where is the vector of estimates at the th branch.
To deal with this problem the log-likelihood maximization will be repeated several times with different starting values and the parameters estimates will be chosen to be the vector of estimates that corresponds to the highest log-likelihood assuming that it does not have the characteristics of a spurious maximizer.
The estimating equation for α can be written as, where is the upper diagonal of the estimated covariance matrix in vector form as, s is an "empirical covariance vector" with (n+ n)/2 elements s ij = (Y i - E Y i )(Y j - E Y j ) corresponding to the elements of, is the (n+ n)/2 length vector of estimates for, and is the (n+ n)/2 x (n+ n)/2 estimated covariance matrix for the vector s.
(hat {X}) or (hat {X} z)) : Vector of estimated state variables.
(hat {X}^) or (hat {X}^ z)) : Vector of estimated state variables after applying the rectification.
(hat {X}^) or (hat {X}^ z)) : Vector of estimated state variables before applying the rectification.
Where α is the intercept, X' are the claims characteristics (injury type, cancer type, body part affected, age of claimant at biopsy, industry type, region), β is a vector of estimated coefficients, and ε is the error term.
The vector of estimated parameters, ( {tilde{boldsymbol{beta}}}_i ), and the predictor, W i, are used to predict the response vector ỹ i for the ith class as begin{array}{cc}hfill {tilde{boldsymbol{y}}}_i={boldsymbol{W}}_i{tilde{boldsymbol{beta}}}_i,hfill & hfill i=1,2,dots, Nhfill end{array} (5).
Consider, for example, the standard linear regression model y = X β + ε, where the vector of estimated parameters is b = X T X − 1 X T y and the fitted values are y * = X b = X X T X − 1 X T y.
The logarithm of per capita income of household i in t = s ((hat{y}_{i,t = s}^{U})) is estimated as follows: hat{y}_{i,t = s}^{U} = hat{beta }_{t = s}^ x_{i,t = 0} + hat{tilde{varepsilon }}_{i,t = s} (2 where the apex U indicates uncorrelated error terms, and (hat{tilde{varepsilon }}_{i,t = s}) is the mean of 50 random draws (with replacement) from the vector of estimated residuals in t = s.
In practice, we stop at mstop, and we denote the final vector of estimated coefficients as λ ^ = λ ^ [ m stop ].
