2.1 First and second kind adjusted one-sided confidence intervals
According to (Ibragimov and Linnik 1971; Linnik 1961), it is said that a random variable X satisfies the Linnik condition of order γ,0<γ<1/2, if
$$ {E}_{\mu} \exp\left\{|X-\mu|^{\frac{4\gamma}{2\gamma+1}}\right\} <\infty. $$
((1))
Let us define the first kind (or first order) adjusted asymptotic Gaussian quantile by
$$z_{1-\alpha(n)}(1)=z_{1-\alpha(n)} +\frac{g_{1}}{6\sqrt{n}} z^{2}_{1-\alpha(n)} $$
where g
1=E(X−E(X))3/σ
3/2 is the skewness of X. Moreover, let the first kind (order) adjusted upper asymptotic confidence interval for μ be defined by
$$ACI^{u}(1)=\left[\left.\bar{X}_{n} -\frac{\sigma}{\sqrt{n}} z_{1-\alpha(n)}(1), \infty\right)\right. $$
and denote a first kind modified non-true local parameter choice by
$$\mu_{1,n}(1)=\mu_{1,n}+\frac{\sigma g_{1}} {6n} \left(z^{2}_{1-\alpha(n)}-z^{2}_{\beta(n)}\right). $$
Let us say that the probabilities α(n) and β(n) satisfy an Osipov-type condition of order γ if
$$ n^{\gamma}\exp\left\{\frac{n^{2\gamma}}{2}\right\} \cdot \min\left\{\alpha(n),\beta(n)\right\}\rightarrow \infty,\; n\rightarrow \infty. $$
((2))
This condition means that neither α(n) nor β(n) tend to zero as fast as or even faster than n
−γ exp{−n
2γ/2}, i.e. min{α(n),β(n)}≫n
−γ exp{−n
2γ/2}, and that max{z
1−α(n),z
1−β(n)}=o(n
γ),n→∞. Here, o(.) stands for the small Landau symbol.
If two functions f,g satisfy the relation \(\lim \limits _{n\rightarrow \infty }f(n)/g(n)=1\) then this asymptotic equivalence will be expressed as f(n)∼g(n),n→∞.
Theorem
1.
If α(n)↓0, β(n)↓0 as n→∞ and conditions (1) and (2) are satisfied for \(\gamma \in \left (\frac {1}{6},\right.\left.\!\!\!\frac {1}{4}\right ]\) then
$$ P_{\mu} (ACI^{u}(1) {\; does\; not\; cover\;} \mu)\sim \alpha(n), \, n\rightarrow \infty $$
and
$$ P_{\mu_{1,n}(1)} (ACI^{u}(1) {\; covers\;} \mu)\sim \beta(n), \, n\rightarrow \infty. $$
Let us define the second kind adjusted asymptotic Gaussian quantile
$$z_{1-\alpha(n)}(2)=z_{1-\alpha(n)}(1) +\frac{3g_{2}-4{g_{1}^{2}}}{72n} z^{3}_{1-\alpha(n)} $$
where g
2=E(X−E(X))4/σ
4−3 is the kurtosis of X, the second kind adjusted upper asymptotic confidence interval for μ
$$ACI^{u}(2)=\left[\vphantom{\frac{0}{0}}\bar{X}_{n} \right.\left.-\frac{\sigma}{\sqrt{n}} z_{1-\alpha(n)}(2), \infty\right), $$
and a second kind modified non-true local parameter choice
$$\mu_{1,n}(2) =\mu_{1,n}(1)+ \frac{\sigma\left(3g_{2}-4{g_{1}^{2}}\right)} {72n^{3/2}}\left(z^{3}_{1-\alpha(n)}-z_{\beta(n)}^{3}\right). $$
Theorem
2.
If α(n)↓0, β(n)↓0 as n→∞ and conditions (1) and (2) are satisfied for \(\gamma \in \left (\frac {1}{4},\right.\left.\!\!\!\frac {3}{10}\right ]\) then
$$ P_{\mu} (ACI^{u}(2) {\; does\; not\; cover\;} \mu)\sim \alpha(n), \, n\rightarrow \infty $$
and
$$ P_{\mu_{1,n}(2)} (ACI^{u}(2) {\; covers\;} \mu)\sim \beta(n), \, n\rightarrow \infty. $$
Remark
1.
Under the same assumptions, analogous results are true for lower asymptotic confidence intervals, i.e. for \(ACI^{l}(s)=\left (-\infty, \bar {X}_{n}+\frac {\sigma }{\sqrt {n}}z^{-}_{1-\alpha }(s)\right), s=1,2:\)
$$P_{\mu}(ACI^{l}(s)\; does\; not\; cover\; \mu) \sim \alpha(n) $$
and
$$P_{\mu^{-}_{1,n}(s)}(ACI^{l}(s)\; covers\; \mu) \sim \beta(n),\, n\rightarrow \infty. $$
Here, \(z^{-}_{1-\alpha }(s)\) means the quantity z
1−α
(s) where g
1 is replaced by −g
1,s=1,2, and
$$\mu^{-}_{1,n}(s)=\mu-\frac{\sigma}{\sqrt{n}} (z_{1-\alpha}-z_{\beta})+\frac{\sigma g_{1}}{6n} \left(z^{2}_{1-\alpha}-z^{2}_{\beta}\right)- \frac{\sigma\left(3g_{2}-4{g_{1}^{2}}\right)} {72n^{3/2}}\left(z^{3}_{1-\alpha}- z^{3}_{\beta}\right)I_{\{2\}}(s). $$
Remark
2.
In many situations where limit theorems are considered as they were in Section 1, the additional assumptions (1) and (2) may, possibly unnoticed, be fulfilled. In such situations, Theorems 1 and 2, together with the following theorem, give more insight into the asymptotic relations stated in Section 1.
Theorem
3.
Large Gaussian quantiles satisfy the asymptotic representation
$$z_{1-\alpha}=\sqrt{-2\ln\alpha-\ln|\ln\alpha|- \ln(4\pi)}\cdot \left(1+O\left(\frac{\ln|\ln\alpha|}{(\ln\alpha)^{2}}\right)\right), \alpha\rightarrow +0. $$
Note that O(.) means the big Landau symbol.
2.2 Two-sided confidence intervals
For s∈{1,2},α>0, put \( L(s;\alpha)=\bar {X}_{n}-\frac {\sigma }{\sqrt {n}}z_{1-\alpha }(s)\) and \( R(s;\alpha)=\bar {X}_{n}+\frac {\sigma }{\sqrt {n}}z^{-}_{1-\alpha }(s).\) Further, let α
i
(n)>0, i=1,2,α
1(n)+α
2(n)<1, and
$$ACI(s;\alpha_{1}(n),\alpha_{2}(n))=\left[L(s;\alpha_{1}(n)), R(s;\alpha_{2}(n))\right]. $$
If conditions (1) and (2) are fulfilled then P
μ
((−∞,L(s;α
1(n))) covers
μ)∼α
1(n) and P
μ
((R(s;α
2(n)),∞) covers
μ)∼α
2(n) as n→∞.
With more detailed notation μ
1,n
(s)=μ
1,n
(s;α,β) and \(\mu ^{-}_{1,n}(s)=\mu ^{-}_{1,n}(s;\alpha,\beta)\),
\(P_{\mu _{1,n}(s;\alpha _{1}(n),\beta _{1}(n))} ((L(s;\alpha _{1}(n)),\infty)\; covers\; \mu)\sim \beta _{1}(n)\),
\(P_{\mu ^{-}_{1,n}(s;\alpha _{2}(n),\beta _{2}(n))} ((-\infty, R(s;\alpha _{2}(n)))\; covers\; \mu)\sim \beta _{2}(n), n\rightarrow \infty.\)
The following corollary has thus been proved.
Corollary
1.
If α
1(n)↓0, α
2(n)↓0 as n→∞ and conditions (1) and (2) are satisfied for \(\gamma \in \left (\frac {1}{6},\!\!\right.\left.\frac {1}{4}\right ]\) if s=1 and for \(\gamma \in \left (\frac {1}{4},\!\!\right.\left.\frac {3}{10}\right ]\) if s=2, and with (α(n),β(n))=(α
1(n),α
2(n)), then
$$ P_{\mu} (ACI(s;\alpha_{1}(n),\alpha_{2}(n)) {\; does\; not\; cover\;} \mu)\sim (\alpha_{1}(n)+\alpha_{2}(n)), \, n\rightarrow \infty. $$
Moreover,
$$ \max\limits_{\nu\in\{\mu_{1,n} (s;\alpha_{1}(n),\beta_{1}(n)), \mu^{-}_{1,n}(s;\alpha_{2}(n),\beta_{2}(n)) \}} P_{\nu} (ACI(s) {\; covers\;} \mu)\leq\max\left\{\beta_{1}(n),\beta_{2}(n)\right\}. $$