Variance of Signum Function

Functional and composite function

Variance for the statistics of the general rank score test

median of a series of numbers minus its median is always zero

1. suppose we only have 3 (odd) number of values x1,x2, x3 ordered already

so the median is x2.

The  new sequence will be x1-x2, 0, x3-x2.

We see the new median is 0.

2. suppose we have 4 (even) number of values y1, y2, y3, y4, ordered already.

the median is (y2+y3)/2.

The new sequence is : y1-(y2+y3)/2, y2-(y2+y3)/2, y3-(y2+y3)/2, y4-(y2+y3)/2.

The new median is: (y2-(y2+y3)/2+ y3-(y2+y3)/2) which is also 0.

This observation is  useful when we estimate the shift estimator of a Sign Scores test.

Numerically solve estimating equations for shift estimator of the Van der Waerden test (normal scores)

#Data from HMC p557.

s1<-c(51.9,56.9,45.2,52.3,59.5,41.4,46.4,45.1,53.9,42.9,41.5,55.2,32.9,54.0,45.0)
s2<-c(59.2,49.1,54.4,47.0,55.9,34.9,62.2,41.6,59.3,32.7,72.1,43.8,56.8,76.7,60.3)

L<-vector("list",21)
L2<-vector("list",21)
L_rank<-vector("list",21) ##21 is from the steps 4 to 6 by 0.1

for(d in seq(from=4, to=6, by=0.1)){
L[[round((d-3.9)/0.1)]]<-(s2-d)
L2[[round((d-3.9)/0.1)]]<-c(s1,L[[round((d-3.9)/0.1)]])
L_rank[[round((d-3.9)/0.1)]]<-rank(c(s1,L[[round((d-3.9)/0.1)]]))
}
L_w<-vector("list",21)
L_ns<-vector("list",21)
L_sum_w<-vector("list",21)

for (i in 1:21){
 for (j in 16:30){

L_w[[i]][j-15]<-L_rank[[i]][j]

L_ns[[i]][j-15]<-qnorm(L_w[[i]][j-15]/31,0,1)

L_sum_w[[i]]<-sum(L_ns[[i]])
 
  }
}


####
[[1]]
[1] 0.6743001

[[2]]
[1] 0.3482201

[[3]]
[1] 0.2545677

[[4]]
[1] 0.2545677

[[5]]
[1] 0.2545677

[[6]]
[1] 0.21268

[[7]]
[1] 0.171218

[[8]]
[1] 0.171218

[[9]]
[1] 0.171218

[[10]]
[1] 0.1301031

[[11]]
[1] 0.08926116

[[12]]
[1] -0.0229044

[[13]]
[1] -0.06941334

[[14]]
[1] -0.1639976

[[15]]
[1] -0.4882569

[[16]]
[1] -0.5844623

[[17]]
[1] -0.6919131

[[18]]
[1] -0.7492664

[[19]]
[1] -0.7492664

[[20]]
[1] -0.7492664

[[21]]
[1] -0.7492664


We can see the 12th  value =-0.0229044 is the closest value to zero, the corresponded delta is 5.1. (form the 4 to 6 by 0.1 step)

Therefore, 5.1 is the solution of the Estimating  Equations.

Difference between logit and probit models

https://stats.stackexchange.com/a/30909/61705

Family	Default Link Function
binomial	(link = "logit")
gaussian	(link = "identity")
Gamma	(link = "inverse")
inverse.gaussian	(link = "1/mu^2")
poisson	(link = "log")
quasi	(link = "identity", variance = "constant")
quasibinomial	(link = "logit")
quasipoisson	(link = "log")

Normal score and rankit

The second meaning of normal score is associated with data values derived from the ranks of the observations within the dataset. A given data point is assigned a value which is either exactly, or an approximation, to the expectation of the order statistic of the same rank in a sample of standard normal random variables of the same size as the observed data set.^[1] Thus the meaning of a normal score of this type is essentially the same as a rankit, although the term "rankit" is becoming obsolete. In this case the transformation creates a set of values which is matched in a certain way to what would be expected had the original set of data values arisen from a normal distribution.