Jackknife Knife Cutting method

First, Introduction

Jackknife (Cutting method) is a re-sampling method proposed by Maurice Quenouille (1949), whose original motive is to reduce the deviation of estimation. Jackknife is a Swiss mini-knife that is easy to carry. By analogy, John W. Tukey (1958) created the term in statistics as a general method of hypothesis testing and confidence interval calculation.

Jackknife is similar to the cross-validation method of "Leave one out." To x= (x1,x2,..., Xn) as the observed sample, define the sample I jackknife as the remaining sample after discarding the sample I

The difference between the resulting Jackknife sample sets is very small, with only two of the original samples in each of the two jackknife samples being different.

Second, the deviation of the Jackknife estimate code is as follows:

function Out=jackbias (theta,orig)%estimate the bias using the Jackknife%theta have to be a character string containg% a Val ID function name[n,p]=size (orig); Lot=feval (Theta,orig (2:n,:)); K=length (lot); Lo=zeros (n,k); Lo (1,:) =lot;lo (n,:) = Feval (Theta,orig (1: (n-1),:)); for i= (2: (n-1))   lo (i,:) =feval (Theta,orig ([1: (I-1), (i+1): n],:)); endthetadot= Mean (LO); out= (n-1) * (Thetadot-feval (Theta,orig));

Jackknife Estimation of standard deviation


The code is as follows:

function out=jackstd (Theta,orig) [N,p]=size (orig); Lot=feval (Theta,orig (2:n,:)); K=length (lot); Lo=zeros (n,k); Lo (1, :) =lot;lo (n,:) =feval (Theta,orig (1: (n-1),:)); for i= (2: (n-1))   lo (i,:) =feval (Theta,orig ([1: (I-1), (i+1): n],:)); Endthetadot=mean (LO); Out=sqrt ((N-1)/n.*sum ((Lo-repmat (thetadot,n,[))). ^2);

Four, jackknife not suitable for the occasion

A statistical function is not a smoothing function: A small change in data results in a large variation of the statistic, such as the Extremum, median. If the results of the data X= (10,27,31,40,46,50,52,104,146) are worth the 48,48,48,48,45,43,43,43,43, The median value of an even number of numbers is the average of the most intermediate two numbers.

V. Relationship between Jackknife and bootstrap self-help law

Efron1979 The article points out the relationship between self-help method and knife-cutting method. Firstly, the self-help method constructs the self-help method through the empirical distribution function, and transforms the problem of ill-defined probability distribution into resampling from a given sample set. Second, the self-help method can solve the problem of non-smooth parameters. when the Smooth parameter estimation is encountered, the cutting method will fail, and the self-help method can be used to estimate the median number effectively. Thirdly, using the Taylor formula to estimate the self-help method, we can get the first order approximation of the method of self-help method. For the problem of estimating variance of linear statistics, the method of cutting or self-help will get the same result. But on the question of variance estimation of nonlinear statistics, the cutter-cutting method relies heavily on the linear fitting degree of statistic, so it is far less effective than self-help method.

To be continued ...

Jackknife Knife Cutting method