Generalized inverse Gaussian distribution

In <a href="/facts/Probability_theory/mFBn51rE">probability theory</a> and <a href="/facts/Statistics/DNPsKGYU">statistics</a>, the generalized inverse Gaussian distribution (GIG) is a three-parameter family of continuous <a href="/facts/Probability_distribution/EpsKKVRu">probability distributions</a> with <a href="/facts/Probability_density_function/zvfybna4">probability density function</a>

f
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        =
        
          
            
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              a
              
                /
              
              b
              
                )
                
                  p
                  
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                  2
                
              
            
            
              2
              
                K
                
                  p
                
              
              (
              
                
                  a
                  b
                
              
              )
            
          
        
        
          x
          
            (
            p
            −
            1
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          e
          
            −
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            a
            x
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            b
            
              /
            
            x
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            2
          
        
        ,
        
        x
        >
        0
        ,
      
    
    {\displaystyle f(x)={\frac {(a/b)^{p/2}}{2K_{p}({\sqrt {ab}})}}x^{(p-1)}e^{-(ax+b/x)/2},\qquad x>0,}

where Kp is a <a href="/facts/Modified_Bessel_function/hSyvFPqC">modified Bessel function</a> of the second kind, a > 0, b > 0 and p a real parameter. It is used extensively in <a href="/facts/Geostatistics/UeChfgNG">geostatistics</a>, statistical linguistics, finance, etc. This distribution was first proposed by <a href="/facts/%25C3%2589tienne_Halphen/FAhRMkcM">Étienne Halphen</a>. 
It was rediscovered and popularised by <a href="/facts/Ole_Barndorff-Nielsen/1Xr71bYs">Ole Barndorff-Nielsen</a>, who called it the generalized inverse Gaussian distribution. Its statistical properties are discussed in Bent Jørgensen's lecture notes.

Generalized inverse Gaussian distribution open-in-new

Generalized inverse Gaussian distribution