Weighted Average Least Squares Method And Its Applications In Generalized Linear Models

In econometrics,many statisticians and economists have begun to study parameter estimation under model uncertainty.Two of the most popular methods are model selection and model average.In model selection,the initial model selection step has a negligible effect on the statistical properties of the resulting estimator.The model average,by calculating the weighted average of the conditional estimates of all possible models,combines all valid information pieces into an unconditional estimate,combining the uncertainties caused by the model selection step and the model estimation step.Bayesian model average estimation(BMA)has become a very popular estimation method,and the new model average method introduced in this article is called weighted average least squares(WALS),which is superior to BMA both theory and practice.This paper introduces the WALS estimation methods in Gaussian linear models and generalized linear models.The WALS estimation method is a Bayesian combination of frequency estimates: the parameters of each model are estimated using the least squares from a frequency perspective,and the weight combination is a posterior distribution from the Bayesian perspective to reflect the confidence of the observations.First,compared with BMA estimates,the use of normal priors will lead to unbounded risks.The prior selection in WALS estimates theoretically considers admissibility,bounded risks,and robustness,which is close to optimal based on the minimum regret criterion,and Treat a priori neutrality.Second,the WALS estimation uses a semi-orthogonal transformation,which can obtain accurate model average estimates of the parameters in a short calculation time.This paper also conducts application research based on the WALS estimation method.In the empirical data analysis of the linear model,the data set describes the relevant data of American baseball players from 1986 to 1987.We study the relationship between the player’s salary and various conditions of the player in this data set to build a regression model.The salary is the dependent variable,and the other 19 variables such as the number of turnovers,age,and number of hits are independent variables.We compared the WALS estimation,the ridge regression,LASSO regression,it is obtained that the results of the WALS estimation are very close to the results of the LASSO regression when the prior information is appropriate,even when some of the prior information is inappropriate,WALS estimation method is better than LASSO regression,indicating that the WALS estimation method will have a good application in real life.There are many data that follow the logical distribution and Poisson distribution in medicine.The use of WALS estimation methods in these medical data can help explore the risk factors that cause disease and predict the probability of disease based on risk factors.In the empirical data analysis of logistic regression,we used a coronary heart disease data set.The 15 features observed were specifically divided into 3categories.One is the inherent attributes of the population,such as age and gender;the other is the life behavior,such as whether smoking,smoking volume;the last category is medical measures such as: whether the patient has taken high blood pressure drugs,whether high blood pressure,whether diabetes,whether there is stroke and cholesterol levels,systolic blood pressure,diastolic blood pressure,body mass index,Heart rate,glucose level.The target variable of logistic regression is whether coronary heart disease occur within ten years.The results obtained significant variables including age,gender,daily smoking volume,cholesterol level,blood systolic blood pressure,and diastolic blood pressure.In the analysis of Poisson regression empirical data,we use the Breslow data set,which is epilepsy data.There are a total of 12 variables in the data set,and the results are shown as Trt(treatment condition),age,and Base(the number of onsets within the first 8 weeks)are all significant for the dependent variable sum Y(the number of onsets after 8 weeks)influences.It can be found that the WALS estimation method performs very well in both logistic regression and Poisson regression.