap n nonparametrc fronter models. Manage Sc 44 (1):49-61 Smth P (1997) Model msspecfcaton n Data Envelopment Analyss. Ann Oper Res 73: Staat M (2001) The effect of sample sze on the mean effcency n DEA: Comment. J Prod Anal 15: Zhang Y, Bartels R (1998) The effect of sample sze on the mean effcency n DEA wth an applcatno to electrcty dstrbuton n Australa, Sweden and New Zealand. J Prod Anal 9: