Skewness-kurtosis adjusted confidence estimators and significance tests

First and second kind modifications of usual confidence intervals for estimating the expectation and of usual local alternative parameter choices are introduced in a way such that the asymptotic behavior of the true non-covering probabilities and the covering probabilities under the modified local non-true parameter assumption can be asymptotically exactly controlled. The orders of convergence to zero of both types of probabilities are assumed to be suitably bounded below according to an Osipov-type condition and the sample distribution is assumed to satisfy a corresponding tail condition due to Linnik. Analogous considerations are presented for the power function when testing a hypothesis concerning the expectation both under the assumption of a true hypothesis as well as under a modified local alternative. Applications are given for exponential families.


Introduction
Asymptotic normality of the distribution of the suitably centered and normalized arithmetic mean of i.i.d. random variables is one of the best studied and most often exploited facts in asymptotic statistics. It is supplemented in local asymptotic normality theory by limit theorems for the corresponding distributions under the assumption that the mean is shifted of order n −1/2 . There are many successful simulations and real applications of both types of central limit theorems, and one may Wolf-Dieter Richter Institute of Mathematics, University of Rostock, Ulmenstraße 69, Haus 3, 18057 Rostock, Germany e-mail: name@email.address ask for a more detailed explanation of those success. The present note is aimed to present such additional theoretical explanation under certain circumstances. Moreover, the note is aimed to further stimulate analogous consideration in more general situations and to stimulate checking the new results by simulation. Moreover, based upon the results presented here, it might become attractive to search for additional explanation to various known simulation results in the area of asymptotic normality which is, however, behind the scope of the present note.
Based upon a large deviation result in [1], skewness-kurtosis modifications of usual confidence intervals for estimating the expectation and of usual local alternative parameter choices are introduced here in a way such that the asymptotic behavior of the true non-covering probabilities and the covering probabilities under the modified local non-true parameter assumption can be exactly controlled. The orders of convergence to zero of both types of probabilities are suitably bounded below by assuming an Osipov-type condition, see [2], and the sample distribution is assumed to satisfy a corresponding Linnik condition.
Analogous considerations are presented for the power function when testing a hypothesis concerning the expectation both under the assumption of a true hypothesis as well as under a local alternative. Finally, applications are given for exponential families.
A concrete situation where the results of this paper apply is the case sensitive preparing of the machine settings of a machine tool. In this case, second and higher order moments of the manipulated variable do not change from one adjustment to another one and may be considered to be known over time.
It might be another aspect of stimulating further research if one asks for the derivation of limit theorems in the future being close to those in [1] but where higher order moments are estimated.
Let X 1 , ..., X n be i.i.d. random variables with the common distribution law from a shift family of distributions, P µ = P(· − µ), where the expectation equals µ, µ ∈ R, and the variance is σ 2 . It is well known that T n = √ n(X n − µ)/σ is asymptotically standard normally distributed, T n ∼ AN(0, 1). Hence, P µ (T n > z 1−α ) → α, and under the local non-true parameter assumption, µ 1,n = µ + σ √ n (z 1−α − z β ), i.e. if one assumes that a sample is drawn with a shift of location (or with an error in the variable), then P µ 1,n (T n ≤ z 1−α ) = P µ 1,n ( √ nX n −µ 1,n σ ≤ z β ) → β as n → ∞, where z q denotes the quantile of order q of the standard Gaussian distribution.
Let ACI u = [X n − σ √ n z 1−α , ∞) denote the upper asymptotic confidence interval for µ where the true non-covering probabilities satisfy the asymptotic relation , the covering probabilities under n −1/2 -locally chosen non-true parameters satisfy The aim of this note is to prove refinements of the latter two asymptotic relations where α = α(n) → 0 and β = β (n) → 0 as n → ∞, and to prove similar results for two-sided confidence intervals and for the power function when testing corresponding hypotheses.

First and second kind adjusted one-sided confidence intervals
According to [1], it is said that a random variable X satisfies the Linnik condition of order γ, γ > 0, if Let us define the first kind (or first order) adjusted asymptotic Gaussian quantile by Moreover, let the first kind (order) adjusted upper asymptotic confidence interval for µ be defined by and denote a first kind modified non-true local parameter choice by Let us say that the probabilities α(n) and β (n) satisfy an Osipov-type condition of order γ if 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 If two functions f , g satisfy the relation lim n→∞ f (n)/g(n) = 1 then this asymptotic equivalence will be expressed as f (n) ∼ g(n), n → ∞. (1) and (2) are satisfied Let us define the second kind adjusted asymptotic Gaussian quantile 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 and a second kind modified non-true local parameter choice Theorem 2. If α(n) ↓ 0, β (n) ↓ 0 as n → ∞ and conditions (1) and (2) are satisfied for γ ∈ ( 1 4 , 3 10 ] then P µ (ACI u (2) does not cover µ) ∼ α(n), n → ∞ and P µ 1,n (2) (ACI u (2) covers µ) ∼ β (n), n → ∞.

Adjusted quantiles
Let us consider the problem of testing the hypothesis H 0 : µ ≤ µ 0 versus the alternative H A : µ > µ 0 . The first and second kind adjusted decision rules of the one-sided asymptotic Gauss test suggest to reject H 0 if T n,0 > z 1−α(n) (s) for s = 1 or s = 2, respectively, where T n,0 = √ n(X n − µ 0 )/σ . Because P µ 0 (re ject H 0 ) = P µ 0 (ACI u (s) does not cover µ 0 ), it follows from Theorems 1 and 2 that under the conditions given there the (sequence of) probabilities of an error of first kind satisfy the asymptotic relation Concerning the power function of this test, because P µ 1,n (s) (do not re ject H 0 ) = P µ 1,n (s) (ACI u (s) covers µ 0 ), it follows under the same assumptions that the probabilities of a second kind error in the case that the sequence of the modified local parameters is (µ 1,n (s)) n=1,2,... , satisfy P µ 1,n (s) (do not re ject H 0 ) ∼ β (n), n → ∞.

Theorem 4. If the conditions
Clearly, the results of this theorem apply to both hypothesis testing and confidence estimation in a similar way as described in the preceding sections.

Application to exponential families
Let ν denote a σ -finite measure and assume that the distribution P ϑ has the Radon-Nikodym density dP ϑ dν (x) = e ϑ x e ϑ x ν(dx) = e ϑ x−B(ϑ ) , say. We assume that X(ϑ ) ∼ P ϑ and X 1 = X(ϑ ) − EX(ϑ ) + µ ∼ P µ where ϑ is known and µ is unknown. In the product-shift-experiment [R n , B n , { P ×n µ , µ ∈ R}], expectation estimation and testing may be done as in Sections 2 and 3, respectively, where g 1 = B ′′′ (ϑ )/(B ′′ (ϑ )) 3/2 and g 2 allows a similar representation. Another problem which can be dealt with is to test the hypothesis H 0 : ϑ ≤ ϑ 0 versus the alternative H 1n : ϑ ≥ ϑ 1n if one assumes that the expectation function ϑ → B ′ (ϑ ) = E ϑ X is strongly monotonous. For this case, we finally present just the following particular result which applies to both estimating and testing. (1) and (2) are satisfied for γ ∈ ( 1 6 , 1 4 ] then
Proof of Theorem 3. We start from the well known relations