Which is the best estimate of weighted least squares?
The weighted least squares estimate is then With this setting, we can make a few observations: Since each weight is inversely proportional to the error variance, it reflects the information in that observation.
Which is a positive constant in weighted least squares?
: Here w1;:::;wn are known positive constants. Weighted least squares is an estimation technique which weights the observations proportional to the reciprocal of the error variance for that observation and so overcomes the issue of non-constant variance.
When to use weights in a regression experiment?
In designed experiments with large numbers of replicates, weights can be estimated directly from sample variances of the response variable at each combination of predictor variables. Use of weights will (legitimately) impact the widths of statistical intervals.
How are standard deviations used in a regression model?
These standard deviations reflect the information in the response Y values (remember these are averages) and so in estimating a regression model we should downweight the obervations with a large standard deviation and upweight the observations with a small standard deviation.
What is the variance of an estimator called?
The variance of an estimator is simply Var(θˆ) where the random variable is the training set The square root of the the variance is called the
Which is more important the variance or the bias?
The UMVUE is, as the name suggests, the estimator that has the minimum variance among the unbiased estimators for the parameter of interest. Among the two basic mea- sures of the quality of an estimator, the bias is more important factor for UMVUE than the variance.
How are the residuals used to estimate the weights?
The residuals are much too variable to be used directly in estimating the weights, w i, so instead we use either the squared residuals to estimate a variance function or the absolute residuals to estimate a standard deviation function. We then use this variance or standard deviation function to estimate the weights.
Is the i-th squared residual an estimate of σ i?
Provided the regression function is appropriate, the i-th squared residual from the OLS fit is an estimate of σ i 2 and the i-th absolute residual is an estimate of σ i (which tends to be a more useful estimator in the presence of outliers).