Confidence interval calculationI need to compute models, which return the probability to reach the goal state per every sample. For some models I have the probability sequence equal to 0,0,0,0,0,0,0,0,0… and 1,1,1,1,1,1,1,1,1,1,1…I need to compute confidence intervals for these extreme conditions as well as for different cases, where I have some zeros and ones in the sequence.I can't use the general formula C I = ( μ − z ∗ σ / n , μ + z ∗ σ / n ) .here, as well as t-distribution formula, because it will return the interval equal to [ μ − 0 , μ + 0 ] immediately after the first sample.

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Confidence interval calculationI need to compute models, which return the probability to reach the goal state per every sample. For some models I have the probability sequence equal to 0,0,0,0,0,0,0,0,0… and 1,1,1,1,1,1,1,1,1,1,1…I need to compute confidence intervals for these extreme conditions as well as for different cases, where I have some zeros and ones in the sequence.I can&#039;t use the general formula  C  I  =  (  μ  −  z  ∗  σ      /        n    ,  μ  +  z  ∗  σ      /        n    )  .here, as well as t-distribution formula, because it will return the interval equal to   [  μ  −  0  ,  μ  +  0  ] immediately after the first sample.

darkflamexivcr · Accepted Answer

Step 1You are using the so-called Wald interval, based on normal approximation and other asymptotics. It is OK for huge n (e.g., nationwide public opinion polls). But it is known to have problems (including the one you mention) for small n.Agresti-Couil Interval: Let             n      ~        =  n  +  4 and estimate             p      ~        =            x      +      2                      n        ~              .. Then a 95% CI for p is of the form            p      ~        ±  1.96                                          p            ~                          (        1        −                              p            ~                          )                              n          ~                      .This amounts to appending two successes and two failures to the actual data before computing the CI.For   x  =  0 successes and   n  =  10 trials, this gives (−.04,.33), which you can interpret as (0,.33). [Similarly for upper endpoints above 1.]The Agresti interval closely approximates the more accurate (but messier) Wilson interval. The Wilson interval &#039;inverts the test&#039; for       H    0    :  p  =      p    0   vs       H    a    :  p  ≠      p    0    , rejecting when       |    Z      |    ≥  1.96  ,, where   Z  =            x              /            n      −              p        0                            p        1            (      1      −              p        0            )              /            n        .Step 2Bayesian Interval from Uniform Prior. Using       U    n    i    f    (  0  ,  1  )  ≡      B    e    t    a    (  1  ,  1  ) as prior and the binomial likelihood       p    x    (  1  −  p      )          n      −      x        , the Bayesian posterior is       B    e    t    a    (  x  +  1  ,  n  −  x  +  1  ) and the 95% posterior probability interval uses quantiles .025 and .975 of the posterior distribution. For   x  =  0 and   n  =  10  ,, this interval is (.002,.282). The Bayesian interpretation differs in philosophy, but the numerical results are sometimes used as a confidence interval by frequentists. This interval can never produce endpoints outside of (0,1).&amp;gt; qbeta(c(.025,.974), 1, 11)[1] 0.002298972 0.282359950Of course, the Bayesian approach has the advantage of being able to take substantive prior information, if available, into account by using an &#039;informative&#039; prior.

Confidence interval calculation. I need to compute models, which return the probability to reach the goal state per every sample. For some models I have the probability sequence equal to 0,0,0,0,0,0,0,0,0… and 1,1,1,1,1,1,1,1,1,1,1…

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