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.
Every time I think I know what's going on, suddenly there's another layer of complications.
2017年6月10日星期六
2017年6月8日星期四
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.
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.
2017年6月6日星期二
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") |
2017年6月4日星期日
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.
2017年5月31日星期三
2017年5月30日星期二
2017年5月29日星期一
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