signed_rank = function( data ){
    #
    # The large sample Wilcoxon signed rank test.
    #
    ad = abs(data)
    r = rank( ad, ties.method='average' )
    z = rep( 0, length(data) )
    z[ data > 0 ] = 1
    w = sum( z * r )
    n = length(data)
    m = n*(n+1)/4 # expected mean under H_0
    v = n*(n+1)*(2*n+1)/24 # expected variance under H_0
    z = ( w - m ) / sqrt(v)
    res = list( w=w, n=n, m=m, v=v, z=z )
    res
}


rank_sum = function( X, y ){
    #
    # The Wilcoxon rank sum test
    #
    n = length(x)
    m = length(y)
    #stopifnot( (n>10) & (m>10) )

    combined = c( x, y )
    r = rank( combined, ties.method='average' )

    # Our statistic is then the sum of the first n (the number of elements in x) ranks:
    w_prime = sum( r[1:n] )

    expected_w_prime = n*(n+m+1)/2
    expected_var_w_prime = n*m*(n+m+1)/12
    z = ( w_prime - expected_w_prime )/sqrt(expected_var_w_prime)

    res = list( w_prime=w_prime, expected_w_prime=expected_w_prime, expected_var_w_prime=expected_var_w_prime, z=z )
    res 
}


number_of_runs_test = function( data ){
    #
    # The number of runs test.
    #
    n = length(data)
    dd = diff(data)
    runs = rle(sign(dd))
    w = length(runs$lengths)
    
    m = (2*n-1)/3 # expected mean under H_0
    v = (16*n-29)/90 # expected variance under H_0
    z = ( w - m ) / sqrt(v)
    p_value = 2*(1-pnorm(abs(z)))
    
    res = list( w=w, n=n, m=m, v=v, z=z, p_value=p_value )
    res
}