MMSE: Minimum mean squared error estimation with additional...

In starima74/AddInf: The package functions implement statistical estimation with the use of additional information

Minimum mean squared error estimation with additional information.

Description

This function implements statistical estimation with the use of additional information. This additional information can come from multiple indepedent data sources. Each source is reported by a mean estimate and its variance. Then, this additional information is used together with an estimating procedure theta.f applied to the dataset dd. A new estimator minimizes mean squared error.

Usage

MMSE(dd, theta.f, Add.Inf, nboots = 500, eig.cutoff = 0.9, ...)

Arguments

dd	a data frame containing the variables used by function theta.f
theta.f	a function applied to the dataset dd to estimate a parameter of interest
Add.Inf	a data frame which consist of a columm of means (Means), a columm of variances (Vars), a vector of functions (Functions) and a vector of indicator of possible bias parameters (Biases). Functions column consists of functions used to calculate means in the additional data source; will be applied to the dataset dd. A indicator of possible bias parameter equal to 0 defines unbiased additional information, 1 corresponds to possible presence of bias.
nboots	a number of boostrap resamples to calculate variances and covariances
eig.cutoff	a percent cutoff for eigen values to approximate a matrix inverse
...	additional parameters. These parameters can be used by function theta.f or by additional functions.

Value

Theta.Est	The estimate of the parameter of interest without the use of addtional information
Theta.Est.Var	Variance covariance matrix of Theta.Est
Theta.Hat	New estimate of the parameter of interest with the use of addtional information
Theta.Hat.Var	Variance covariance matrix of Theta.Hat

Author(s)

Sergey Tarima ([e-mail])

References

Tarima S.S. and Dmitriev Y.G. Statistical estimation with possibly incorrect model assumptions. The Bulletin of Tomsk State University: control, computing, informatics 2009, 3(8):87-99

Tarima S.S., Vexler A., Singh S., Robust mean estimation under a possibly incorrect log-normality assumption. Communications in Statistics - Simulation and Computation, 42: 316-326, 2013

Examples

### data (an available dataset of patient ages)

dd <- data.frame(age = c(66,59,61,76,35,59,37,47,71,63,
                         41,56,49,58,48,79,87,50,57,58,32,
                         52,57,61,62,42,67,77,66,79))
                         
### the objective is to estimate proportion of pateints 
### younder than 40, 50, 60 and 70 years of age      
times1 <- c(40,50,60,70)

### this function estimates these proportions on a dataset d, 
### defined in its first argument
theta.f <- function(d,times) {
  ecdf(c(d)[[1]])(times)
}

### (Empty) Additional Information
Add.Info.Means <- list()
Add.Info.Vars <- list()
Add.Info.Functions <- list()
Add.Info.Biases <- list()

#### the first additional data source says that average of patients
### is 58.58 years of age with variance = 0.09, this information is possibly biased
Add.Info.Means[[1]] <- 58.58
Add.Info.Vars[[1]] <- 0.09
Add.Info.Functions[[1]] <- function(d) mean(d[[1]],na.rm = TRUE)
Add.Info.Biases[[1]] <- 1 # biased additional information

### create the data frame where the additional information will be saved
Add.Info <- data.frame(Means = rep(NA,1), 
                       Vars = rep(NA,1), 
                       Functions = rep(NA,1),
                       Biases = rep(NA,1))

### Adding the additional information to the data frame
Add.Info$Means = Add.Info.Means
Add.Info$Vars = Add.Info.Vars
Add.Info$Functions = Add.Info.Functions
Add.Info$Biases = Add.Info.Biases

### using both available dataset and additional information for estimation
res <- MMSE(dd, theta.f, Add.Info, nboots = 500, 
               eig.cutoff = 1, times = times1)

### calculating new estimates and pointwise confidence intervals
lo  <- res$Theta.Est - 1.96*sqrt(diag(res$Theta.Est.Var))
cdf <- res$Theta.Est
hi  <- res$Theta.Est + 1.96*sqrt(diag(res$Theta.Est.Var))

### calculating empirical (no additional information) estimates 
### and pointwise confidence intervals
theta_lo  <- res$Theta.Hat - 1.96*sqrt(diag(res$Theta.Hat.Var))
theta_cdf <- res$Theta.Hat
theta_hi  <- res$Theta.Hat + 1.96*sqrt(diag(res$Theta.Hat.Var))

### plot
plot(cdf ~ times1, type = "l", ylab="Proportions", 
          main = paste("Means and 95 percent confidence intervals\n", 
                   "with (dashed) and without (solid) additional information"), 
          xlab="Age", lty = 2, ylim= c(0,1))
lines(lo ~ times1, type = "l", lty = 2)
lines(hi ~ times1, type = "l", lty = 2)
points(cdf ~ times1)
points(lo ~ times1)
points(hi ~ times1)

lines(theta_cdf ~ times1, type = "l")
lines(theta_lo ~ times1, type = "l")
lines(theta_hi ~ times1, type = "l")
points(theta_cdf ~ times1)
points(theta_lo ~ times1)
points(theta_hi ~ times1)

starima74/AddInf documentation built on June 13, 2022, 12:08 p.m.