Maximum score estimator

Usually some specific distribution assumption on the error term is imposed, such that the parameter ${\displaystyle \beta }$ 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;[2] if it is assumed to be a Gumbel distribution, then the model becomes a multinomial logit model. The parametric model[3] is convenient for computation but might not be consistent once the distribution of the error term is misspecified.[4]

For example, suppose that C only contains two items. This is the latent utility representation[5] of a binary choice model. In this model, the choice is: ${\displaystyle Y_{t}=1[X_{1,t}\beta +\varepsilon _{1}>X_{2,t}\beta +\varepsilon _{2}]}$, where ${\displaystyle X_{1,t},X_{2,t}}$ are two vectors of the explanatory covariates, ${\displaystyle \varepsilon _{1}}$ and ${\displaystyle \varepsilon _{2}}$ are i.i.d. response errors,

${\displaystyle X_{1,t}\beta +\varepsilon _{1}{\text{ and }}X_{2,t}\beta +\varepsilon _{2}}$

are latent utility of choosing choice 1 and 2. Then the log likelihood function can be given as:

${\displaystyle Q=\sum _{i-1}^{N}Y_{t}\log(P[X_{1,t}\beta -X_{2,t}\beta >\varepsilon _{2}-\varepsilon _{1}])+(1-Y_{t})\log(1-P[X_{1,t}\beta -X_{2,t}\beta >\varepsilon _{2}-\varepsilon _{1}])}$

If some distributional assumption about the response error is imposed, then the log likelihood function will have a closed-form representation.[2] For instance, if the response error is assumed to be distributed as: ${\displaystyle N(0,\sigma ^{2})}$, then the likelihood function can be rewritten as:

${\displaystyle Q=\sum _{i-1}^{N}Y_{t}\log \left(\Phi \left[{\frac {X_{1,t}\beta -X_{2,t}\beta }{\surd 2\sigma }}\right]\right)+(1-Y_{t})\log \left(\Phi \left[{\frac {X_{2,t}\beta -X_{1,t}\beta }{\surd 2\sigma }}\right]\right)}$

where ${\displaystyle \Phi }$ is the cumulative distribution function (CDF) for the standard normal distribution. Here, even if ${\displaystyle \Phi }$ doesn't have a closed-form representation, its derivative does. This is the probit model.

This model is based on a distributional assumption about the response error term. Adding a specific distribution assumption into the model can make the model computationally tractable due to the existence of the closed-form representation. But if the distribution of the error term is misspecified, the estimates based on the distribution assumption will be inconsistent.

The basic idea of the distribution-free model is to replace the two probability term in the log-likelihood function with other weights. The general form of the log-likelihood function can written as:

${\displaystyle Q=\sum _{i-1}^{N}Y_{t}\cdot \log(W_{1}(X_{1,t}\beta ,X_{2,t}\beta ))+(1-Y_{t})\log(W_{0}(X_{1,t}\beta ,X_{2,t}\beta ))}$

In the binary context, the maximum score estimator can be represented as:

${\displaystyle W_{1}(X_{1,t}\beta ,X_{2,t}\beta )=w_{1}1[X_{1,t}\beta -X_{2,t}\beta >0]+w_{0}1[X_{1,t}\beta -X_{2,t}\beta <0],}$

where

${\displaystyle W_{0}(X_{1,t}\beta ,X_{2,t}\beta )=1-W_{1}(X_{1,t}\beta ,X_{2,t}\beta )}$

and ${\displaystyle w_{1}}$ and ${\displaystyle w_{0}}$ are two constants in (0,1). The intuition of this weighting scheme is that the probability of the choice depends on the relative order of the certainty part of the utility.