# PikHol: Optimal Inclusion Probabilities Under Multi-purpose Sampling In TeachingSampling: Selection of Samples and Parameter Estimation in Finite Population

## Description

Computes the population vector of optimal inclusion probabilities under the Holmbergs's Approach

## Usage

 1 PikHol(n, sigma, e, Pi) 

## Arguments

 n Vector of optimal sample sizes for each of the characteristics of interest. sigma A matrix containing the size measures for each characteristics of interest. e Maximum allowed error under the ANOREL approach. Pi Matrix of first order inclusion probabilities. By default, this probabilites are proportional to each sigma.

## Details

Assuming that all of the characteristic of interest are equally important, the Holmberg's sampling design yields the following inclusion probabilities

π_{(opt)k}=\frac{n^*√{a_{qk}}}{∑_{k\in U}√{a_{qk}}}

where

n^*≥q \frac{(∑_{k\in U}√{a_{qk}})^2}{(1+c)Q+∑_{k\in U}a_{qk}}

and

a_{qk}= ∑_{q=1}^Q \frac{σ^2_{qk}}{∑_{k\in U}≤ft( \frac{1}{π_{qk}}-1\right)σ^2_{qk}}

Note that σ^2_{qk} is a size measure associated with the k-th element in the q-th characteristic of interest.

## Value

The function returns a vector of inclusion probabilities.

## Author(s)

Hugo Andres Gutierrez Rojas hagutierrezro@gmail.com

## References

Holmberg, A. (2002), On the Choice of Sampling Design under GREG Estimation in Multiparameter Surveys. RD Department, Statistics Sweden.
Sarndal, C-E. and Swensson, B. and Wretman, J. (1992), Model Assisted Survey Sampling. Springer.
Gutierrez, H. A. (2009), Estrategias de muestreo: Diseno de encuestas y estimacion de parametros. Editorial Universidad Santo Tomas

## Examples

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 ####################### #### First example #### ####################### # Uses the Lucy data to draw an otpimal sample # in a multipurpose survey context data(Lucy) attach(Lucy) # Different sample sizes for two characteristics of interest: Employees and Taxes N <- dim(Lucy)[1] n <- c(350,400) # The size measure is the same for both characteristics of interest, # but the relationship in between is different sigy1 <- sqrt(Income^(1)) sigy2 <- sqrt(Income^(2)) # The matrix containign the size measures for each characteristics of interest sigma<-cbind(sigy1,sigy2) # The vector of optimal inclusion probabilities under the Holmberg's approach Piks<-PikHol(n,sigma,0.03) # The optimal sample size is given by the sum of piks n=round(sum(Piks)) # Performing the S.piPS function in order to select the optimal sample of size n res<-S.piPS(n,Piks) sam <- res[,1] # The information about the units in the sample is stored in an object called data data <- Lucy[sam,] attach(data) names(data) # Pik.s is the vector of inclusion probability of every single unit # in the selected sample Pik.s <- res[,2] # The variables of interest are: Income, Employees and Taxes # This information is stored in a data frame called estima estima <- data.frame(Income, Employees, Taxes) E.piPS(estima,Pik.s) ######################## #### Second example #### ######################## # We can define our own first inclusion probabilities data(Lucy) attach(Lucy) N <- dim(Lucy)[1] n <- c(350,400) sigy1 <- sqrt(Income^(1)) sigy2 <- sqrt(Income^(2)) sigma<-cbind(sigy1,sigy2) pikas <- cbind(rep(400/N, N), rep(400/N, N)) Piks<-PikHol(n,sigma,0.03, pikas) n=round(sum(Piks)) n res<-S.piPS(n,Piks) sam <- res[,1] data <- Lucy[sam,] attach(data) names(data) Pik.s <- res[,2] estima <- data.frame(Income, Employees, Taxes) E.piPS(estima,Pik.s) 

TeachingSampling documentation built on April 22, 2020, 1:05 a.m.