### 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,

$$ \max\limits_{\nu\in\{\mu_{1,n} (s;\alpha_{1}(n),\beta_{1}(n)), \mu^{-}_{1,n}(s;\alpha_{2}(n),\beta_{2}(n)) \}} P_{\nu} (ACI(s) {\; covers\;} \mu)\leq\max\left\{\beta_{1}(n),\beta_{2}(n)\right\}. $$