Different formats of Fisher Information

Different forms of integration by parts

Derivatives of Inverse Functions

Difference between 'link function' and 'canonical link function' for GLM

https://stats.stackexchange.com/questions/40876/difference-between-link-function-and-canonical-link-function-for-glm

Community structure in social and biological networks

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC122977/pdf/pq1202007821.pdf

Introduction to Tensors

https://www.grc.nasa.gov/www/k-12/Numbers/Math/documents/Tensors_TM2002211716.pdf

interchange of the expected value and summation

The expectation of a sum of nonnegative random variables indexed by the integers is equal to the sum of the expectations, whether the resulting number is finite or infinite. This follows from Fubini's theorem for nonnegative functions (often called Tonelli's theorem) applied to the product measure which is the product of P by the counting measure on the integers.

Square of summation

Create an empty list and flatten the list

x<-c(2.3,0.3,5.2,3.1,1.1,0.9,2.0,0.7,1.4,0.3)
y<-c(0.8,2.8,4.0,2.4,1.2,0.0,6.2,1.5,28.8,0.7)

L<-vector("list",10)

for(i in 1:10){

L[[i]]<-y[i]-x #key is [[]] operator here
}

d<-unlist(L)
median(d)  # This is Hodges-Lehmann estimator

Hodges-Lehmann Estimation of Location Shift

data e1041;
input group $ x;
datalines;
1 2.3
1 0.3
1 5.2
1 3.1
1 1.1
1 0.9
1 2.0
1 0.7
1 1.4
1 0.3
2 0.8
2 2.8
2 4.0
2 2.4
2 1.2
2 0.0
2 6.2
2 1.5
2 28.8
2 0.7
;
run;
 proc npar1way hl alpha=.05 data=e1041;
      class group;
      var x;
      exact hl;
      ods select WilcoxonScores HodgesLehmann;
  run;

Expectation of an indicator function

linear vs nolinear

A regression model is called nonlinear, if the derivatives of the model with respect to the model parameters depends on one or more parameters. This definition is essential to distinguish nonlinear from curvilinear regression. A regression model is not necessarily nonlinear if the graphed regression trend is curved. A polynomial model such as y = b₀ + b₁x + b₂x² + e appears curved when y is plotted against x. It is, however, not a nonlinear model. To see this, take derivatives of y with respect to the parameters b₀, b₁, and b₂: dy/db₀ = 1, dy/db₁ = x, dy/db₂ = x² None of these derivatives depends on a model parameter, the model is linear. In contrast, consider the log-logistic model y = d + (a - d)/(1 + exp{b log(x/g)}) + e Take derivatives with respect to d, for example: dy/dd = 1 - 1/(1 + exp{b log(x/g)}). The derivative involves other parameters, hence the model is nonlinear

.http://www.ats.ucla.edu/stat/sas/library/SASNLin_os.htm

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