메뉴 건너뛰기




Volumn 32, Issue 8, 2012, Pages 1293-1308

Quantile Uncertainty and Value-at-Risk Model Risk

Author keywords

Basel II; Maximum entropy; Model risk; Quantile; Risk capital; Value at risk

Indexed keywords

BASEL II; MAXIMUM ENTROPY; QUANTILE; RISK CAPITAL; VALUE AT RISK;

EID: 84865068828     PISSN: 02724332     EISSN: 15396924     Source Type: Journal    
DOI: 10.1111/j.1539-6924.2012.01824.x     Document Type: Article
Times cited : (63)

References (83)
  • 1
    • 1642329557 scopus 로고    scopus 로고
    • Amendment to the capital accord to incorporate market risks
    • Basel Committee on Banking Supervision.
    • Basel Committee on Banking Supervision. Amendment to the capital accord to incorporate market risks. Bank for International Settlements, Basel, 1996.
    • (1996) Bank for International Settlements, Basel
  • 2
    • 4644355177 scopus 로고    scopus 로고
    • International convergence of capital measurement and capital standards: A revised framework
    • Basel Committee on Banking Supervision.
    • Basel Committee on Banking Supervision. International convergence of capital measurement and capital standards: A revised framework. Bank for International Settlements, Basel, 2006.
    • (2006) Bank for International Settlements, Basel
  • 3
    • 84865083275 scopus 로고    scopus 로고
    • Revisions to the Basel II market risk framework
    • Basel Committee on Banking Supervision.
    • Basel Committee on Banking Supervision. Revisions to the Basel II market risk framework. Bank for International Settlements, Basel, 2009.
    • (2009) Bank for International Settlements, Basel
  • 4
    • 0002386952 scopus 로고    scopus 로고
    • Evaluation of Value-at-Risk models using historical data
    • Hendricks D. Evaluation of Value-at-Risk models using historical data. FRBNY Economic Policy Review, 1996:39-69.
    • (1996) FRBNY Economic Policy Review , pp. 39-69
    • Hendricks, D.1
  • 5
    • 70449106060 scopus 로고    scopus 로고
    • The level and quality of Value-at-Risk disclosure by commercial banks
    • Perignon C, Smith D. The level and quality of Value-at-Risk disclosure by commercial banks. Journal of Banking and Finance, 2010; 34(2):362-377.
    • (2010) Journal of Banking and Finance , vol.34 , Issue.2 , pp. 362-377
    • Perignon, C.1    Smith, D.2
  • 6
    • 32944462440 scopus 로고    scopus 로고
    • The hidden dangers of historical simulation
    • Pritsker M. The hidden dangers of historical simulation. Journal of Banking and Finance, 2006; 30:561-582.
    • (2006) Journal of Banking and Finance , vol.30 , pp. 561-582
    • Pritsker, M.1
  • 8
    • 85021287120 scopus 로고    scopus 로고
    • Value at risk when daily changes in market variables are not normally distributed
    • Hull J, White A. Value at risk when daily changes in market variables are not normally distributed. Journal of Derivatives, 1998; 5(3):9-19.
    • (1998) Journal of Derivatives , vol.5 , Issue.3 , pp. 9-19
    • Hull, J.1    White, A.2
  • 9
    • 0001535268 scopus 로고    scopus 로고
    • Conditional density and Value-at-Risk: Prediction of Asian currency exchange rates
    • Mittnik S, Paolella M. Conditional density and Value-at-Risk: Prediction of Asian currency exchange rates. Journal of Forecasting, 2000; 19:313-333.
    • (2000) Journal of Forecasting , vol.19 , pp. 313-333
    • Mittnik, S.1    Paolella, M.2
  • 10
    • 0005789445 scopus 로고    scopus 로고
    • Risk estimation using the normal inverse Gaussian distribution
    • Venter J, de Jongh P. Risk estimation using the normal inverse Gaussian distribution. Journal of Risk, 2002; 4:1-23.
    • (2002) Journal of Risk , vol.4 , pp. 1-23
    • Venter, J.1    de Jongh, P.2
  • 13
    • 67349106729 scopus 로고    scopus 로고
    • Assessing Value-at-Risk with CARE: The conditional autoregressive expectile models
    • Kuan C-M, Yeh J-H, Hsu Y-C. Assessing Value-at-Risk with CARE: The conditional autoregressive expectile models. Journal of Econometrics, 2009; 150(2):261-270.
    • (2009) Journal of Econometrics , vol.150 , Issue.2 , pp. 261-270
    • Kuan, C.-M.1    Yeh, J.-H.2    Hsu, Y.-C.3
  • 14
    • 0001925391 scopus 로고
    • Techniques for verifying the accuracy of risk measurement models
    • Kupiec P. Techniques for verifying the accuracy of risk measurement models. Journal of Derivatives, 1995; 3(2):73-84.
    • (1995) Journal of Derivatives , vol.3 , Issue.2 , pp. 73-84
    • Kupiec, P.1
  • 17
    • 4444289240 scopus 로고    scopus 로고
    • CAViaR: Conditional autoregressive Value at Risk by regression quantile
    • Engle R, Manganelli S. CAViaR: Conditional autoregressive Value at Risk by regression quantile. Journal of Business and Economic Statistics, 2004; 22:367-381.
    • (2004) Journal of Business and Economic Statistics , vol.22 , pp. 367-381
    • Engle, R.1    Manganelli, S.2
  • 18
    • 77953774534 scopus 로고    scopus 로고
    • The Turner review: A regulatory response to the global banking crisis
    • Turner L. The Turner review: A regulatory response to the global banking crisis. Financial Services Authority, London, 2009.
    • (2009) Financial Services Authority, London
    • Turner, L.1
  • 19
    • 34548568292 scopus 로고    scopus 로고
    • The black swan: The Impact of The Highly Improbable
    • Taleb N. The black swan: The Impact of The Highly Improbable. London: Penguin, 2007.
    • (2007) London: Penguin
    • Taleb, N.1
  • 20
    • 0002339024 scopus 로고
    • VAR: Seductive but dangerous
    • Beder T. VAR: Seductive but dangerous. Financial Analysts Journal, 1995; 51(5):12-24.
    • (1995) Financial Analysts Journal , vol.51 , Issue.5 , pp. 12-24
    • Beder, T.1
  • 22
    • 34250884839 scopus 로고
    • Kendall's Advanced Theory of Statistics
    • London: Edward Arnold
    • O'Hagan A. Kendall's Advanced Theory of Statistics. Vol 2B: Bayesian Inference. London: Edward Arnold, 1994.
    • (1994) Vol 2B: Bayesian Inference
    • O'Hagan, A.1
  • 25
    • 84865083676 scopus 로고    scopus 로고
    • Estimation error in the assessment of financial risk exposure. SSRN-id424363, version of June 29
    • Figlewski S. Estimation error in the assessment of financial risk exposure. SSRN-id424363, version of June 29, 2003.
    • (2003)
    • Figlewski, S.1
  • 26
    • 34248160827 scopus 로고    scopus 로고
    • Model risk
    • Derman E. Model risk. Risk, 1996; 9(5):139-145.
    • (1996) Risk , vol.9 , Issue.5 , pp. 139-145
    • Derman, E.1
  • 27
    • 84865081284 scopus 로고    scopus 로고
    • Model error: Evaluation of various finance models. New England Economic Review
    • Simons K. Model error: Evaluation of various finance models. New England Economic Review, 1997, 17-28.
    • (1997) , pp. 17-28
    • Simons, K.1
  • 29
    • 0009942755 scopus 로고    scopus 로고
    • Market risk and model risk for a financial institution writing options
    • Green T, Figlewski S. Market risk and model risk for a financial institution writing options. Journal of Finance, 1999; 54: 1465-1499.
    • (1999) Journal of Finance , vol.54 , pp. 1465-1499
    • Green, T.1    Figlewski, S.2
  • 31
    • 84865083677 scopus 로고    scopus 로고
    • Managing Model Risk
    • Alexander C. (ed.). UK: Pearson
    • Rebonato R. Managing Model Risk. Vol. 2 in Alexander C (ed.). Mastering Risk. UK: Pearson, 2001.
    • (2001) Mastering Risk , vol.2
    • Rebonato, R.1
  • 32
    • 0002899375 scopus 로고    scopus 로고
    • Measuring the risk in value at risk
    • Jorion P. Measuring the risk in value at risk. Financial Analysts Journal, 1996; 52:47-56.
    • (1996) Financial Analysts Journal , vol.52 , pp. 47-56
    • Jorion, P.1
  • 33
    • 30244543364 scopus 로고    scopus 로고
    • Worst case model risk management
    • Talay D, Zheng Z. Worst case model risk management. Finance and Stochastics, 2002; 6:517-537.
    • (2002) Finance and Stochastics , vol.6 , pp. 517-537
    • Talay, D.1    Zheng, Z.2
  • 36
    • 0034649483 scopus 로고    scopus 로고
    • A discussion of parameter and model uncertainty in insurance
    • Cairns A. A discussion of parameter and model uncertainty in insurance. Insurance: Mathematics and Economics, 2000; 27: 313-330.
    • (2000) Insurance: Mathematics and Economics , vol.27 , pp. 313-330
    • Cairns, A.1
  • 38
    • 33744788678 scopus 로고    scopus 로고
    • After VaR: The theory, estimation and insurance applications of quantile-based risk measures
    • Dowd K, Blake D. After VaR: The theory, estimation and insurance applications of quantile-based risk measures. Journal of Risk and Insurance, 2009; 73(2): 193-229.
    • (2009) Journal of Risk and Insurance , vol.73 , Issue.2 , pp. 193-229
    • Dowd, K.1    Blake, D.2
  • 39
    • 33745038284 scopus 로고    scopus 로고
    • Model uncertainty and its impact on the pricing of derivative instruments
    • Cont R. Model uncertainty and its impact on the pricing of derivative instruments. Mathematical Finance, 2006; 16(3): 519-547.
    • (2006) Mathematical Finance , vol.16 , Issue.3 , pp. 519-547
    • Cont, R.1
  • 40
    • 0035998717 scopus 로고    scopus 로고
    • A methodology for assessing model risk and its application to the implied volatility function model
    • Hull J, Suo W. A methodology for assessing model risk and its application to the implied volatility function model. Journal of Financial and Quantitative Analysis, 2002; 37(2):297-318.
    • (2002) Journal of Financial and Quantitative Analysis , vol.37 , Issue.2 , pp. 297-318
    • Hull, J.1    Suo, W.2
  • 41
    • 76249092394 scopus 로고    scopus 로고
    • Model risk: A conceptual framework for risk measurement and hedging
    • Available from SSRN
    • Branger N, Schlag C. Model risk: A conceptual framework for risk measurement and hedging. Working Paper, EFMA Basel Meetings. Available from SSRN, 2004.
    • (2004) Working Paper, EFMA Basel Meetings
    • Branger, N.1    Schlag, C.2
  • 42
    • 0039580836 scopus 로고    scopus 로고
    • Ordering univariate distributions by entropy and variance
    • Ebrahimi N, Maasoumi E, Soofi E. Ordering univariate distributions by entropy and variance. Journal of Econometrics, 1999; 90:317-336.
    • (1999) Journal of Econometrics , vol.90 , pp. 317-336
    • Ebrahimi, N.1    Maasoumi, E.2    Soofi, E.3
  • 43
    • 84856043672 scopus 로고
    • A mathematical theory of communication
    • July, (Oct): 623-656.
    • Shannon, C. A mathematical theory of communication. Bell System Technical Journal, 1948; 27 (July): 379-423 and (Oct): 623-656.
    • (1948) Bell System Technical Journal , vol.27 , pp. 379-423
    • Shannon, C.1
  • 44
    • 0002684291 scopus 로고
    • Maximal data information prior distributions
    • Aykac A. and Brumat C. (eds). Amsterdam: North-Holland
    • Zellner A. Maximal data information prior distributions. In: Aykac A. and Brumat C. (eds). New methods in the Applications of Bayesian Methods. Amsterdam: North-Holland, 1977.
    • (1977) New methods in the Applications of Bayesian Methods
    • Zellner, A.1
  • 45
    • 84865083674 scopus 로고
    • Papers on Probability, Statistics and Statistical Physics. In: Rosenkrantz R.D. (ed.), Dordrecht: Reidel Publishing Company
    • Jaynes E. Papers on Probability, Statistics and Statistical Physics. In: Rosenkrantz R.D. (ed.), Dordrecht: Reidel Publishing Company, 1983.
    • (1983)
    • Jaynes, E.1
  • 46
    • 0344669945 scopus 로고    scopus 로고
    • Entropy densities with an application to autoregressive conditional skewness and kurtosis
    • Rockinger M, Jondeau E. Entropy densities with an application to autoregressive conditional skewness and kurtosis. Journal of Econometrics, 2002; 106:119-142.
    • (2002) Journal of Econometrics , vol.106 , pp. 119-142
    • Rockinger, M.1    Jondeau, E.2
  • 47
    • 0346093801 scopus 로고    scopus 로고
    • Calculation of maximum entropy densities with application to income distribution
    • Wu X. Calculation of maximum entropy densities with application to income distribution. Journal of Econometrics, 2003; 115(2):347-354.
    • (2003) Journal of Econometrics , vol.115 , Issue.2 , pp. 347-354
    • Wu, X.1
  • 48
    • 67349083561 scopus 로고    scopus 로고
    • Modelling time-varying higher moments with maximum entropy density
    • Chan F. Modelling time-varying higher moments with maximum entropy density. Mathematics and Computers in Simulation, 2009; 79(9):2767-2778.
    • (2009) Mathematics and Computers in Simulation , vol.79 , Issue.9 , pp. 2767-2778
    • Chan, F.1
  • 50
    • 67349084720 scopus 로고    scopus 로고
    • Maximum entropy autoregressive conditional heteroskedasticity model
    • Park S, Bera A. Maximum entropy autoregressive conditional heteroskedasticity model. Journal of Econometrics, 2009; 150(2):219-230.
    • (2009) Journal of Econometrics , vol.150 , Issue.2 , pp. 219-230
    • Park, S.1    Bera, A.2
  • 51
    • 54249121221 scopus 로고    scopus 로고
    • Value-at-risk in US stock indices with skewed generalized error distribution
    • Lee M-C, Su J-B, Liu H-C. Value-at-risk in US stock indices with skewed generalized error distribution. Applied Financial Economics Letters, 2008; 4(6):425-431.
    • (2008) Applied Financial Economics Letters , vol.4 , Issue.6 , pp. 425-431
    • Lee, M.-C.1    Su, J.-B.2    Liu, H.-C.3
  • 52
    • 0001504360 scopus 로고
    • The variation of certain speculative prices
    • Mandelbrot B. The variation of certain speculative prices. Journal of Business, 1963; 36:394-419.
    • (1963) Journal of Business , vol.36 , pp. 394-419
    • Mandelbrot, B.1
  • 53
    • 0000375581 scopus 로고
    • A conditionally heteroskedastic time series model for speculative prices and rates of return
    • Bollerslev T. A conditionally heteroskedastic time series model for speculative prices and rates of return. Review of Economics and Statistics, 1987; 69(3):542-547.
    • (1987) Review of Economics and Statistics , vol.69 , Issue.3 , pp. 542-547
    • Bollerslev, T.1
  • 54
    • 0041853844 scopus 로고    scopus 로고
    • How accurate are Value-at-Risk models at commercial banks?
    • Berkowitz J, O'Brien J. How accurate are Value-at-Risk models at commercial banks? Journal of Finance, 2002; 55:1093-1111.
    • (2002) Journal of Finance , vol.55 , pp. 1093-1111
    • Berkowitz, J.1    O'Brien, J.2
  • 56
    • 0010917718 scopus 로고    scopus 로고
    • Incorporating volatility updating into the historical simulation method for Value-at-Risk
    • Hull J, White A. Incorporating volatility updating into the historical simulation method for Value-at-Risk. Journal of Risk, 1998; 1:5-19.
    • (1998) Journal of Risk , vol.1 , pp. 5-19
    • Hull, J.1    White, A.2
  • 57
    • 51349087307 scopus 로고    scopus 로고
    • Developing a stress testing framework based on market risk models
    • Alexander C, Sheedy E. Developing a stress testing framework based on market risk models. Journal of Banking and Finance, 2008; 32(10):2220-2236.
    • (2008) Journal of Banking and Finance , vol.32 , Issue.10 , pp. 2220-2236
    • Alexander, C.1    Sheedy, E.2
  • 59
    • 11144322968 scopus 로고    scopus 로고
    • Families of distributions arising from distributions of order statistics
    • Jones M. Families of distributions arising from distributions of order statistics. Test, 2004; 13(1):1-43.
    • (2004) Test , vol.13 , Issue.1 , pp. 1-43
    • Jones, M.1
  • 60
    • 67650879319 scopus 로고    scopus 로고
    • On families of beta- and generalized gamma-generated distributions and associated inference
    • Zografos K, Balakrishnan, N. On families of beta- and generalized gamma-generated distributions and associated inference. Statistical Methodology, 2009; 6(4):344-362.
    • (2009) Statistical Methodology , vol.6 , Issue.4 , pp. 344-362
    • Zografos, K.1    Balakrishnan, N.2
  • 61
    • 0000641348 scopus 로고
    • Conditional heteroskedasticity in asset returns: A new approach
    • Nelson DB. Conditional heteroskedasticity in asset returns: A new approach. Econometrica, 1991; 59:347-370.
    • (1991) Econometrica , vol.59 , pp. 347-370
    • Nelson, D.B.1
  • 62
    • 0034032267 scopus 로고    scopus 로고
    • Regional flood quantile estimation under linear transformation of the data
    • Arsenault M, Ashkar F. Regional flood quantile estimation under linear transformation of the data. Water Resources Research, 2000; 36(6):1605-1610.
    • (2000) Water Resources Research , vol.36 , Issue.6 , pp. 1605-1610
    • Arsenault, M.1    Ashkar, F.2
  • 63
    • 78851469671 scopus 로고    scopus 로고
    • Multivariate quantiles in hydrological frequency analysis
    • Chebana F, Ouarda T. Multivariate quantiles in hydrological frequency analysis. Environmetrics, 2011; 22(1):63-78.
    • (2011) Environmetrics , vol.22 , Issue.1 , pp. 63-78
    • Chebana, F.1    Ouarda, T.2
  • 65
    • 69349083757 scopus 로고    scopus 로고
    • Assessing and modelling changes in rainfall erosivity at different climate scales
    • Diodato N, Bellocchi G. Assessing and modelling changes in rainfall erosivity at different climate scales. Earth Surface Processes and Landforms, 2009; 34(7):969-980.
    • (2009) Earth Surface Processes and Landforms , vol.34 , Issue.7 , pp. 969-980
    • Diodato, N.1    Bellocchi, G.2
  • 66
    • 14344255817 scopus 로고    scopus 로고
    • Probabilistic wind power forecasts using local quantile regression
    • Bremnes J. Probabilistic wind power forecasts using local quantile regression. Wind Energy, 2004; 7(1):47-54.
    • (2004) Wind Energy , vol.7 , Issue.1 , pp. 47-54
    • Bremnes, J.1
  • 67
    • 34249911725 scopus 로고    scopus 로고
    • Safety prediction in nuclear power
    • Rumyantsev A. Safety prediction in nuclear power. Atomic Energy, 2007; 102(2):94-99.
    • (2007) Atomic Energy , vol.102 , Issue.2 , pp. 94-99
    • Rumyantsev, A.1
  • 68
    • 34250624938 scopus 로고    scopus 로고
    • Statistical process control methods from the viewpoint of industrial application
    • Anghel C. Statistical process control methods from the viewpoint of industrial application. Economic Quality Control, 2001; 16(1):49-63.
    • (2001) Economic Quality Control , vol.16 , Issue.1 , pp. 49-63
    • Anghel, C.1
  • 69
    • 50049088493 scopus 로고    scopus 로고
    • Non-parametric, unconditional quantile estimation for efficiency analysis with an application to Federal Reserve check processing operations
    • Wheelock D, Wilson P. Non-parametric, unconditional quantile estimation for efficiency analysis with an application to Federal Reserve check processing operations. Journal of Econometrics, 2009; 145:209-225.
    • (2009) Journal of Econometrics , vol.145 , pp. 209-225
    • Wheelock, D.1    Wilson, P.2
  • 71
    • 0030274088 scopus 로고    scopus 로고
    • Uncertainty analysis of flood quantile estimates with reference to Tanzania
    • Mkhandi S, Kachroo R, Guo S. Uncertainty analysis of flood quantile estimates with reference to Tanzania. Journal of Hydrology, 1996; 185, 317-333.
    • (1996) Journal of Hydrology , vol.185 , pp. 317-333
    • Mkhandi, S.1    Kachroo, R.2    Guo, S.3
  • 72
    • 14344258897 scopus 로고    scopus 로고
    • On-line assessment of prediction risk for wind power production forecasts
    • Pinson P, Kariniotakis G. On-line assessment of prediction risk for wind power production forecasts. Wind Energy, 2004; 7(2):119-132.
    • (2004) Wind Energy , vol.7 , Issue.2 , pp. 119-132
    • Pinson, P.1    Kariniotakis, G.2
  • 73
    • 15244343969 scopus 로고    scopus 로고
    • Quantile regression reveals hidden bias and uncertainty in habitat models
    • Noon B, Cade B, Flather C. Quantile regression reveals hidden bias and uncertainty in habitat models. Ecology, 2009; 86(3):786-800.
    • (2009) Ecology , vol.86 , Issue.3 , pp. 786-800
    • Noon, B.1    Cade, B.2    Flather, C.3
  • 74
    • 23644437803 scopus 로고    scopus 로고
    • Assessing uncertainty in simulation-based maritime risk assessment
    • Merrick J, Van Dorp R, Dinesh V. Assessing uncertainty in simulation-based maritime risk assessment. Risk Analysis, 2005; 25(3):731-743.
    • (2005) Risk Analysis , vol.25 , Issue.3 , pp. 731-743
    • Merrick, J.1    Van Dorp, R.2    Dinesh, V.3
  • 75
    • 33845302985 scopus 로고    scopus 로고
    • Confidence regions for high quantiles of a heavy tailed distribution
    • Peng L, Qi Y. Confidence regions for high quantiles of a heavy tailed distribution. Annals of Statistics, 2006; 34(4):1964-1986.
    • (2006) Annals of Statistics , vol.34 , Issue.4 , pp. 1964-1986
    • Peng, L.1    Qi, Y.2
  • 76
    • 84858658461 scopus 로고    scopus 로고
    • A modified quantile estimator using extreme-value theory with applications
    • Vermaat M, Does R, Steerneman A. A modified quantile estimator using extreme-value theory with applications. Economic Quality Control, 2005; 20(1):31-39.
    • (2005) Economic Quality Control , vol.20 , Issue.1 , pp. 31-39
    • Vermaat, M.1    Does, R.2    Steerneman, A.3
  • 77
    • 1542533065 scopus 로고    scopus 로고
    • Uncertainty of quantile estimators using the population index flood method
    • 1206, doi:10.1029/2002WR001594
    • Sveinsson O, Salas J, Boes D. Uncertainty of quantile estimators using the population index flood method. Water Resources Research, 2003; 39(8), 1206, doi:10.1029/2002WR001594
    • (2003) Water Resources Research , vol.39 , Issue.8
    • Sveinsson, O.1    Salas, J.2    Boes, D.3
  • 79
    • 0038128274 scopus 로고    scopus 로고
    • On maximum entropy characterization of Pearson's type II and VII multivariate distributions
    • Zografos K. On maximum entropy characterization of Pearson's type II and VII multivariate distributions. Journal of Multivariate Analysis, 1999; 71:67-75.
    • (1999) Journal of Multivariate Analysis , vol.71 , pp. 67-75
    • Zografos, K.1
  • 80
    • 11144274009 scopus 로고    scopus 로고
    • A maximum entropy characterization of symmetric Kotz type and multivariate Burr distributions
    • Aulogiaris G, Zografos K. A maximum entropy characterization of symmetric Kotz type and multivariate Burr distributions. Test, 2004; 13:65-83.
    • (2004) Test , vol.13 , pp. 65-83
    • Aulogiaris, G.1    Zografos, K.2
  • 81
    • 33747591696 scopus 로고    scopus 로고
    • Maximum entropy characterizations of the multivariate Liouville distributions
    • Bhattacharya B. Maximum entropy characterizations of the multivariate Liouville distributions. Journal of Multivariate Analysis, 2006; 97:1272-1283.
    • (2006) Journal of Multivariate Analysis , vol.97 , pp. 1272-1283
    • Bhattacharya, B.1
  • 82
    • 0024666706 scopus 로고
    • Entropy expressions and their estimators for multivariate distributions
    • Ahmed NA, Gokhale DV. Entropy expressions and their estimators for multivariate distributions. IEEE Transactions on Information Theory, 1989; 35:688-692.
    • (1989) IEEE Transactions on Information Theory , vol.35 , pp. 688-692
    • Ahmed, N.A.1    Gokhale, D.V.2
  • 83
    • 27444437823 scopus 로고    scopus 로고
    • Expressions for Renyi and Shannon entropies for multivariate distributions
    • Zografos K, Nadarajah S. Expressions for Renyi and Shannon entropies for multivariate distributions. Statistics and Probability Letters, 2005; 71:71-84.
    • (2005) Statistics and Probability Letters , vol.71 , pp. 71-84
    • Zografos, K.1    Nadarajah, S.2


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.