R/mhde.R

In mhde: Minimum Hellinger Distance Test for Normality

mhde.grid<- function( Xmin, Xrange, MinScaEst, MaxScaEst, NGauss, Nsubs, WG, Xint, GpdfSqrt, Xdel2 ) {
  #-------------------------------------------------------------------------------------------------------------
  #
  #'mhde.grid
  #
  #'@description
  #'  This function evalutes the Hellinger distance on a 21 by 21 grid of location and scale values.  The
  #'  purpose is to find an initial (location,scale) pair where the Newton-Rhapson method will converge.
  #'  The grid is defined as equally spaced divisions of the reasonable range in location and scale
  #
  # History:
  #    Paul W. Eslinger : 17 Aug 2015 : Original source
  #
  # Parameters:
  #'   @param Xmin        Minimum sample value
  #'   @param Xrange      Range of the sample data
  #'   @param MinScaEst   Minimum reasonale scale estimate
  #'   @param MaxScaEst   Maximum reasonable scale estimate
  #'   @param NGauss      Number of Gauss integration intervals
  #'   @param Nsubs       Number of substeps in each Gauss integration interval
  #'   @param WG          Vector of weights for the Gauss-Legendre integration
  #'   @param Xint        Vector of X values for evaluating the Hellinger distance integral
  #'   @param GpdfSqrt    Vector of the square root of the data-based kernel density function
  #'   @param Xdel2       Delta step used in setting up the integration grid.  Needed for scaling purposes.
  #
  # Returned values:
  #'   @return List containing
  #'   \describe{
  #'   \item{BestLoc} {Location value for the minimum Hellinger distance on the grid}
  #'   \item{BestSca} {Scale value for the minimum Hellinger distance on the grid}
  #'   }
  #
  #-------------------------------------------------------------------------------------------------------------
  #
  # Define the number of grid points and the resulting spacing
  Nloc <- 21
  Nsca <- 21
  Dloc <- Xrange/(Nloc+1)
  Dsca <- (MaxScaEst-MinScaEst)/(Nsca+1)
  # Loop over grid points and calculate the minimum Hellinger distance
  MinHdis <- 9999.99
  for( Iloc in 1:Nloc ) {
  # Define a location
    Hloc <- Xmin + Iloc*Dloc
    for( Isca in 1:Nsca ) {
  #   Define a scale
      Hsca <- MinScaEst + Isca*Dsca
  #   Define the normal density for this grid location and scale
      Npdf <- dnorm(Xint,Hloc,Hsca)
      NpdfSqrt <- sqrt( Npdf )
  #   Calculate the Hellinger distance for this starting point
      Hdis <- mhde.hdis( NGauss, Nsubs, WG, GpdfSqrt, NpdfSqrt, Xdel2 )
  #   Set new initial values to the minimum distance over the grid search
      if( Hdis < MinHdis ) {
        BestLoc <- Hloc
        BestSca <- Hsca
        MinHdis <- Hdis
       }
    }
  }
  return(list(BestLoc,BestSca))
}

mhde.hdis <- function( NGauss, Nsubs, WG, GpdfSqrt, NpdfSqrt, Xdel2 ) {
  #-------------------------------------------------------------------------------------------------------------
  #
  #' mhde.hdis
  #
  #'@description
  #'   This function evaluates the Hellinger distance between the kernel-based density function and the normal
  #'   distribution function.  The normal distribution function is defined at a location and scale in the calling
  #'   function.  Both the kernel density and the normal density are provided to this function as square roots.
  #
  # History:
  #    Paul W. Eslinger : 17 Aug 2015 : Original source
  #
  # Parameters:
  #' @param   NGauss      Number of Gauss integration intervals
  #' @param   Nsubs       Number of substeps in each Gauss integration interval
  #' @param   WG          Vector of weights for the Gauss-Legendre integration
  #' @param   GpdfSqrt    Vector of the square root of the data-based kernel density function
  #' @param   NpdfSqrt    Vector of the normal distribution
  #' @param   Xdel2       Delta step used in setting up the integration grid.  Needed for scaling purposes.
  #
  # Returned values:
  #' @return   Hdis        Value of the Hellinger distance
  #
  #-------------------------------------------------------------------------------------------------------------
#
  Hdis <- 0.0
  idx <- 0
  for(i in 1:NGauss ){
    for(j in 1:Nsubs) {
      idx <- idx + 1
      Hdis <- Hdis + WG[j]*GpdfSqrt[idx]*NpdfSqrt[idx]
    }
  }
  Hdis <- Hdis * Xdel2
  Hdis <- sqrt( 1.0 - Hdis )
  return(Hdis)
}


mhde.checkparam <- function( EstLoc, EstSca, MinLocEst, MaxLocEst, MinScaEst, MaxScaEst ) {
  #-------------------------------------------------------------------------------------------------------------
  #
  #'mhde.checkparam
  #
  #'@description
  #'   This function returns a numeric error code that is nonzero if the location or scale estimates are
  #'   outside reasonable rnges.
  #
  # History:
  #   Paul W. Eslinger : 17 Aug 2015 : Original source
  #
  # Parameters:
  #' @param  EstLoc      Location estimate
  #' @param   EstSca      Scale estimate
  #' @param   MinLocEst   Minimum reasonable location estimate
  #' @param   MaxLocEst   Maximum reasonable Location estimate
  #' @param   MinScaEst   Minimum reasonable scale estimate
  #' @param   MaxScaEst   Maximim reasonable scale estimate
  #
  # Returned values:
  #' @return   ierr        Numeric error code (0=reasonable estimates)
  #
  #-------------------------------------------------------------------------------------------------------------
  ierr <- 0
  if( EstLoc < MinLocEst ) ierr <- 16
  if( EstLoc > MaxLocEst ) ierr <- 17
  if( EstSca < MinScaEst ) ierr <- 18
  if( EstSca > MaxScaEst ) ierr <- 19
  return(ierr)
}


mhde.calcest <- function(EstLoc, EstSca, MaxIter, NGauss, Nsubs, Xint, WG, GpdfSqrt, Xdel2, EpsLoc, EpsSca,
                         MinLocEst, MaxLocEst, MinScaEst, MaxScaEst ) {
  #-------------------------------------------------------------------------------------------------------------
  #
  #' mhde.calcest
  #
  #' @description
  #'   This function calculates the location and scale estimates corresponding to the minimum Hellinger
  #'   distance.  A Newton-Rhapson method with analytical derivatives is used.
  #
  # History:
  #    Paul W. Eslinger : 17 Aug 2015 : Original source
  #
  # Parameters:
  #' @param   EstLoc      Location estimate
  #' @param   EstSca      Scale estimate
  #' @param   MaxIter     The maximum number of iterations allowed
  #' @param   NGauss      Number of Gauss integration intervals
  #' @param   Nsubs       Number of substeps in each Gauss integration interval
  #' @param   Xint        Vector of X values for evaluating the Hellinger distance integral
  #' @param   WG          Vector of weights for the Gauss-Legendre integration
  #' @param   GpdfSqrt    Vector of the square root of the data-based kernel density function
  #' @param   Xdel2       Delta step used in setting up the integration grid.  Needed for scaling purposes.
  #' @param   EpsLoc      The location epsilon for convergence
  #' @param   EpsSca      The scale epsilon for convergence
  #' @param   MinLocEst   Minimum reasonable location estimate
  #' @param   MaxLocEst   Maximum reasonable Location estimate
  #' @param   MinScaEst   Minimum reasonable scale estimate
  #' @param   MaxScaEst   Maximim reasonable scale estimate
  #
  #' @return list containing
  #' \describe{
  #'  \item {Hloc}        {Location estimate for the minimized distance}
  #'   \item{Hsca}        {Scale estimate for the minimized distance}
  #'   \item{myiter}      {Number of iterations performed}
  #'   \item{ierr}        {Numeric error code (0=reasonable estimates)}
  #'   }
  #
  #-------------------------------------------------------------------------------------------------------------
  #
  ierr <- 0
  Hloc <- EstLoc
  Hsca <- EstSca
  #
  # Iterative loop on the estimates
  for( i in 1:MaxIter ) {
    myiter <- i
    # Zero the integration sums
    sdotl  <- 0.0
    sdots  <- 0.0
    sdotll <- 0.0
    sdotls <- 0.0
    sdotss <- 0.0
    # Integrate using a composite Gauss-Legendre rule
    Idx <- 0
    for( J in 1:NGauss ){
      for( K in 1:Nsubs ){
        Idx <- Idx + 1
        X   <- Xint[Idx]
        GX  <- GpdfSqrt[Idx]
        parts <- mhde.partial( X, Hloc, Hsca )
        sdotl  <- sdotl  + parts[[1]] * GX * WG[K]
        sdots  <- sdots  + parts[[2]] * GX * WG[K]
        sdotll <- sdotll + parts[[3]] * GX * WG[K]
        sdotls <- sdotls + parts[[4]] * GX * WG[K]
        sdotss <- sdotss + parts[[5]] * GX * WG[K]
      }
    }
    # The nominal multiplicative integration scaling of Xdel2 on sdotl, etc cancels out in update step
    # Start update procedure
    denom <- sdotss*sdotll - sdotls**2
    # Check for a singular Jacobian
    if( abs(denom) < 1.0E-25 ){
      ierr <- 21
      return(list(Hloc,Hsca,myiter,ierr))
    }
    # Compute the delta step for both location and scale
    dell <- -( sdotss*sdotl - sdotls*sdots ) / denom
    dels <- -( sdotll*sdots - sdotls*sdotl ) / denom
    # Negative scale is not allowed
    # Don't let any step move the scale estimate more than half the distance to zero
    delta <- 1.0
    if( (Hsca+dels) <= 0.0 ){ delta <- abs(Hsca/dels)/2.0 }
    # Always take a smaller first step
    if( i == 1 ) { delta <- min(delta,0.5) }
    # Update the estimates
    Hloc <- Hloc + delta * dell
    Hsca <- Hsca + delta * dels
    # Check if the values meet reasonable bounds
    ierr <- mhde.checkparam( Hloc, Hsca, MinLocEst, MaxLocEst, MinScaEst, MaxScaEst )
    if( ierr > 0 ) return(list(Hloc,Hsca,myiter,ierr))
    # If converged
    if( abs(dell)<EpsLoc & abs(dels)<EpsSca ) return(list(Hloc,Hsca,myiter,ierr))
  } # End of iteration loop
  # Iteration limit without convergence
  ierr <- 22
  return(list(Hloc,Hsca,myiter,ierr))
}

mhde.partial <- function(X,Mean,Sigma) {
  #-------------------------------------------------------------------------------------------------------------
  #
  #' mhde.partial
  #
  #' @description
  #'   This function evaluates the first and second partial derivatives of the square root of the normal density
  #'   function with respect to location and scale.
  #
  # History:
  #    Paul W. Eslinger : 17 Aug 2015 : Original source
  #
  # Parameters:
  #' @param   X       Location (X value) for evaluating the partial derivative
  #' @param   Mean    Mean of the normal density function
  #' @param   Sigma   Standard deviation of the normal density function
  #
  #' @return a list containing
  #' \describe{
  #'   \item{pl}    {Partial derivative with respect to location}
  #'   \item{ps}    {Partial derivative with respect to scale}
  #'   \item{pll}   {Second partial derivative with respect to location}
  #'   \item{pls}   {Second partial derivative with respect to location and scale}
  #'   \item{pss}   {Second partial derivative with respect to scale}
  #'   }
  #
  #-------------------------------------------------------------------------------------------------------------
  # const = 2**(-5/4) * PI**(-1/4)
  const <- 0.31580938887
  #
  z <- (X-Mean)/Sigma
  z2 <- z * z
  z3 <- z2 * z
  z4 <- z3 * z
  zexp <- exp( -0.25 * z2 )
  esm15 <- Sigma**(-1.5)
  esm25 <- esm15 / Sigma
  # Derivative with respect to location
  pl <- const * esm15 * z * zexp
  # Derivative with respect to scale
  ps <- const * esm15 * (z2-1.0) * zexp
  # Second derivative with respect to location
  pll <- const * esm25 * (0.5*z2-1.0) * zexp
  # Second derivative with respect to location and scale
  pls <- const * esm25 * (-0.25*z+0.5*z3 ) * zexp
  # Second derivative with respect to scale
  pss <- const * esm25 * (1.5-4.0*z2+0.5*z4) * zexp
  return(list(pl,ps,pll,pls,pss))
}

#-------------------------------------------------------------------------------------------------------------
#' Utility function for retrieving the critical value for any sample size and desired \eqn{\alpha} level
#
#' @description
#'   This function calculates the critical value for the minimized Hellinger distance for a given sample size
#'   and probability level for a goodness of fit test for normality.
#'
#'   P values below 0.0001 return the value for p=0.0001
#'   P values above 0.9999 return the value for p=0.9999
#'
#'   This utility function is not used by the other functions.  It is provided as a "curiosity" for users.
#
# History:
#   Paul W. Eslinger : 17 Aug 2015 : Original source
#
# Parameters:
#' @param   Nsize   The sample size is defined using \code{Nsize}.  No default value is set.
#' @param   Plevel  The probability level associated with the critical value is set using \code{Plevel}.  The default value is 0.95.  The \eqn{\alpha} level of the test is 1-\code{Plevel}.
#
# Returned values:
#' @return   pval   Critical value for the minimized Hellinger distance
#'
#'
#' @examples
#' ## example using default alpha level of .05
#' mhde.crit(45)
#'
#' ## example using alpha level of .01
#' mhde.crit(45, 0.99)
#'
#
#' @export
#
#-------------------------------------------------------------------------------------------------------------
mhde.crit <- function(Nsize,Plevel=0.95) {
  #
  # Load the coefficients from a data file
  pcoef <- mhde.loadpval()
  # Explicity define the alpha levels corresponding to the coefficients in the file
  nalpha <- 63
  alpha <- c(0.0001,0.001,0.005,0.01,0.015,0.02,0.025,0.03,0.035,0.04,
             0.045,0.05,0.055,0.06,0.065,0.07,0.075,0.08,0.085,0.09,
             0.095,0.1,0.125,0.15,0.175,0.2,0.25,0.3,0.35,0.4,0.45,0.5,
             0.55,0.6,0.65,0.7,0.75,0.8,0.825,0.85,0.875,0.9,0.905,0.91,
             0.915,0.92,0.925,0.93,0.935,0.94,0.945,0.95,0.955,0.96,
             0.965,0.97,0.975,0.98,0.985,0.99,0.995,0.999,0.9999)
  #
  pcrit <- numeric(nalpha)
  # If the sample size is 40 or smaller, use row Nsize-4 for the explicit coeffs
  if( Nsize <= 40 ) {
    nr <- Nsize-4
    for( i in 1:nalpha ){ pcrit[i] <- pcoef[nr,i] }
  } else {
    # Sample size greater than 40 pick the rows for the intercept and slope coefficients
    pintercept <- numeric(nalpha)
    pslope <- numeric(nalpha)
    # Pick the row for the intercept
    nr <- 0
    if( Nsize >=   41 & Nsize <=   60 ) nr <- 37
    if( Nsize >=   61 & Nsize <=   80 ) nr <- 38
    if( Nsize >=   81 & Nsize <=  100 ) nr <- 39
    if( Nsize >=  101 & Nsize <=  150 ) nr <- 40
    if( Nsize >=  151 & Nsize <=  200 ) nr <- 41
    if( Nsize >=  201 & Nsize <=  300 ) nr <- 42
    if( Nsize >=  301 & Nsize <=  400 ) nr <- 43
    if( Nsize >=  401 & Nsize <=  600 ) nr <- 44
    if( Nsize >=  601 & Nsize <=  800 ) nr <- 45
    if( Nsize >=  801 & Nsize <= 1000 ) nr <- 46
    if( Nsize >= 1001 & Nsize <= 1500 ) nr <- 47
    if( Nsize >= 1501 & Nsize <= 2000 ) nr <- 48
    if( Nsize >= 2001 & Nsize <= 3000 ) nr <- 49
    if( Nsize >= 3001 & Nsize <= 4000 ) nr <- 50
    if( Nsize >= 4001 & Nsize <= 5000 ) nr <- 51
    if( Nsize >= 5001 & Nsize <= 6000 ) nr <- 52
    if( Nsize >= 6001 & Nsize <= 8000 ) nr <- 53
    if( Nsize > 8001 ) nr <- 54
    for( i in 1:nalpha ) pintercept[i] <- pcoef[nr,i]
    # Pick the row for the slope (18 groups of sample sizes)
    nr <- nr + 18
    for( i in 1:nalpha ) pslope[i] <- pcoef[nr,i]
    # Calculate the critical values from intercept and slope (log-log fit)
    for( i in 1:nalpha ) pcrit[i] <- pintercept[i] * Nsize ** pslope[i]
  }
  # Have critical values for all alpha levels for this sample size
  # Now calculate the critical value for the p value using linear interpolation
  if( Plevel <= alpha[1] ) return(pcrit[1])
  if( Plevel >= alpha[nalpha] ) return(pcrit[nalpha])
  for( i in 2:nalpha){
    if( Plevel <= alpha[i] ) {
      Adel <- (Plevel-alpha[i-1])/(alpha[i]-alpha[i-1])
      Cdel <- pcrit[i] - pcrit[i-1]
      pval <- pcrit[i-1] + Cdel*Adel
      return(pval)
    }
  }
}

mhde.loadpval <- function() {
  #-------------------------------------------------------------------------------------------------------------
  #
  #' mhde.loadpval
  #
  #' @description
  #'   Load the coeficient data from which to calculate the p values
  #
  # History:
  #   Paul W. Eslinger :  1 Sep 2015 : Original source
  #
  # Parameters:
  #   None
  #
  #' @return
  #'   pcoef   A matrix of coefficients
  #
  #-------------------------------------------------------------------------------------------------------------
  ctmp <- c(
  0.068768,0.069260,0.070177,0.070900,0.071478,0.071973,0.072421,0.072828,0.073208,
  0.073563,0.073902,0.074223,0.074527,0.074820,0.075100,0.075368,0.075628,0.075881,
  0.076125,0.076359,0.076584,0.076805,0.077795,0.078768,0.079773,0.080808,0.082944,
  0.085148,0.087535,0.090375,0.094229,0.099106,0.105246,0.113042,0.123122,0.136439,
  0.154508,0.179753,0.196043,0.215781,0.240597,0.273833,0.281910,0.290574,0.300073,
  0.309949,0.319304,0.325324,0.329478,0.331887,0.333230,0.334382,0.336194,0.340463,
  0.347939,0.360425,0.382265,0.418823,0.476528,0.480221,0.481625,0.482073,0.484554,
  0.070946,0.071527,0.072608,0.073402,0.074003,0.074503,0.074938,0.075322,0.075669,
  0.075988,0.076281,0.076553,0.076813,0.077055,0.077286,0.077499,0.077707,0.077906,
  0.078100,0.078291,0.078479,0.078664,0.079577,0.080493,0.081433,0.082401,0.084496,
  0.086818,0.089417,0.092341,0.095702,0.099653,0.104484,0.110755,0.119086,0.130080,
  0.144813,0.165012,0.178135,0.194141,0.214123,0.240408,0.246722,0.253405,0.260501,
  0.268027,0.275886,0.283599,0.290491,0.296148,0.300432,0.302942,0.305067,0.307489,
  0.311029,0.317882,0.329682,0.349041,0.386603,0.430627,0.434181,0.436483,0.439434,
  0.059792,0.062558,0.065618,0.067300,0.068469,0.069394,0.070180,0.070877,0.071496,
  0.072063,0.072585,0.073068,0.073518,0.073940,0.074342,0.074724,0.075091,0.075444,
  0.075789,0.076119,0.076440,0.076757,0.078247,0.079655,0.081014,0.082356,0.085020,
  0.087758,0.090736,0.094153,0.098236,0.103268,0.109548,0.117418,0.127323,0.139909,
  0.156152,0.177560,0.191089,0.207456,0.227907,0.253312,0.258833,0.264303,0.269527,
  0.274307,0.278426,0.281339,0.284044,0.287503,0.292333,0.298541,0.306275,0.315878,
  0.327916,0.342948,0.363811,0.392288,0.399501,0.413997,0.491903,0.521524,0.796988,
  0.061364,0.063539,0.066358,0.068151,0.069430,0.070463,0.071337,0.072100,0.072791,
  0.073417,0.073990,0.074528,0.075034,0.075507,0.075961,0.076389,0.076800,0.077193,
  0.077569,0.077929,0.078278,0.078616,0.080146,0.081482,0.082723,0.083928,0.086311,
  0.088797,0.091527,0.094633,0.098256,0.102591,0.107890,0.114497,0.122854,0.133410,
  0.146985,0.164746,0.175919,0.189299,0.205808,0.226785,0.231575,0.236513,0.241581,
  0.246672,0.251681,0.256423,0.260669,0.263926,0.267266,0.271534,0.277112,0.284440,
  0.293702,0.305283,0.320284,0.340857,0.365852,0.374630,0.422481,0.467202,0.627102,
  0.055744,0.058644,0.062168,0.064360,0.065912,0.067148,0.068192,0.069105,0.069934,
  0.070679,0.071367,0.072004,0.072607,0.073173,0.073706,0.074216,0.074703,0.075167,
  0.075613,0.076044,0.076459,0.076859,0.078710,0.080395,0.081966,0.083476,0.086418,
  0.089397,0.092555,0.096046,0.100073,0.104871,0.110739,0.117933,0.126794,0.137757,
  0.151517,0.169258,0.180298,0.193474,0.209521,0.228958,0.233158,0.237318,0.241413,
  0.245333,0.248813,0.252237,0.256159,0.260896,0.266515,0.273105,0.280836,0.289934,
  0.300656,0.313443,0.329086,0.344995,0.355213,0.382787,0.433691,0.519531,0.776109,
  0.055855,0.058908,0.062557,0.064819,0.066434,0.067744,0.068852,0.069830,0.070705,
  0.071501,0.072237,0.072918,0.073555,0.074149,0.074712,0.075247,0.075756,0.076243,
  0.076706,0.077150,0.077580,0.077988,0.079880,0.081553,0.083086,0.084523,0.087263,
  0.089983,0.092832,0.095978,0.099590,0.103849,0.108979,0.115260,0.122972,0.132489,
  0.144414,0.159720,0.169187,0.180296,0.193884,0.210656,0.214412,0.218278,0.222251,
  0.226280,0.230260,0.234073,0.237604,0.241267,0.245609,0.250866,0.257129,0.264534,
  0.273382,0.284017,0.297077,0.313202,0.328836,0.344608,0.399421,0.478652,0.561384,
  0.052491,0.055794,0.059773,0.062258,0.064008,0.065413,0.066609,0.067655,0.068588,
  0.069440,0.070227,0.070955,0.071636,0.072281,0.072888,0.073464,0.074013,0.074543,
  0.075052,0.075541,0.076013,0.076471,0.078573,0.080463,0.082210,0.083867,0.087029,
  0.090138,0.093367,0.096825,0.100715,0.105203,0.110544,0.116984,0.124815,0.134361,
  0.146188,0.161198,0.170439,0.181339,0.194378,0.210017,0.213446,0.216913,0.220411,
  0.223874,0.227263,0.230785,0.234697,0.239149,0.244233,0.250066,0.256754,0.264453,
  0.273401,0.283928,0.296405,0.309841,0.322722,0.346593,0.390338,0.491213,0.669400,
  0.052647,0.055928,0.059889,0.062416,0.064218,0.065677,0.066907,0.067992,0.068959,
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  -0.263000,-0.274090,-0.280560,-0.282420,-0.284620,-0.286040,-0.287360,-0.288750,-0.289710,
  -0.290470,-0.291560,-0.292350,-0.292680,-0.293100,-0.294040,-0.294270,-0.294630,-0.295160,
  -0.295770,-0.296590,-0.296240,-0.297570,-0.299260,-0.301110,-0.302320,-0.303960,-0.306610,
  -0.308920,-0.311380,-0.314280,-0.316230,-0.318140,-0.320380,-0.322070,-0.323990,-0.326610,
  -0.328790,-0.331720,-0.332860,-0.334530,-0.336360,-0.338790,-0.338990,-0.339550,-0.339760,
  -0.340120,-0.340860,-0.341530,-0.342060,-0.342930,-0.342980,-0.343640,-0.344320,-0.345220,
  -0.346340,-0.347410,-0.348570,-0.350300,-0.352840,-0.355370,-0.359100,-0.364730,-0.382340)
  pcoef <- matrix(ctmp,nrow=72,ncol=63,byrow=TRUE )
  return(pcoef)
}

mhde.pval <- function(pcoef,Nsize,Hdis) {

  #-------------------------------------------------------------------------------------------------------------
  #
  #'mhde.pval
  #
  #' @description
  #'   This function computes the upper tail probability level for a computed Hellinger distance given the
  #'   sample size.
  #
  # History:
  #    Paul W. Eslinger : 17 Aug 2015 : Original source
  #
  # Parameters:
  #' @param   pcoef   A matrix of coefficients used in functional fits for the tail probabilities
  #' @param   Nsize   Number of data values (sample size)
  #' @param   Hdis    The value of the Hellinger distance
  #
  # Returned values:
  #' @return   pval  Upper tail p value.
  #
  #-------------------------------------------------------------------------------------------------------------
#
  # Explicity define the alpha levels corresponding to the coefficients in the data file
  nalpha <- 63
  alpha <- c(0.0001,0.001,0.005,0.01,0.015,0.02,0.025,0.03,0.035,0.04,
             0.045,0.05,0.055,0.06,0.065,0.07,0.075,0.08,0.085,0.09,
             0.095,0.1,0.125,0.15,0.175,0.2,0.25,0.3,0.35,0.4,0.45,0.5,
             0.55,0.6,0.65,0.7,0.75,0.8,0.825,0.85,0.875,0.9,0.905,0.91,
             0.915,0.92,0.925,0.93,0.935,0.94,0.945,0.95,0.955,0.96,
             0.965,0.97,0.975,0.98,0.985,0.99,0.995,0.999,0.9999)
  #
  pcrit <- numeric(nalpha)
  # If the sample size is 40 or smaller, use row Nsize-4 for the explicit coeffs
  if( Nsize <= 40 ) {
    nr <- Nsize-4
    for( i in 1:nalpha ){ pcrit[i] <- pcoef[nr,i] }
  } else {
    pintercept <- numeric(nalpha)
    pslope <- numeric(nalpha)
    # Pick the row for the intercept
    nr <- 0
    if( Nsize >=   41 & Nsize <=   60 ) nr <- 37
    if( Nsize >=   61 & Nsize <=   80 ) nr <- 38
    if( Nsize >=   81 & Nsize <=  100 ) nr <- 39
    if( Nsize >=  101 & Nsize <=  150 ) nr <- 40
    if( Nsize >=  151 & Nsize <=  200 ) nr <- 41
    if( Nsize >=  201 & Nsize <=  300 ) nr <- 42
    if( Nsize >=  301 & Nsize <=  400 ) nr <- 43
    if( Nsize >=  401 & Nsize <=  600 ) nr <- 44
    if( Nsize >=  601 & Nsize <=  800 ) nr <- 45
    if( Nsize >=  801 & Nsize <= 1000 ) nr <- 46
    if( Nsize >= 1001 & Nsize <= 1500 ) nr <- 47
    if( Nsize >= 1501 & Nsize <= 2000 ) nr <- 48
    if( Nsize >= 2001 & Nsize <= 3000 ) nr <- 49
    if( Nsize >= 3001 & Nsize <= 4000 ) nr <- 50
    if( Nsize >= 4001 & Nsize <= 5000 ) nr <- 51
    if( Nsize >= 5001 & Nsize <= 6000 ) nr <- 52
    if( Nsize >= 6001 & Nsize <= 8000 ) nr <- 53
    if( Nsize > 8001 ) nr <- 54
    for( i in 1:nalpha ){ pintercept[i] <- pcoef[nr,i] }
    # Pick the row for the slope (18 groups of sample sizes)
    nr <- nr + 18
    for( i in 1:nalpha ){ pslope[i] <- pcoef[nr,i] }
    for( i in 1:nalpha ){ pcrit[i] <- pintercept[i] * Nsize ** pslope[i] }
  }
  # Have critical values for all alpha levels for this sample size
  # Now calculate the pvalue using linear interpolation
  if( Hdis <= pcrit[1] ) return(1.0-alpha[1])
  if( Hdis >= pcrit[nalpha] ) return(1.0-alpha[nalpha])
  for( i in 2:nalpha){
    if( Hdis <= pcrit[i] ) {
       Cdel <- (Hdis-pcrit[i-1])/(pcrit[i]-pcrit[i-1])
       Adel <- alpha[i] - alpha[i-1]
       pval <- alpha[i-1] + Cdel*Adel
       return(1.0-pval)
    }
  }
  return(1.0-pval)
}


#-------------------------------------------------------------------------------------------------------------
#' Utility function for retrieving \eqn{c_n}{Cn} value for any sample size
#
#'@description
#'   This function evaluates the \eqn{c_n}{Cn} bandwidth value for the Epanechnikov kernel based on the sample size.
#
# History:
#    Paul W. Eslinger : 17 Aug 2015 : Original source
#
# Parameters:
#' @param   Nsize   Number of values in the sample data set
#
# Returned value:
#' @return   The Epanechnikov kernel bandwidth parameter for the sample size.
#
#' @export
#
#-------------------------------------------------------------------------------------------------------------
mhde.cn <- function(Nsize) {

  if( Nsize ==  5 ) return(1.7738)
  if( Nsize ==  6 ) return(1.6583)
  if( Nsize ==  7 ) return(1.4884)
  if( Nsize ==  8 ) return(1.4221)
  if( Nsize ==  9 ) return(1.3273)
  if( Nsize == 10 ) return(1.2826)
  if( Nsize == 11 ) return(1.2213)
  if( Nsize == 12 ) return(1.1888)
  if( Nsize == 13 ) return(1.1443)
  if( Nsize == 14 ) return(1.1188)
  if( Nsize == 15 ) return(1.0848)
  if( Nsize == 16 ) return(1.0649)
  if( Nsize == 17 ) return(1.0380)
  if( Nsize == 18 ) return(1.0209)
  if( Nsize == 19 ) return(0.99949)
  if( Nsize == 20 ) return(0.98423)
  if( Nsize == 21 ) return(0.96641)
  if( Nsize == 22 ) return(0.95340)
  if( Nsize == 23 ) return(0.93794)
  if( Nsize == 24 ) return(0.92604)
  if( Nsize == 25 ) return(0.91282)
  if( Nsize == 26 ) return(0.90227)
  if( Nsize == 27 ) return(0.89025)
  if( Nsize == 28 ) return(0.88118)
  if( Nsize == 29 ) return(0.87021)
  if( Nsize == 30 ) return(0.86210)
  if( Nsize == 31 ) return(0.85202)
  if( Nsize == 32 ) return(0.84479)
  if( Nsize == 33 ) return(0.83597)
  if( Nsize == 34 ) return(0.82846)
  if( Nsize == 35 ) return(0.82136)
  if( Nsize == 36 ) return(0.81407)
  if( Nsize == 37 ) return(0.80666)
  if( Nsize == 38 ) return(0.80058)
  if( Nsize == 39 ) return(0.79402)
  if( Nsize == 40 ) return(0.78806)
  if( Nsize >=   41 & Nsize <=   70 ) return(2.3519*Nsize**(-0.29653))
  if( Nsize >=   71 & Nsize <=  100 ) return(2.2697*Nsize**(-0.28809))
  if( Nsize >=  101 & Nsize <=  150 ) return(2.2268*Nsize**(-0.28394))
  if( Nsize >=  151 & Nsize <=  200 ) return(2.1904*Nsize**(-0.28064))
  if( Nsize >=  201 & Nsize <=  300 ) return(2.1882*Nsize**(-0.28046))
  if( Nsize >=  301 & Nsize <=  400 ) return(2.1835*Nsize**(-0.28008))
  if( Nsize >=  401 & Nsize <=  600 ) return(2.1983*Nsize**(-0.28121))
  if( Nsize >=  601 & Nsize <=  800 ) return(2.2070*Nsize**(-0.28183))
  if( Nsize >=  801 & Nsize <= 1000 ) return(2.2267*Nsize**(-0.28317))
  if( Nsize >= 1001 & Nsize <= 1500 ) return(2.2501*Nsize**(-0.28467))
  if( Nsize >= 1501 & Nsize <= 2000 ) return(2.2822*Nsize**(-0.28662))
  if( Nsize >= 2001 & Nsize <= 3000 ) return(2.3029*Nsize**(-0.28780))
  if( Nsize >= 3001 & Nsize <= 4000 ) return(2.3330*Nsize**(-0.28942))
  if( Nsize >= 4001 & Nsize <= 5000 ) return(2.3502*Nsize**(-0.29031))
  if( Nsize >= 5001 & Nsize <= 6000 ) return(2.3596*Nsize**(-0.29078))
  if( Nsize >= 6001 & Nsize <= 8000 ) return(2.3929*Nsize**(-0.29239))
  if( Nsize > 8001 ) return(2.4130*Nsize**(-0.29332))
}

#-------------------------------------------------------------------------------------------------------------
#
#' Minimum Hellinger Distance Test for Normality
#
#' @description
#'   This function fits a normal distribution to a data set using a mimimum Hellinger distance approach and
#'   then performs a test of hypothesis that the data are from a normal distribution.
#
#' @details
#'   Let \emph{f(x)} and \emph{g(x)} be absolutely continuous probability density functions.  The square of the
#'   Hellinger distance can be written as \eqn{H^2 = 1 - \int\sqrt{f(x)g(x)}dx}.  For this package, \emph{f(x)}
#'   denotes the family of normal densities and \emph{g} is a data-based density obtained by using the Ephanechnikov
#'   kernel. The kernel has the form \eqn{w(z)=0.75(1-z^2 )} for \samp{-1<z<1} and 0 elsewhere.  Let the
#'   \emph{n} sample data be denoted by \eqn{x_1}{X1}, ..., \eqn{x_n}{Xn}.  The data-based kernel density at any
#'   point \emph{y} is calculated from \deqn{g_n(y) = \frac{1}{n s_n c_n }\sum\limits_{i=1}^n w(\frac{y-x_i}{s_n c_n})}
#'
#'   A Newton-Rhapson method with analytical derivatives is to determine the minimum Hellinger distance.
#'   Numerical integration is done using a 6-point composite Gauss-Legendre technique.
#
# Parameters:
#' @param   DataVec       The data are supplied by the user in a numeric vector.  The length of the vector determines the number of data values.
#
#' @param   NGauss        The number of subintervals for the Gauss-Legendre integration techniques is controlled by \code{NGauss}.  A default value of 100 is used.  A minimum of 25 is enforced.
#
#' @param   MaxIter       The maximum number of iterations that can occur in evaluating the minimum Hellinger distance is controlled by \code{MaxIter}.  A default of 25 is used.  A minimum of 1 is enforced.
#
#' @param   InitLocation  An optional initial location estimate can be defined using \code{InitLocation}.  The data median is the default value.
#
#' @param   InitScale     An optional initial scale estimate can be defined using \code{InitScale}. The data median absolute deviation is the default value.
#
#' @param   EpsLoc        The epsilon (in data units) below which the iterative minimization approach declares convergence in the location estimate is controlled by \code{EpsLoc}.  \code{EpsLoc} should be set to give approximately 5 digits of accuracy in the location estimate. A default value of 0.0001 is used.
#
#' @param   EpsSca        The epsilon (in data units) below which the iterative minimization approach declares convergence in the SCALE estimate is controlled by \code{EpsSca}.  \code{EpsSca} should be set to give approximately 5 digits of accuracy in the scale estimate. A default value of 0.0001 is used.
#
#' @param   Silent        A value of FALSE for \code{Silent} writes several results to the R console.  Use \code{Silent}=TRUE to eliminate the output.
#
#' @param   Small         A value of FALSE for \code{Small} returns a list of 11 objects.  Use \code{Small}=TRUE to return a shorter list containing only the Hellinger distance and the p-value.
#
# Returned values:
#' @return   Values returned in a list include the following items:
#' \itemize{
#'     \item {Minimized Hellinger distance}
#'     \item {p-value for the minimized distance}
#'     \item {Initial location used in the iterative solution}
#'     \item {Initial scale used in the iterative solution}
#'     \item {Final location estimate}
#'     \item {Final scale estimate}
#'     \item {Sample size}
#'     \item {Kernel density bandwidth parameter}
#'     \item {Vector of x values used in the integration for the Hellinger distance}
#'     \item {Vector of nonparametric density values at the x values used in the integration}
#'     \item {Vector of normal density values for the estimated location and scale at the x values used in integration}
#' }
#
# References:
#
#' @references
#'   Epanechnikov, VA. 1969. "Non-Parametric Estimation of a Multivariate Probability Density."
#'   Theory of Probability and its Applications 14(1):153-156. doi \url{http://dx.doi.org/10.1137/1114019}
#'
#'   Hellinger, E. 1909. "Neue Begrundung Der Theorie Quadratischer Formen Von Unendlichvielen
#'   Veranderlichen." Journal fur die reine und angewandte Mathematik 136:210-271.
#'   doi \url{http://dx.doi.org/10.1515/crll.1909.136.210}
#
#   Eslinger, Paul W, and Wayne A Woodward. 1991. "Minimum Hellinger Distance Estimation for Normal Models."
#   Journal of Statistical Computation and Simulation 39(1-2):95-114. doi http://dx.doi.org/10.1080/00949659108811342
#
#   Tjalling J. Ypma. 1995. "Historical development of the Newton-Raphson method", SIAM Review 37(4), 531-551.
#   doi:http://dx.doi.org/10.1137/1037125.
#
#   Mike "Pomax" Kamermans, 2011.  "Primer on Bezier curves"
#   http://pomax.github.io/bezierinfo/legendre-gauss.html
#
#' @author
#'   Paul W. Eslinger and Heather M. Orr
#
#' @examples
#' ## example with a normal data set
#' mhde.test(rnorm(20,0.0,1.0),Small=TRUE)
#'
#' ## example with a uniform data set including example plot
#' MyList <- mhde.test(runif(25,min=2,max=4))
#' mhde.plot(MyList)
#
#' @export
#' @importFrom stats dnorm mad median sd
#-------------------------------------------------------------------------------------------------------------
mhde.test <- function(DataVec,NGauss=100,MaxIter=25,InitLocation,InitScale,EpsLoc=0.0001,EpsSca=0.0001,
  Silent=FALSE,Small=FALSE ) {

  # Argument checking
  if(missing(DataVec)) {
    stop("The mhde.test function expects a numeric vector on input",call.=FALSE)
  }
  # Get the number of data values from the length of the input vector
  Nsize <- length(DataVec)
  # Error terminate if Nsize < 5
  if(Nsize < 5) stop("At least 5 data values are required",call.=FALSE)
  #
  # Check the maximum number of iterations
  if(!missing(MaxIter)) {
    if( MaxIter < 1 ) stop("At least one iteration must be performed",call.=FALSE)
  }
  #
  # Check values for number of integration subintervals
  if(!missing(NGauss)) {
    if( NGauss < 25 ) stop("At least 25 integration subintervals must be specified",call.=FALSE)
  }
  #
  # Any general checks on location and scale convergence epsilon must depend on scale of the inputs.
  #
  #--------------------------------------------------------------------------
  # Set up values for integration
  # Reference for coefficients is http://pomax.github.io/bezierinfo/legendre-gauss.html
  Nsubs <- 6
  XG <- c(-0.9324695142031521,-0.6612093864662645,-0.2386191860831969,0.2386191860831969,0.6612093864662645,0.9324695142031521)
  WG <- c( 0.1713244923791704, 0.3607615730481386, 0.4679139345726910,0.4679139345726910,0.3607615730481386,0.1713244923791704)
  Nsteps <- NGauss * Nsubs
  #
  # The coding cycles through indices on XG and WG in reverse order
  # Set up a vector of indices to use in "for loops"
  Jsubs <- sort(c(1:Nsubs),decreasing=TRUE)
  #
  #--------------------------------------------------------------------------
  # Sort the incoming data and store in a vector
  x <- sort(DataVec)
  #
  # Calculate summary statistics on the data
  Xmin <- x[1]
  Xmax <- x[Nsize]
  Xrange <- Xmax - Xmin
  Xmedian <- median(x)
  Xstdev <- sd(x)
  #
  # Determine reasonable parameter bounds
  # Constrain the location estimate to the data range
  MinLocEst <- Xmin
  MaxLocEst <- Xmax
  # Constrain the scale maximum to the data range
  MaxScaEst <- Xrange
  # Constrain the minimum scale to 2.5% of the 5% trimmed range
  Ntmp <- as.integer(0.025*Nsize + 1)
  Ns1 <- max( Ntmp, 2 )
  Ns2 <- Nsize - Ns1 + 1
  MinScaEst <- (x[Ns2]-x[Ns1]) / 40.0
  #
  # Set bandwidth values needed for the Epanechnikov kernel
  Sne <- mad(x)
  Cne <- mhde.cn(Nsize)
  Hne <- Cne * Sne
  #
  # Set up initial values
  if(missing(InitLocation)) { InitLocation <- Xmedian }
  if(missing(InitScale))    { InitScale <- mad(x) }
  #
  #--------------------------------------------------------------------------
  # Set up the Gauss-Legendre integration stuff
  # Don't need to look beyond data range adjusted for the kernel width
  Xmini <- Xmin - Hne
  Xmaxi <- Xmax + Hne
  Xdel <- ( Xmaxi - Xmini ) / NGauss
  Xdel2 <- 0.5 * Xdel
  Xint <- numeric(Nsteps)
  Idx <- 0
  for( i in 1:NGauss ){
    Xhalf <- Xmini + (i-1)*Xdel + Xdel2
    for ( j in Jsubs ){
      Idx <- Idx + 1
      Xint[Idx] <- Xhalf - Xdel2*XG[j]
    }
  }
  #
  #--------------------------------------------------------------------------
  # Generate the data-based kernel density function
  # Define the kernel density on the integration grid
  Gpdf <- numeric(Nsteps)
  #
  # Increment the density at every location for influence of every data value
  for( i in 1:Nsteps ){
    for( j in 1:Nsize ){
      Diff <- abs(Xint[i]-x[j])
      if( Diff < Hne ) Gpdf[i] <- Gpdf[i] + (1.0-(Diff/Hne)**2)
    }
  }
  # Final scaling for the density estimate
  for(i in 1:Nsteps ){ Gpdf[i] <- 0.75 * Gpdf[i] / (Hne*Nsize) }
  #
  # Need the square root of the density in the integration scheme
  GpdfSqrt <- sqrt( Gpdf )
  #
  #--------------------------------------------------------------------------
  # Do the parameter estimation using a Newton-Rhapson minimization method
  # with analytical derivatives
  #
  # Initial values
  EstLoc <- InitLocation
  EstSca <- InitScale
  #
  # Estimation via minimization of the Hellinger distance
  Estimate <- mhde.calcest(EstLoc, EstSca, MaxIter, NGauss, Nsubs, Xint, WG,
                           GpdfSqrt, Xdel2, EpsLoc, EpsSca, MinLocEst, MaxLocEst,
                           MinScaEst, MaxScaEst )
  #
  # Check for a valid estimation result. Valid if Estimate[[4]] is 0
  # If an invalid estimate, use a grid search to determine new initial values
  if( Estimate[[4]] != 0 ) {
    NewInitial <- mhde.grid( Xmin, Xrange, MinScaEst, MaxScaEst, NGauss, Nsubs,
                             WG, Xint, GpdfSqrt, Xdel2 )
    EstLoc <- NewInitial[[1]]
    EstSca <- NewInitial[[2]]
    Estimate <- mhde.calcest(EstLoc, EstSca, MaxIter, NGauss, Nsubs, Xint, WG,
                             GpdfSqrt, Xdel2, EpsLoc, EpsSca, MinLocEst, MaxLocEst,
                             MinScaEst, MaxScaEst )
  #
  # If didn't converge, use the solution for smallest Hellinger distance in the grid
    if( Estimate[[4]] != 0 ) {
      Estimate[[1]] <- NewInitial[[1]]
      Estimate[[2]] <- NewInitial[[2]]
    }
  }
  #
  #--------------------------------------------------------------------------
  # Use the location and scale estimates in a goodness of fit test for normality
  #
  # Evaluate the normal distribution at the estimated location and scale
  Npdf <- dnorm(Xint,Estimate[[1]],Estimate[[2]])
  NpdfSqrt <- sqrt( Npdf )
  #
  # Calculate the minimized Hellinger distance
  Hdis <- mhde.hdis( NGauss, Nsubs, WG, GpdfSqrt, NpdfSqrt, Xdel2 )
  #
  # Convert the minimum distance into a probability level (p value) for
  # the goodness of fit test for normality
  # Load the data from which to calculate the p values
  pcoef <- mhde.loadpval()
  pvalue <- mhde.pval(pcoef,Nsize,Hdis)
  #
  #--------------------------------------------------------------------------
  # Write the final summary information to the R console
  if( !Silent ) {
    cat("Minimum Hellinger Distance normality test","\n")
    cat("Data set          ",deparse(substitute(DataVec)),"\n")
    cat("Sample size       ",Nsize,"\n")
    cat("Location estimate ",Estimate[[1]],"\n")
    cat("Scale estimate    ",Estimate[[2]],"\n")
    cat("Hellinger Distance",Hdis,"\n")
    cat("p-value           ",pvalue,"\n")
  }
  #
  # Return the results in a list
  if( Small ) {return(list(Hdis,pvalue))}
  return(list(Hdis,pvalue,InitLocation,InitScale,Estimate[[1]],Estimate[[2]],Nsize,Cne,Xint,Gpdf,Npdf))
}

#-------------------------------------------------------------------------------------------------------------
#' Plot the non-parametric and normal densities
#
#' @description
#'   Simple plot function to display the data-based kernel density and the normal density for the final location and scale estimates.
#
# History:
#    Paul W. Eslinger : 17 Aug 2015 : Original source
#
# Parameters:
#' @param   ListOut    The output list from \code{mhde.test}.  The argument \code{Small} for \code{mhde.test} must have the value FALSE.
#
# Returned values:
#   None.
#' @export
#' @importFrom graphics legend lines plot
#-------------------------------------------------------------------------------------------------------------
mhde.plot <- function(ListOut) {
#
# Plot range
xrange <- range( ListOut[[9]] )
yrange <- range( c(ListOut[[10]]),(ListOut[[11]]) )
yrange[1] <- 0.0
plot( xrange, yrange, type="n", xlab="Sample Data Range", ylab="Proability Density",
  main="Kernel and Normal Densities", col.lab="black", col.main="black" )
lines(ListOut[[9]], ListOut[[10]], type="l", lwd=2.5, lty=1, pch=18)
lines(ListOut[[9]], ListOut[[11]], type="l", lwd=2.5, lty=1, pch=19, col="red")
legend(xrange[1], yrange[2], legend=c("normal","kernel"), cex=0.9, col=c("red","black"),
  pch=c(18,19), lty=c(1,1), title="Densities")
}