Estimators and Confidence intervalsI was curious as to what the relationship between probabilistic values, estimators and confidence intervals are. I was wondering, if I have an estimator of some parameter λ, and a probability value that depended on λ and the estimator, what would be the relationship between those two and a confidence interval for an arbitrary distribution?

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Estimators and Confidence intervalsI was curious as to what the relationship between probabilistic values, estimators and confidence intervals are. I was wondering, if I have an estimator of some parameter   λ, and a probability value that depended on   λ and the estimator, what would be the relationship between those two and a confidence interval for an arbitrary distribution?

Klecanlh · Accepted Answer

Step 1Confidence intervals are formed by inverting probability statements pertaining to pivotal quantities. In the present context, a pivotal quantity is a function   f  (      X    ,  λ  ) with a probability distribution that is fixed (i.e., does not depend on   λ). Pivotal quantities are often formed by taking an estimator of   λ and then standardising that estimator or doing some other thing to it that yields a quantity with a fixed distribution. This means that there is often (but not always) a relationship between a pivotal quantity used to form the confidence interval for   λ, and an estimator of   λ.For an arbitrary distribution it is not possible to determine if this is the case, but in many particular problems it holds. Here is an example of a case where there is a connection between these things.Step 2An example: A simple example of the connection between an estimator and a confidence interval is for IID normal data. If       X    1    ,  .  .  .  ,      X    n    ∼  N  (  μ  ,  σ  ) and we want to estimate   μ then we can form the pivotal quantity:                                          X            ¯                          n            −      μ                      S        n        2                    /                    n              ∼  Student-T  (  d  f  =  n  −  1  )  .This pivotal quantity makes use of the point estimator                     X        ¯              n   and &quot;studentizes&quot; this estimator by subtracting its mean and dividing by its estimated variance. We can use this pivotal quantity to obtain the probability statement:  1  −  α  =      P              (        −      t          n      −      1      ,      α              /            2        ⩽                                          X            ¯                          n            −      μ                      S        n        2                    /                    n              ⩽      t          n      −      1      ,      α              /            2                  )        .Inverting this probability statement gives:  1  −  α  =      P              (                          X        ¯              n    −            t              n        −        1        ,        α                  /                2                    n        ⋅      S    n    2    ⩽  μ  ⩽                    X        ¯              n    +            t              n        −        1        ,        α                  /                2                    n        ⋅      S    n    2              )        ,which gives us the standard confidence interval formula:  CI  (  1  −  α  )  =            [                          x        ¯              n    ±            t              n        −        1        ,        α                  /                2                    n        ⋅      s    n    2              ]        .

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