Numerical Maximum Likelihood Estimator

Description

The parameters Maximum Likelihood Estimation is obtained by numerical optimization.

Usage

MLE(nvData, strDistribution, lDensityExpr)

Arguments

Value

A list with distribution name, distribution parameters, value of the objective function corresponding to the parameters, additional information returned by the optimizer, convergence of the algorithm.

Negative Log-Likelihood

Description

The function compute the Negative Log-Likelihood value that has to be used for optimization in MLE function.

Usage

NlLike(nvTheta, nvData, lDensityExpr)

Arguments

Value

Negative log likelihood value.

Optimal B-Robust Estimator

Description

Function for obtaining the Optimal B-Robust Estimates starting by a vector of data and a two parameters distribution.

Usage

OBRE(
  nvData,
  strDistribution,
  nCParOBRE,
  dfParOBRE = data.frame(nEta = 1e-06, nMaxIterLoopWc = 10, nMaxIterLoopA = 10, nRelTol =
    0.001, nAbsTol = 0.5, stringsAsFactors = FALSE),
  nTheta1Init = NA,
  nTheta2Init = NA,
  eDensityFun = NA
)

Arguments

Value

A list with the vector containing the final parameters, the exit OBRE message, the values of vector a and matrix A, the OBRE tuning parameter c, the initial values of the parameters (if unspecified by the user, the values of MLE are reported), the vector of data, the density expression.

References

Bellio, R. (2007). Algorithms for bounded-influence estimation. Comput. Stat. Data Anal. 51, 2531-2541.

Hampel F (1968). Contributions to the theory of robust estimation. University of California.

Hampel, F., Ronchetti, E., Rousseeuw, P. & Stahel, W. (1985). Robust Statistics. The approach based on influence function. John Wiley and Sons Ltd., Chichester, UK.

Victoria-Feser, M.P. & Ronchetti, E. (1994). Robust methods for personal-income distribution models. Canadian Journal of Statistics 22, 247-258.

Examples

# Using the densityExpressions function for initialize the distribution
distrForOBRE <- densityExpressions(strDistribution = "normal")
simData = c(rnorm(1000, 12, 2),200,150)
try({estOBRE <- OBRE(nvData = simData, strDistribution = distrForOBRE, nCParOBRE = 3)
# Launching the generation of the density expression directly from OBRE
simData = c(rnorm(1000, 12, 2),200,150)
estOBRE <- OBRE(nvData = simData, strDistribution = "normal", nCParOBRE = 3)
# Using the "custom" option and using the normal distribution
simData = c(rnorm(1000, 12, 2),200,150)
estOBRE <- OBRE(nvData = simData, strDistribution = "custom", nCParOBRE = 3,
eDensityFun = expression((exp( -((nvData - nTheta1)^2) / (2 * nTheta2^2)) /
(sqrt(2 * pi) * nTheta2))))})

Check if OBRE matrix A and vector a are final.

Description

The function compute the relative distance from the past to the current iteration of matrix A, with respect to the relative tolerance if at the current iteration matrix A is not null. Otherwise the absolute error is checked. Then the vector a is checked in the same way.

Usage

OBRECheckTolParameters(matANew, matAOld, nvANew, nvAOld, nRelTol, nAbsTol)

Arguments

Value

A flag indicating if condition on matrix A and vector a are both satisfied.

Function that computes the OBRE covariance matrix.

Description

The function computes matrices M (Jacobian) and Q (Variability) and uses them to evaluate the covariance matrix V.

Usage

OBRECovarianceMatrix(lOBRE)

Arguments

Value

A list containing Jacobian of the estimate function, variability and asymptotic covariance matrices, as well as the relative efficiency with respect to Maximum Likelihood Estimator

References

Hampel, F., Ronchetti, E., Rousseeuw, P. & Stahel, W. (1985). Robust Statistics. The approach based on influence function. John Wiley and Sons Ltd., Chichester, UK.

Heritier S, Cantoni E, Copt S, Victoria-Feser M (2011). Robust Methods in Biostatistics. John Wiley and Sons Ltd., Chichester, UK.

Examples

try({distrForOBRE <- densityExpressions(strDistribution = "normal")
simData = c(rnorm(1000, 12, 2),200,150)
estOBRE <- OBRE(nvData = simData, strDistribution = distrForOBRE, nCParOBRE = 3)
lOBRECov = OBRECovarianceMatrix(estOBRE)})

Function computing the OBRE matrix M.

Description

The function evaluates integrals used to compute the M_1 and M_2 OBRE matrices. Element (1,1) uses argument (A,B,F); element (1,2) uses argument (B,D,E,F); elements (2,2) uses arguments (C,D,F).

Usage

OBREMatMComputation(
  nvData,
  nTheta1,
  nTheta2,
  lDensityExpr,
  nCParOBRE,
  matA,
  nvA,
  nK
)

Arguments

Value

OBRE M matrix (M_1 if nK = 1; M_2 if nK = 2).

Element [1, 1] of matrix M.

Description

Function computing element [1, 1] of matrix M, for the computation of asymptotic covariance matrix V.

Usage

OBREMatVMatMEl11(nvData, nTheta1, nTheta2, lDensityExpr, nCParOBRE, matA, nvA)

Arguments

Element [1, 2] of matrix M.

Description

Function computing element [1, 2] of matrix M, for the computation of asymptotic covariance matrix V.

Usage

OBREMatVMatMEl12(nvData, nTheta1, nTheta2, lDensityExpr, nCParOBRE, matA, nvA)

Arguments

Element [2, 1] of matrix M.

Description

Function computing element [2, 1] of matrix M, for the computation of asymptotic covariance matrix V.

Usage

OBREMatVMatMEl21(nvData, nTheta1, nTheta2, lDensityExpr, nCParOBRE, matA, nvA)

Arguments

Element [2, 2] of matrix M.

Description

Function computing element [2, 2] of matrix M, for the computation of asymptotic covariance matrix V.

Usage

OBREMatVMatMEl22(nvData, nTheta1, nTheta2, lDensityExpr, nCParOBRE, matA, nvA)

Arguments

Element [1, 1] of matrix Q.

Description

Function computing argument element [1, 1] of matrix Q of asymptotic covariance matrix V.

Usage

OBREMatVMatQEl11(nvData, nTheta1, nTheta2, lDensityExpr, nCParOBRE, matA, nvA)

Arguments

Element [1, 2] of matrix Q.

Description

Function computing argument element [1, 2] of matrix Q of asymptotic covariance matrix V.

Usage

OBREMatVMatQEl12(nvData, nTheta1, nTheta2, lDensityExpr, nCParOBRE, matA, nvA)

Arguments

Element [2, 2] of matrix Q.

Description

Function computing argument element [2, 2] of matrix Q of asymptotic covariance matrix V.

Usage

OBREMatVMatQEl22(nvData, nTheta1, nTheta2, lDensityExpr, nCParOBRE, matA, nvA)

Arguments

OBRE weights.

Description

Function for computing OBRE weights. The function computes the score function for both parameters and build the score matrix. The score matrix is then modified using OBRE parameters A matrix and a vector and an euclidean norm is derived. The weights are finally found as the minimum between the normalized nCParOBRE and 1.

Usage

OBREWeightsFun(nvData, nTheta1, nTheta2, lDensityExpr, nCParOBRE, matA, nvA)

Arguments

Value

A numeric vector containing OBRE weights.

Argument A for OBRE matrix M integrals.

Description

Function computing argument A for OBRE matrix M integrals.

Usage

OBREmatMArgumentA(
  nvData,
  nTheta1,
  nTheta2,
  lDensityExpr,
  nCParOBRE,
  matA,
  nvA,
  nK
)

Arguments

Argument B for OBRE matrix M integrals.

Description

Function computing argument B for OBRE matrix M integrals.

Usage

OBREmatMArgumentB(
  nvData,
  nTheta1,
  nTheta2,
  lDensityExpr,
  nCParOBRE,
  matA,
  nvA,
  nK
)

Arguments

Argument C for OBRE matrix M integrals.

Description

Function computing argument C for OBRE matrix M integrals.

Usage

OBREmatMArgumentC(
  nvData,
  nTheta1,
  nTheta2,
  lDensityExpr,
  nCParOBRE,
  matA,
  nvA,
  nK
)

Arguments

OBRE vector a.

Description

The function evaluates integrals used to compute the components of OBRE a vector.

Usage

OBREnvAComputation(
  nvData,
  nTheta1,
  nTheta2,
  lDensityExpr,
  nCParOBRE,
  matA,
  nvA
)

Arguments

Value

The OBRE a vector.

Denominator for nvA

Description

Function computing denominator for OBRE numeric vector nvA evaluation.

Usage

OBREnvADen(nvData, nTheta1, nTheta2, lDensityExpr, nCParOBRE, matA, nvA)

Arguments

First part numerator for nvA

Description

Function computing first part numerator for OBRE numeric vector nvA evaluation.

Usage

OBREnvANum1(nvData, nTheta1, nTheta2, lDensityExpr, nCParOBRE, matA, nvA)

Arguments

Second part numerator for nvA

Description

Function computing second part numerator for OBRE numeric vector nvA evaluation.

Usage

OBREnvANum2(nvData, nTheta1, nTheta2, lDensityExpr, nCParOBRE, matA, nvA)

Arguments

Distributions formulas for OBRE

Description

Function containing expressions of density and cumulative functions, plus the first and second derivatives.

Usage

densityExpressions(strDistribution = "normal", eDensityFun = NA)

Arguments

Value

Returns list containing all the symbolic functions.

Examples

# Generates the Normal distribution input for OBRE
distrForOBRE <- densityExpressions(strDistribution = "normal")
# The same result can be generated by inserting manually the formula
distrForOBRE <- densityExpressions(strDistribution = "custom",
eDensityFun = expression((exp( -((nvData - nTheta1)^2) / (2 * nTheta2^2)) /
(sqrt(2 * pi) * nTheta2))))

Part 1 of element [1, 1] for Fisher Information matrix

Description

Function computing part 1 of element [1, 1] for Fisher Information matrix computation. The Fisher Information matrix is splitted in the four elements ([1, 1], [1, 2], [2, 1], [2, 2]). Each element is split in part 1 and part 2

Usage

fisherEl11Part1(nvData, nTheta1, nTheta2, lDensityExpr)

Arguments

Part 2 of element [1, 1] for Fisher Information matrix

Description

Function computing part 2 of element [1, 1] for Fisher Information matrix computation. The Fisher Information matrix is splitted in the four elements ([1, 1], [1, 2], [2, 1], [2, 2]). Each element is split in part 1 and part 2

Usage

fisherEl11Part2(nvData, nTheta1, nTheta2, lDensityExpr)

Arguments

Part 1 of element [1, 2] for Fisher Information matrix

Description

Function computing part 1 of element [1, 2] for Fisher Information matrix computation. The Fisher Information matrix is splitted in the four elements ([1, 1], [1, 2], [2, 1], [2, 2]). Each element is split in part 1 and part 2

Usage

fisherEl12Part1(nvData, nTheta1, nTheta2, lDensityExpr)

Arguments

Part 2 of element [1, 1] for Fisher Information matrix

Description

Function computing part 2 of element [1, 1] for Fisher Information matrix computation. The Fisher Information matrix is splitted in the four elements ([1, 1], [1, 2], [2, 1], [2, 2]). Each element is split in part 1 and part 2

Usage

fisherEl12Part2(nvData, nTheta1, nTheta2, lDensityExpr)

Arguments

Part 1 of element [2, 2] for Fisher Information matrix

Description

Function computing part 1 of element [2, 2] for Fisher Information matrix computation. The Fisher Information matrix is splitted in the four elements ([1, 1], [1, 2], [2, 1], [2, 2]). Each element is split in part 1 and part 2

Usage

fisherEl22Part1(nvData, nTheta1, nTheta2, lDensityExpr)

Arguments

Part 2 of element [2, 2] for Fisher Information matrix

Description

Function computing part 2 of element [2, 2] for Fisher Information matrix computation. The Fisher Information matrix is splitted in the four elements ([1, 1], [1, 2], [2, 1], [2, 2]). Each element is split in part 1 and part 2

Usage

fisherEl22Part2(nvData, nTheta1, nTheta2, lDensityExpr)

Arguments

Fisher information matrix

Description

Function calculating the Fisher information matrix.

Usage

matFisherComputation(nTheta1, nTheta2, lDensityExpr)

Arguments

Value

The Fisher information matrix.

Function that plot an OBREresult object.

Description

The function computes the plot of the OBRE computation

Usage

## S3 method for class 'OBREresult'
plot(x, ...)

Arguments

Value

A graphical representation of an OBREresult obect. The plot is composed by four plots: the value of input data in logaritmic scale, the values of score function evaluated in the input data, the OBRE weights, the values of OBRE components.

Examples

try({# Generates the Normal distribution input for OBRE
distrForOBRE <- densityExpressions(strDistribution = "normal")
# Generates input data
simData = c(rnorm(100, 12, 1), rnorm(10, 10, 10))
# Estimates OBREresult object
estOBRE = OBRE(nvData = simData, strDistribution = "normal", nCParOBRE = 3)
plot(estOBRE)})

First component of the score function.

Description

The function evaluates the formula used to compute the first component of the score function. The missing elements are imputed with 0.

Usage

scoreComponent(nvData, nTheta1, nTheta2, lDensityExpr, nParIndex)

Arguments

Value

The first component of the score function.

Generic summary method

Description

Generic summary method

Usage

summary(object)

Arguments

Function that summarize the results contained in an OBREresult object.

Description

The function shows the estimated parameters, the OBRE tuning parameter, the proportion of data weighted and the relative efficiency with respect to MLE of an OBREresult object.

Usage

## S3 method for class 'OBREresult'
summary(object)

Arguments

Value

The summary an OBREresult obect with the estimated parameters, the OBRE tuning parameter, the proportion of data weighted and the relative efficiency with respect to MLE.

Examples

try({# Generates the Normal distribution input for OBRE
distrForOBRE <- densityExpressions(strDistribution = "normal")
# Generates input data
simData = c(rnorm(100, 12, 1), rnorm(10, 10, 10))
# Estimates OBREresult object
estOBRE <- OBRE(nvData = simData, strDistribution = distrForOBRE, nCParOBRE = 3)
# Summary of the results
summary(estOBRE)})