Package popkorn. R topics documented: February 20, Type Package

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1 Type Pacage Pacage popkor February 20, 2015 Title For iterval estimatio of mea of selected populatios Versio Date Author Vi Gopal, Claudio Fuetes Maitaier Vi Gopal Depeds R (>= 3.0.0), boot Suggests plotrix Provides a suite of tools for various methods of estimatig cofidece itervals for the mea of selected populatios. Licese MIT + file LICENSE LazyLoad yes NeedsCompilatio o Repository CRAN Date/Publicatio :45:25 R topics documeted: popkor-pacage asymmetricitervals boferroiitervals bootstrapitervals eqformi exactcoverageprob gedelmat itegrad itegrate optimalc optimallambda optimallambdac Idex 13 1

2 2 asymmetricitervals popkor-pacage Iterval estimatio for selected populatios Iterval estimatio for selected populatios Pacage: popkor Type: Pacage Versio: Date: Licese: MIT LazyLoad: yes This pacage provides routies for estimatig the iterval for the mea of selected populatios. Author(s) Claudio Fuetes, Vi Gopal asymmetricitervals Compute Asymmetric Itervals This fuctio will compute asymmetric itervals for the mea of the selected populatios. asymmetricitervals(x, = 0.05, = 2, var = NULL, eps = 0.1) X var eps is a matrix or data frame that cotais the resposes. Each colum represets a differet populatio. deotes the sigificace level of the itervals to be formed. correspods to the umber of populatios to be selected. deotes the commo variace of the populatios from which the data is draw. If this is NULL (the default), the the variace will be estimated from the data. If it is ow, the it should be provided as a scalar. The grid size that is to be set up.

3 boferroiitervals 3 This fuctio will compute the optimal lambda ad c to be used to shri the iterval for the selected populatios. This fuctio returs a matrix with rows ad 3 colums. This is similar to the output of the predict.lm fuctio of R. Refereces Claudio Fuetes, George Casella ad Marti Wells (2013). Iterval estimatio for the mea of the selected populatios (Submitted). Vi Gopal ad Claudio Fuetes (2013). Pop: A R pacage for iterval estimatio of selected populatios. user! boferroiitervals, bootstrapitervals Examples set.seed(18) p <- 10; <- 10 Xmat <- matrix(rorm(p*), row=, col=p) colames(xmat) <- paste("p.", 1:p, sep="") asymmetricitervals(xmat, =0.1, =4) boferroiitervals Compute Boferroi Itervals This fuctio will compute apply the Boferroi correctio to the selected populatios. boferroiitervals(x, = 0.05, = 2) X is a matrix or data frame that cotais the resposes. Each colum represets a differet populatio. deotes the sigificace level of the itervals to be formed. correspods to the umber of populatios to be selected.

4 4 bootstrapitervals If there are p populatios, the the Boferroi correctio will be applied by usig α/p istead of just p. The fuctio returs a matrix with rows ad 3 colums. This is similar to the output of the predict.lm fuctio of R. asymmetricitervals, bootstrapitervals Examples set.seed(18) p <- 10; <- 10 Xmat <- matrix(rorm(p*), row=, col=p) colames(xmat) <- paste("p.", 1:p, sep="") boferroiitervals(xmat, =0.1, =4) bootstrapitervals Compute Bootstrap Itervals This fuctio will apply the Boferroi correctio to bootstrap itervals of the mea of the selected populatios. bootstrapitervals(x, = 0.05, = 2, R = 10, retur.obj = "itervals", type = "basic",...) X R retur.obj type is a matrix or data frame that cotais the resposes. Each colum represets a differet populatio. deotes the sigificace level of the itervals to be formed. correspods to the umber of populatios to be selected. deotes the umber of bootstrap replicate samples to produce. is a character vector of legth 1, idicatig if this fuctio should retur the output of the boot() fuctio, or proceed to compute the bootstrap cofidece itervals ad retur those. is a character vector of legth oe. It should be oe of the followig: "orm","basic", "stud", "perc" or "bca".... deotes further argumets that will be passed to the boot fuctio.

5 eqformi 5 The bootstrap that is carried out is the stratified bootstrap, sice there are p populatio i cosideratio. Withi each populatio, samplig with replacemet is carried out, ad the largest sample meas are retured. The user ca use ay of the 5 cofidece iterval methods that are preset i the boot.ci fuctio. However, a Boferroi correctio will be carried out i order to esure that the itervals hold simultaeously. If retur.obj is set to be "boot", the the fuctio returs a object of class "boot". Otherwise, if retur.obj is set to be "itervals", the this fuctio returs a matrix with rows ad 3 colums. This is similar to the output of the predict.lm fuctio of R. boferroiitervals, asymmetricitervals Examples set.seed(18) p <- 10; <- 10 Xmat <- matrix(rorm(p*), row=, col=p) colames(xmat) <- paste("p.", 1:p, sep="") bootstrapitervals(xmat, =0.1, =4) eqformi Evaluate the fuctio to be miimised This will evaluate the appropriate fuctio for determiig the c value i the cofidece iterval for X ( 1). eqformi(c.val, lambda = 0.5, = 0.05, mi.loc = "ifty",, p, = 1, var.ow = TRUE) c.val lambda mi.loc The value at which to evaluate the appropriate fuctio. The value of lambda uder cosideratio. This must be a scalar betwee 0 ad 1. The desired cofidece coefficiet. The locatio of the miimum, either at zero or ifty.

6 6 exactcoverageprob p var.ow The umber of replicatios per populatio. The umber of populatios cosidered. This must be preset if mi.loc is equal to zero. The umber of populatios selected. A logical flag idicatig if the variace of the observatios is ow exactly. It is TRUE by default. This fuctio will choose the correct equatio to use for determiig the smallest c-value that maitais the desired coverage probability. Note that this fuctio does *ot* do the miimizatio. That procedure is doe by optimalc. There are essetially 8 differet cases to cosider. They correspod to the cases whe the variace is ow or uow, whe the umber of populatios selected is greater tha 1 or equal to 1, ad whe the miimum of the equatio is located at ifiity or 0. The fuctio returs a scalar value. exactcoverageprob Evaluate exact coverage probability This fuctio will evaluate the exact coverage probability, as give i equatio (4) o page 7 of the paper. See sectio. exactcoverageprob(c.vec, theta.diff, lambda, c.val, sigma.2 = 1, = 1) c.vec theta.diff lambda c.val sigma.2 This is a vector of legth 2. It cosists of the lower ad upper limits i the itegral. Checig is carried out to esure that is of legth two, ad that 0 <= c.vec[1] <= c.vec[2]. This parameter is igored if lambda is ot missig. A vector of legth p-1, where p is the umber of populatios of treatmets. Coordiate [i] i theta.diff correspods to θ i θ i+1. See gedelmat. I case the user wishes to use the shriage versio, this parameter should be specified. It must be betwee 0 ad 1. I case lambda is specified, this must ot be missig. This will be combied with lambda to create a c.vec. This very fuctio will the call itself. The ow variace of the error terms. The umber of replicatios per populatio.

7 gedelmat 7 This fuctio evaluates the coverage probability for a iterval defied by (X (1) c 2, X (1) + c 1 ). Note that, as specified i the referece paper, we must have that 0 c 1 c 2. This fuctio will call itegrate2. Please ote the orderig of the elemets i the c.vec argumet: the first elemet correspods to the upper limit of the iterval, ad to the egative of the lower limit of the itegral. The fuctio returs a scalar value that is the value of the exact coverage coverage probability defied i equatio (4) of page 7. itegrate2, itegrad Examples del1 <- c(2, 4) exactcoverageprob(c(1.1,1.3), del1) exactcoverageprob(theta.diff=c(2,3,4), lambda=0.9, c.val=2) gedelmat Geerate delta matrix This fuctio will geerate the matrix of deltas, as specified i the paper. See sectio. gedelmat(theta.diff, sigma.2 = 1, = 1) theta.diff sigma.2 A vector of legth p-1, where p is the umber of populatios of treatmets. Coordiate [i] i theta.diff correspods to θ i θ i+1. The ow variace of the error terms. The umber of replicatios i each populatio. As specified i the paper, we ca assume that the thetas are i a decreasig order, meaig that θ 1 θ 2,..., θ. It follows that all the compoets of the theta.diff vector must be positive. Note that the delta matrix i the paper is a scaled versio of the differeces betwee the thetas.

8 8 itegrad The fuctio returs a matrix with p rows ad p colums, that cotais the delta ij s, as described i the paper. exactcoverageprob, itegrad Examples del1 <- c(2, 4) gedelmat(del1) itegrad Evaluate itegrad aloe This fuctio will evaluate the itegrad i the expressio for the exact coverage probability. itegrad(z, mat1) z mat1 This is a real umber betwee -If ad If. This is the matrix of the delta values. This matrix ca be geerated usig the gedelmat fuctio. The fuctio returs a scalar value. exactcoverageprob, itegrate2

9 itegrate2 9 itegrate2 Evaluate itegral This fuctio will evaluate the itegral i equatio (4) o page 7 of the paper. See sectio. itegrate2(limit.vec = c(-3, 3), theta.diff, sigma.2 = 1, = 1) limit.vec theta.diff sigma.2 This is a vector of legth 2. It cosists of the lower ad upper limits i the itegral. Checig is carried out to esure that is of legth two, ad that limit.vec[1] <= limit.vec[2]. A vector of legth p-1, where p is the umber of populatios of treatmets. Coordiate [i] i theta.diff correspods to θ i θ i+1. See gedelmat. The ow variace of the error terms. The umber of replicatios per populatio. This fuctio evaluates the itegral, ad wors with the lower ad upper limits that it is give. If oe desires to compute the coverage probability for a iterval defied by X (1) ± c, the the user should loo at the fuctio exactcoverageprob i this pacage. The fuctio returs a scalar value that is the value of the itegral i equatio (4) of page 7, defied by the lower ad upper limits provided here. exactcoverageprob, itegrad Examples del1 <- c(2, 4) itegrate2(c(-1.1,1.3), del1)

10 10 optimalc optimalc Derive the optimal c Derives the optimal c value for a give lambda. optimalc(lambda, = 0.05, mi.loc = "ifty",, p, = 1, var.ow = TRUE) lambda mi.loc p var.ow The value of lambda uder cosideratio. This must be a vector with values betwee 0 ad 1. The desired cofidece coefficiet. The locatio of the miimum, either at zero or ifty. The umber of replicatios per populatio. The umber of populatios cosidered. This must be preset if mi.loc is equal to zero. The umber of populatios selected. A logical flag idicatig if the variace of the observatios is ow exactly. It is TRUE by default. This fuctio will choose the correct equatio to use for ad use uiroot to fid the c-value that correspods to the desired -level. The fuctio returs a vector of legth equal to that of lambda. optimallambda

11 optimallambda 11 optimallambda Derive the optimal lambda Derives the optimal lambda for a give. optimallambda(,, p, = 1, var.ow = TRUE) p var.ow The desired cofidece coefficiet. The umber of replicatios per populatio. The umber of populatios cosidered. This must be preset if mi.loc is equal to zero. The umber of populatios selected. A logical flag idicatig if the variace of the observatios is ow exactly. It is TRUE by default. This will fid the optimal lambda to be used for the shriage of cofidece itervals. The fuctio returs a scalar value. optimalc optimallambdac Derive the optimal lambda ad c value Derives the optimal lambda ad c-value for a give cofiguratio. optimallambdac( = 0.05,, p, = 1, var.ow = TRUE, eps = 0.1)

12 12 optimallambdac p var.ow eps The desired cofidece coefficiet. The umber of replicatios per populatio. The umber of populatios cosidered. This must be preset if mi.loc is equal to zero. The umber of populatios selected. A logical flag idicatig if the variace of the observatios is ow exactly. It is TRUE by default. The grid size that is to be set up. This fuctio will retur the optimal lambda ad c-value to be used, usig a grid search. There are essetially 2 differet cases to cosider. They correspod to the cases whe the variace is ow or uow. The fuctio returs a list with two compoets, lambda ad c.val that are optimal.

13 Idex Topic pacage popkor-pacage, 2 asymmetricitervals, 2, 4, 5 boferroiitervals, 3, 3, 5 boot.ci, 5 bootstrapitervals, 3, 4, 4 eqformi, 5 exactcoverageprob, 6, 8, 9 gedelmat, 6, 7, 8, 9 itegrad, 7, 8, 8, 9 itegrate2, 7, 8, 9 optimalc, 6, 10, 11 optimallambda, 10, 11 optimallambdac, 11 popkor-pacage, 2 13