Cressie and Read objective functions {ATE} R Documentation

## Cressie and Read objective functions

### Description

Cressie and Read class of objective functions parameterized by a scalar θ.

### Usage

```cr.rho(v,theta)
d.cr.rho(v,theta)
dd.cr.rho(v,theta)
```

### Arguments

 `v` A vector of points at which the function will be evaluated at. `theta` A user defined value for θ. This must be a scalar. See details.

### Details

This set of functions can be used for the main `ATE` function for the arguments `theta`. The class of functions, are defined by (see Cressie and Read (1984))

ρ(v) = -(1+θ v)^{(θ+1)/θ}/(θ+1) ,

where θ is scalar. The expression is interpreted as limits for θ = 0 or θ = -1 (see Newey and Smith (2004)).

### Value

The output of the real values function for a given parameter θ.

### References

Chan, K. C. G. and Yam, S. C. P. and Zhang, Z. (2015). "Globally Efficient Nonparametric Infernence of Average Treatment Effects by Emperical Balancing Calibration Weighting", under revision.

Chan, K. C. G. and Yam, S. C. P. (2014). "Oracle, Multiple Robust and Multipurpose Calibration in a Missing Response Problem". Statistical Science 29 (3): 380-386.

Cressie, N. and Read, T. R. C. (1984). "Multinomial goodness-of-fit tests". J. Roy. Statist. Soc. Ser. B 46 440-464.

Newey, W. K. and Smith, R. J. (2004). Higher order properties of GMM and generalized empirical likelihood estimators. Econometrica 72 219-255.

### Examples

```#Cressie-Read Family example
data("nsw")
Y<-nsw\$re78[301:500]
X<-nsw[301:500,-c(1,9)]
treat<-nsw\$treat[301:500]

#Exponential tilting
ATE(Y,treat,X,theta = 0)

#Empirical likelihood
ATE(Y,treat,X,theta = -1)