# Theil Mixed Estimator

This post deals with the Theil mixed estimator which uses the prior information as well as the sample information.

Theil mixed estimator incorpoate stochastic non-sample information into a linear model. This estimator mixes sample and prior information in a generalized least squares sense. Information regarding GLS estimator can be found at the following previous post.

### Linear model with sample information

Sample information is represented by the following linear model.

### Linear model with sample and prior information

Prior information has the following form

where R is J × K and r is J × 1ν is a J × 1 normally distributed random error vector.

Incorporating the prior information into the sample information leads to the following model specification.

### GLS estimator

The GLS estimator is as follows.

By using matrix multiplications, this result can be simplified to

where ϕ stands for the precision of the regression model : ϕ = 1/σ2.

By inverting and its variance can be obtained.

This is the Theil mixed estimator.

### Theil mixed estimator

More generally, when is used instead of the Theil mixed estimator can also be represented

### Concluding Remarks

This post derived the Theil mixed estimator which uses the prior information as well as the sample information. This formulation will be used when deriving the Black-Litterman model.