I am trying to write a C++ program for parameter estimation(with Confidence Interval information) of an Exponentially distributed data set. I understand that λ X ¯ ∼ Γ ( n , n ). To come up with numeric values of lower/higher confidence intervals for a specified α, I need to be able to compute Inverse Gamma. Could you please point me to some algorithms?

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I am trying to write a C++ program for parameter estimation(with Confidence Interval information) of an Exponentially distributed data set. I understand that   λ            X      ¯        ∼  Γ  (  n  ,  n  ). To come up with numeric values of lower/higher confidence intervals for a specified   α, I need to be able to compute Inverse Gamma. Could you please point me to some algorithms?

Isaiah Haynes · Accepted Answer

Step 1Either I misunderstand the problem, or you are making it more difficult than necessary.Let L and U cut probability   α      /    2 from the lower and upper tails of   G  a  m  m  a  (  s  h  a  p  e  =  n  ,  r  a  t  e  =  n  )  ,, Then a   1  −  α confidence interval for   λ is found as follows:  P  (  L  &amp;lt;  λ            X      ¯        &amp;lt;  U  )  =  P  (  L      /              X      ¯        &amp;lt;  λ  &amp;lt;  U      /              X      ¯        )  =  1  −  α  ,,so the desired CI is   (  L      /              X      ¯        ,    U      /              X      ¯        ).It is not difficult to program numerical integration to find U and L because the PDF of Gamma(n,n) can be evaluated. Alternatively, I&#039;m sure there must be a compiled FORTRAN program in the IMSL library of statistical algorithms that can be called from C++.In particular, if       X    1    ,      X    2    ,  …  ,      X          25       is a random sample from   E  x  p  (  r  a  t  e  =  .2  )  ,, we might have the following sample (rounded to 3 places):                    1.515                    0.187                    6.270                    3.291                    3.597                    10.680                    1.572                    22.555                    4.924                    0.063                            0.214                    0.465                    19.042                    2.467                    4.663                    2.101                    7.597                    10.053                    0.017                    5.627                            12.124                    7.579                    0.735                    11.290                    4.153            The sample mean is             X      ¯        =  5.711.. The cut-off points for a 95% CI from R areqgamma(c(.025,.975), 25, 25)## 0.6471473 1.4284039so that the 95% CI isqgamma(c(.025,.975), 25, 25)/mean(x)## 0.1133112 0.2501040Step 2The (slightly biased) maximum likelihood estimator of of   λ in this example is             λ      ^        =  1      /              X      ¯        =  0.175  ,, which is contained in the CI, but not exactly at the center of it, owing to the skewness of the distributions involved.

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