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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Maximum score estimator ： ウィキペディア英語版 Maximum score estimator When modelling discrete choice model, it is always assumed that the choice is determined by the comparison of the underlying latent utility 〔For more example, refer to: Smith, Michael D. and Brynjolfsson, Erik, Consumer Decision-Making at an Internet Shopbot (October 2001). MIT Sloan School of Management Working Paper No. 4206-01.〕. Denote the population of the agents as $T$, the common choice set for each agent as $C$ . For agent $t\; \backslash in\; T$ , denote her choice as $y\_$ , which is equal to 1 if choice $i$ is chosen and 0 otherwise. Assume the latent utility is linear with the parameters and the error term is additive, then for an agent $t\; \backslash in\; T$ , $y\_\; =\; 1\; \backslash leftrightarrow\; x\_\backslash beta\; +\; \backslash epsilon\_\; >\; x\_\backslash beta\; +\; \backslash epsilon\_\; ,\; \backslash forall\; j\; \backslash neq\; i$ and $j\; \backslash in\; C$ where $x\_$ and $x\_$ are the $q$ -dimensional observable covariates about the agent and the choice, and $\backslash epsilon\_$ and $\backslash epsilon\_$ are the decision errors caused by some cognitive reasons or information incompleteness. The construction of the observable covariates is very general. For instance, if $C$ is a set of different brands of coffee, then $x\_$ includes the characteristics both of the agent $t$ , such as age, gender, income and ethnicity, and of the coffee $i$ , such as price, taste and whether it is local or imported. All of the error terms are assumed i.i.d and we need estimate $\backslash beta$ which characterize the effect of different factors on the agent’s choice. Usually some specific distribution assumption on the error term is imposed, such that the parameter $\backslash beta$ is estimated parametrically. For instance, if the distribution of error term is assumed to be normal, then the model is just a multinomial probit model 〔Wooldridge, J. (2002): Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, Mass, pp 457-460.〕 ; if it is assumed to be an extreme value distribution, then the model becomes a multinomial logit model. The parametric model 〔For an concrete example, refer to: Tetsuo Yai, Seiji Iwakura, Shigeru Morichi, Multinomial probit with structured covariance for route choice behavior, Transportation Research Part B: Methodological, Volume 31, Issue 3, June 1997, Pages 195-207, ISSN 0191-2615〕 is convenient for computation but might not be consistent once the distribution of the error term is misspecified 〔Jin Yan (2012), “A Smoothed Maximum Score Estimator for Multinomial Discrete Choice Models”, Working Paper.〕 . To make the estimator more robust to the distributional assumption, Manski (1975) proposed a non-parametric model to estimate the parameters. In this model, denote the number of the elements of the choice set as $J$ , the total number of the agents as $N$, and $W\; (J\; -1)\; >\; W\; (J\; -\; 2)\; >\; \backslash dots\; >\; W\; (1)\; >\; W\; (0)$ is a sequence of real numbers. The Maximum Score Estimator 〔Charles F. Manski (1975), “Maximum Score Estimation of the Stochastic Utility Model of Choice”, Journal of Econometrics 3, pp. 205-228.〕 is defined as: $\backslash hat=\; \backslash sum\_^N\; \backslash sum\_^J\; y\_\; W\; (\backslash sum\backslash nolimits\_\; 1\; (x\_b\; >\; x\_b))$ Here, $\backslash sum\backslash nolimits\_\; 1\; (x\_b\; >\; x\_b)$ is the ranking of the certainty part of the underlying utility of choosing $i$ . The intuition in this model is that the ranking is higher, the more weight will be assigned to the choice, based on which, the optimization objective function similar to the likelihood function in parametric model is constructed. For more about the consistency and asymptotic property about the maximum score estimator, refer to Manski (1975). == References == 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Maximum score estimator」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.