We define equivalence of asymptotic Gaussian expectation tests when error probabilities of first kind are approaching zero at the same restricted speed for both tests and if the same holds true for the error probabilities of second type which are measured at a moderate locally chosen alternative. To ensure such equivalence, the inﬂuence of skewness and kurtosis parameters is studied.
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Manuscript submitted on 8-11-2015 |
Original Manuscript | Skewness-Kurtosis Controlled Higher Order Equivalent Decisions |
Asymptotic relative efficiency of one test with respect to (w.r.t.) another is extensively studied in the literature. For an introduction and overview we refer to Nikitin [1Y. Nikitin, Asymptotic Efficiency of Nonparametric Tests, Cambridge University Press: Cambridge, England, 1995.
[http://dx.doi.org/10.1017/CBO9780511530081] ]. Several notions of efficiency may be distinguished w.r.t. how the probabilities of first and second kind test errors behave in the case of increasing sample sizes. Roughly spoken, studies of Pitman type are dealing with situations where both kinds of probabilities of test errors stabilize asymptotically at some fixed positive levels while studies of different other types take into consideration that both error probabilities are tending to zero or that one of them stabilizes asymptotically at a positive value and the other one tends to zero. Moreover, one may take into account different speeds of convergence of the two error probabilities.
The present study deals with specific situations where probabilities of both error types are tending to zero not faster than at a certain moderate speed. Note that there are different notions of large and moderate deviations in the literature of probability theory and mathematical statistics. Zons of large deviations concerned here are often called to be of the so called Linnik type, and the speed at which error probabilities are tending to zero is controlled here by a so called Osipov condition.
Second type error probabilities of a test depend on the alternatives taken under consideration. While local asymptotic normality theory is in case of sample size n concerned with differences of parameters under the hypothesis and under the alternative being of the type here we are dealing with such differences of the type where and as . Alternatives of the latter type will be called moderate local alternatives.
Numerous statistical and probabilistic results have been derived for such situations, see Inglot and Ledwina [2T. Inglot, and T. Ledwina, "On probabilities of excessive deviations for Kolmogorov-Smirnov, Cram’er-von Mises and chi-square statistics", Ann. Stat., vol. 18, pp. 1491-1495, 1990.
[http://dx.doi.org/10.1214/aos/1176347764] , 3T. Inglot, and T. Ledwina, "Moderately large deviations and expansions of large deviations for some functionals of weighted empirical process", Ann. Probab, vol. 21, no. 3, pp. 1691-1705, 1993.], Kallenberg [4W.C.M. Kallenberg, "Intermediate efficiency, theory and examples", Ann. Stat., vol. 11, no. 1, pp. 170-182, 1983., 5W.C.M. Kallenberg, "On moderate deviation theory in estimation", Ann. Stat., vol. 11, no. 2, pp. 498-504, 1983.
[http://dx.doi.org/10.1214/aos/1176346156] ], Kallenberg and Ledwina [6W.C. Kallenberg, and T. Ledwina, "On local and non-local measures of efficiency", Ann. Stat., vol. 15, pp. 1401-1420, 1987.
[http://dx.doi.org/10.1214/aos/1176350601] ], Ledwina, Inglot and Kallenberg [7T. Ledwina, T. Inglot, and W.C. Kallenberg, "Strong moderate deviation theorems", Ann. Probab., vol. 20, pp. 987-1003, 1992.
[http://dx.doi.org/10.1214/aop/1176989814] ], Richter [8W-D. Richter, "Remark on moderate deviations in the multidimensional central limit theorem", Math. Nachr., vol. 122, pp. 167-173, 1985.
[http://dx.doi.org/10.1002/mana.19851220117] -10W-D. Richter, "Skewness-kurtosis adjusted confidence estimators and significance tests", J. Stat. Distrib. Appl., 2016. (arXiv:1504.02553)
[http://dx.doi.org/10.1186/s40488-016-0042-3] ] and Wood [11T.A. Wood, "Bootstrap relative errors and subexponential distributions", Bernoulli, vol. 6, no. 5, pp. 809-834, 2000.
[http://dx.doi.org/10.2307/3318757] ].
The notion of probabilities of moderate deviations partly used in those papers should be distinguished from that used for deviations in logarithmic zones, e.g., in Amosova [12N.N. Amosova, "On the probabilities of moderate deviations for sums of independent random variables. Teor", Veroyatnost. i Primenen, vol. 24, no. 4, pp. 858-865, 1979.] and Richter [8W-D. Richter, "Remark on moderate deviations in the multidimensional central limit theorem", Math. Nachr., vol. 122, pp. 167-173, 1985.
[http://dx.doi.org/10.1002/mana.19851220117] , 13W-D. Richter, "Moderate deviations in special sets of R^{k}", Math. Nachr., vol. 113, pp. 339-354, 1983.
[http://dx.doi.org/10.1002/mana.19831130130] , 14W-D. Richter, "Multidimensional narrow integral domains of normal attraction", In: Limit Theorems in Probability Theory and Related Fields, TU Dresden: Germany, 1987, pp. 163-184.] for the one and multi-dimensional case, respectively.
Recent extensions of related considerations to martingale sample schemes are to be found in Fan, Grama and Liu [15X. Fan, I. Grama, and Q. Liu, "A generalization of Cramér large deviations for martingales", Comptes Rendus Mathematique, vol. 352, no. 10, pp. 853-858, 2015. arXiv: 1503.06627v1].
The paper is organized as follows. Section 2.1 provides an introduction to skewness-kurtosis adjustments of the classical asymptotic Gauss test following Richter [10W-D. Richter, "Skewness-kurtosis adjusted confidence estimators and significance tests", J. Stat. Distrib. Appl., 2016. (arXiv:1504.02553)
[http://dx.doi.org/10.1186/s40488-016-0042-3] ], and a comparison to the well known Cornish-Fisher expansion. The equivalence of tests in the sense of Pitman is discussed in Section 2.2. The main result of this paper concerning skewness-kurtosis higher order equivalent decisions between a hypothesis and a moderate-local alternative is derived then in Section 3.
Let X_{1}, ..., X_{n} be independent and identically as X distributed 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}. Assume we are interested in deciding between a producer’s hypothesis and a customer’s apprehension,
(1) |
The test partners are aware of the general circumstance that the values of the power function of a test are only larger than a reasonable bound if the argument of the function is chosen sufficiently far from those arguments representing the null hypothesis. They agree therefore to use a test reflecting this situation from the very beginning. The size of the gap between µ_{ 0} and µ_{1,n} depends on what the customer may be willing to tolerate in a given practical situation, both w.r.t. the absolute value of µ_{1,n}-µ_{0} and w.r.t. the costs, expressed through the sample size n.
It is well known that the statistic where stands for the sample mean is asymptotically for standard normally distributed, T_{n} ~ AN (0, 1). Hence,
and under the n^{-1/2}-local non-true parameter assumption,
i.e. if one assumes that a sample is drawn with a shift of location or with an error in the variable, then
where z_{q} denotes the quantile of order q of the standard Gaussian distribution, and α, β are from the interval (0, 1/2). Thus, the first and second type error probabilities of the decision rule of the asymptotic Gauss test satisfy the asymptotic relations
Refinements of
and of these two asymptotic relations where α = α(n) → 0 and β = β(n) → 0 as n → ∞ were proved in Richter [10W-D. Richter, "Skewness-kurtosis adjusted confidence estimators and significance tests", J. Stat. Distrib. Appl., 2016. (arXiv:1504.02553)
[http://dx.doi.org/10.1186/s40488-016-0042-3] ] under suitable additional assumptions.
It is often said that a random variable X satisfies the Linnik condition of order γ, 0 < γ < 1/2, if
(2) |
This condition and far reaching consequences from it for probabilities of large deviations have been studied in Ibragimov and Linnik [16I.A. Ibragimov, and Y.W. Linnik, Independent and Stationary Sequence, Russian edition 1965 Walters, Nordhoff. Translation from, 1971.] (for a more general condition see Linnik [17Yu.V. Linnik, "Limit theorems for sums of independent variables taking into account large deviations", I-III. Theor. Probab Appl., vol. 6, pp. 131-148, 1961. 345-360; 7 (1962), 115-129.]) and a subsequent series of papers by many authors of whom we refer here to Nagajev [18S.V. Nagaev, "Some limit theorems for large deviations", Theory Probab. Appl., vol. 10, pp. 231-254, 1965.] and Richter [19W.-D. Richter, "Probabilities of large deviations of sums of independent identically distributed random vectors", Lith. Math J., vol. 19, no. 3, pp. 333-343, 1979.
[http://dx.doi.org/10.1007/BF00969969] ] where condition (2) was fundamentally generalized in two steps.
Two sequences of probabilities, (α(n)) _{n=1,2,...} and (β(n)) _{n=1,2,...}, are said to satisfy an Osipov-type condition of order γ if
(3) |
This condition was originally introduced in Osipov [20L.V. Osipov, "Multidimensional limit theorems for large deviations", Theory Probab. Appl, vol. 20, pp. 38-56, 1975.
[http://dx.doi.org/10.1137/1120004] ] for considering large deviations of multivariate random vectors. Several consequences following from (3) for probabilities of moderate and large deviations in finite dimensions have been studied, e.g., in Richter [8W-D. Richter, "Remark on moderate deviations in the multidimensional central limit theorem", Math. Nachr., vol. 122, pp. 167-173, 1985.
[http://dx.doi.org/10.1002/mana.19851220117] , 13W-D. Richter, "Moderate deviations in special sets of R^{k}", Math. Nachr., vol. 113, pp. 339-354, 1983.
[http://dx.doi.org/10.1002/mana.19831130130] , 21W-D. Richter, "Multidimensional domains of large deviations", In: Colloquia Mathematica Societatis János Bolyai, vol. 57. 1990. Limit Theorems in Probability and Statistics, Pecs: Hungary, 1989, pp. 443-458.].
Condition (3) means that neither α(n) nor β(n) tends to zero as fast as or even faster than n^{-γ} exp{-n^{2γ} /2}, i.e.
On using Gaussian quantiles this condition may be written
where o(.) stands for the small Landau symbol. Note that large Gaussian quantiles satisfy the asymptotic representation
see Ittrich, Krause and Richter [22C. Ittrich, D. Krause, and W-D. Richter, "Probabilities and large quantiles of noncentral generalized chi-square distributions", Statistics, vol. 34, pp. 53-101, 2000.
[http://dx.doi.org/10.1080/02331880008802705] ] and Richter [10W-D. Richter, "Skewness-kurtosis adjusted confidence estimators and significance tests", J. Stat. Distrib. Appl., 2016. (arXiv:1504.02553)
[http://dx.doi.org/10.1186/s40488-016-0042-3] ].
Skewness-kurtosis adjustments of the asymptotic Gauss test make use of the following quantities. The first and second kind (or order) adjusted asymptotic Gaussian quantiles are defined by
(4) |
and
(5) |
respectively. Here, and are skewness and kurtosis of X, respectively. Similarly, the first and second kind modified non-true moderate local parameter choices are
and
respectively. The first and second kind adjusted decision rules of the one-sided asymptotic Gauss test determine to reject H_{ 0} if T_{n,0} > z_{1-α(n)}(s) for s = 1 or s = 2 respectively, where
Let us recall that if two functions f, g satisfy the asymptotic
relation
then this asymptotic equivalence will be written f (n) ~ g (n), n → ∞.
It was proved in Richter [10W-D. Richter, "Skewness-kurtosis adjusted confidence estimators and significance tests", J. Stat. Distrib. Appl., 2016. (arXiv:1504.02553)
[http://dx.doi.org/10.1186/s40488-016-0042-3] ] that if the conditions (2) and (3) are satisfied for a certain γ,
(6) |
then the error probabilities of the adjusted decisions satisfy
(7) |
These results have been equivalently reformulated in Richter [10W-D. Richter, "Skewness-kurtosis adjusted confidence estimators and significance tests", J. Stat. Distrib. Appl., 2016. (arXiv:1504.02553)
[http://dx.doi.org/10.1186/s40488-016-0042-3] ] with the help of the first and second kind adjusted asymptotically Gaussian test statistics
respectively. The hypothesis H_{ 0} will be rejected according to decision rule if T_{n}(s) > z_{1-α}(n), and the first and second kind error probabilities of this decision still behave as in (7).
Similar consequences for testing H_{ 0} : µ > µ_{ 0} or H_{ 0} : µ ≠ µ_{ 0}, as well as for constructing confidence intervals, are omitted here.
The material of this section surveys the condensed content of the basic ’testing-part’ of what was presented by the author at the Conference of European Statistics Stakeholders, Rome 2014, (see Abstracts of Communication, p. 90 and Richter [10W-D. Richter, "Skewness-kurtosis adjusted confidence estimators and significance tests", J. Stat. Distrib. Appl., 2016. (arXiv:1504.02553)
[http://dx.doi.org/10.1186/s40488-016-0042-3] ]) where, however, the equivalent ’language’ of confidence estimation is used. The advanced ’testing-part’ of this talk is presented in Section 3 of the present paper.
Remark 2.1. From a formal point of view, the first and second kind adjusted asymptotic Gaussian quantiles defined in (4) and (5), respectively, have the same analytical structure as the coefficients of a Cornish-Fisher quantile-expansion (CFE), see Fisher and Cornish [23R.A. Fisher, and E.A. Cornish, "The percentile points of distributions having known cumulants", Technometrics, vol. 2, no. 2, pp. 209-225, 1960.
[http://dx.doi.org/10.1080/00401706.1960.10489895] ] and Bolschev and Smirnov [24L.N. Bolschev, and N.V. Smirnov, Tables of Mathematical Statistics (in Russian), Nauka: Moscow, 1983.]. However, CFE is valid for fixed or stabilizing values of α while the relations in (4) and (5) apply to the case
α ( n) → 0 as n → ∞. Taking this into account, the large deviation results in their form presented here through a new light onto the CFE. Note that the CFE itself is based upon an Edgeworth type expansion of a corresponding cumulative distribution function. Theoretical and numerical comparisons of normal and large deviation approximations for tail probabilities were presented in Field and Ronchetti [25C. Field, and E. Ronchetti, Small Sample Asymptotics, vol. 13. Institute of Mathematical Statistics Lecture Notes- Monograph Series: Hayward California, 1990.], Fu,Len and Peng [26J.C. Fu, C.M. Len, and C.Y. Peng, "A numerical comparison of normal and large deviation approximation for tail probabilities", J.Japan. Statist. Soc, vol. 20, no. 1, pp. 61-67, 1990.], Ittrich, Krause and Richter [22C. Ittrich, D. Krause, and W-D. Richter, "Probabilities and large quantiles of noncentral generalized chi-square distributions", Statistics, vol. 34, pp. 53-101, 2000.
[http://dx.doi.org/10.1080/02331880008802705] ] and Jensen [27J.L. Jensen, Saddlepoint Approximations, Clarendon Press: Oxford, 1995.].
In production process control, assume two methods are available for measuring the dimension µ of a workpiece which is serially made on a machine. In general, either methods work at different levels of costs and precision, the latter being expressed in terms of variances of measurements, σ_{1}^{2} and σ_{2}^{2}. One may be interested then in knowing for which sample size n_{2} = n_{2}(n_{1}) the second method works as good as the first one works for sample size n_{1}. For comparing the two methods of process control being based upon the two methods of measuring a workpiece one may compare both first and second kind error probabilities when dealing with problem (1) where we are given now the i.i.d. samples satisfying with for any probability distribution law from a family having shift parameter equal to expectation µ, and variance σ_{i}^{2}, i = 1, 2. Throughout what follows in the present paper all statistics built on using the random variables X^{(i)}_{j} , as well as quantiles of their distribution functions, will be indicated by an upper subscript, and their higher order moments and semi-invariants as well as corresponding sample sizes will be indicated by the (possibly second) lower index i.
In this sense, denotes the decision function based upon the asymptotically Gaussian statistic T_{ni}^{(i)} evaluated for sample of size n_{i}, i = 1, 2.
We recall that Pitman’s strategy of defining equivalence of two tests is one of several such strategies which are commonly formulated for rather general tests, see, e.g., in Nikitin [1Y. Nikitin, Asymptotic Efficiency of Nonparametric Tests, Cambridge University Press: Cambridge, England, 1995.
[http://dx.doi.org/10.1017/CBO9780511530081] ]. We restrict our consideration here, however, to pairs of tests based upon the statistics T_{ni}^{(i)} being the suitably centered and normalized means of the i.i.d.-samples
, i = 1, 2.
Two such test sequences based upon samples of sizes n_{1}, n_{2} where n_{2} = n_{2}(n_{1}) → ∞ as n_{1} → ∞ and decisions
are called asymptotically (α, β)-Pitman equivalent for deciding the problem (1) with
(8) |
if
(9) |
We shall refer to this as first order equivalence of decisions and . Note that here
(10) |
and α and β are assumed to belong to the interval (0, 1/2). It turns out that asymptotic (α, β)-Pitman equivalence holds if
(11) |
Let us assume throughout this section that as before n_{2} = n_{2}(n_{1}) → ∞ as n_{1} → ∞ but, differently from Section 2, first and second type error probabilities are tending now to zero, and
The latter assumption means that, for n_{1} → ∞, the sequences of alternative moderate local non-true parameters are tending not as fast to µ_{0 }as before.
Let the first and second order adjusted decision functions be defined as to decide between
(12) |
by rejecting H_{0 }if For a discussion of the meaning and the size of the gap between µ_{0 } and , we refer to what was said just following (1). We recall that the moderate local alternatives allow according to Section 2 the representation
where Note that the sizes of the first and second kind error probabilities α(.) and β(.), respectively, are chosen independently of i, i {1, 2}.
Definition 3.1. The decisions , i = 1, 2 are said to be asymptotically moderate locally equivalent of order s + 1, s {1, 2} for deciding the problem (12) with
(13) |
if, for n_{2}(n_{1}) → ∞ as n_{1} → ∞,
and
where α(n_{1}), β(n_{1}) satisfy the Osipov type condition (3) with γ chosen according to (6).
Theorem 3.1. Let X_{1}^{(1)} and X_{1}^{(2)} satisfy the Linnik condition (2) for one and the same γ = γ(s) fulfilling (6). Moreover, let α(n_{1}), β(n_{1}) satisfy the Osipov condition (3) for the same γ = γ(s), and assume that
(14) |
If in the case s = 1 skewness g_{1,1} and g_{1,2} of X^{(1)} and X^{(2)}, respectively, satisfy
(15) |
and if in the case s= 2 additionally the corresponding kurtosis values satisfy
(16) |
then and are equivalent of order s + 1, s {1, 2}, respectively.
Proof. The proof of the first property of Definition 3.1 follows directly from the results in Richter [10W-D. Richter, "Skewness-kurtosis adjusted confidence estimators and significance tests", J. Stat. Distrib. Appl., 2016. (arXiv:1504.02553)
[http://dx.doi.org/10.1186/s40488-016-0042-3] ], see Section 2.1. We recall that if condition (3) is satisfied then x = z_{1-α(n)} = o(n^{γ}), n → ∞ for γ
(^{1}/_{6} , ^{3}/_{10}], and if (2) is satisfied then, according to Nagajev [18S.V. Nagaev, "Some limit theorems for large deviations", Theory Probab. Appl., vol. 10, pp. 231-254, 1965.] and Petrov [28V.V. Petrov, "Asymptotic behaviour of probabilities of large deviations", Theory Probab. Appl., vol. 13, pp. 408-420, 1968.
[http://dx.doi.org/10.1137/1113050] ],
(17) |
where
and s is an integer satisfying (6), i.e. s = 1 if γ (^{1}/_{6}, ^{1}/_{4}] and s = 2 if γ (^{1}/_{4}, ^{3}/_{10}].
Here, the constants depend on skewness g_{1} and kurtosis g_{2} of X.
Thus, according to the construction of the skewness-kurtosis adjusted quantiles and moderate local alternatives , i {1, 2}, s {1, 2},
(18) |
Hence, for proving the second property of Definition 3.1 it remains to show that one can replace with in (18) if there holds i = 2.
As a consequence of (17), and in accordance with the proof of Theorems 1 and 2 in Richter [10W-D. Richter, "Skewness-kurtosis adjusted confidence estimators and significance tests", J. Stat. Distrib. Appl., 2016. (arXiv:1504.02553)
[http://dx.doi.org/10.1186/s40488-016-0042-3] ],
It remains therefore to show that,
Equivalently, we prove that,
(19) |
with
and
According to Lemma 3.1 below, for proving (19) it is sufficient to prove that k_{s} = o(h_{s}^{-1}). Note that,
and
By (14), and z_{β(n1)} → -∞ as n_{1} → ∞ where max = max {z_{1 - α(n1)}, -z_{β(n1)}}. It follows, symbolically, that,
thus h_{0 }k_{0 } = o(1) as n_{1} → ∞. Moreover,
and
If s = 1 then because of (15) and (14), and thus h_{1}k_{1} = o(1). Similarly, on using (16) and (14), in the case s = 2, h_{2}k_{2} = o(1).
Lemma 3.1. If then Proof. Let then
~ f_{n,s}(x ) if (i) , (ii) and (iii) . Note that assumption (i) is a stronger one than (ii) and (iii), and that it is fulfilled under the assumptions of Lemma 3.1.
The author confirms that this article content has no conflict of interest.
The author is grateful to the Reviewers for carefully reading the paper and giving valuable hints leading to an improvement of the paper.
[1] | Y. Nikitin, Asymptotic Efficiency of Nonparametric Tests, Cambridge University Press: Cambridge, England, 1995. [http://dx.doi.org/10.1017/CBO9780511530081] |
[2] | T. Inglot, and T. Ledwina, "On probabilities of excessive deviations for Kolmogorov-Smirnov, Cram’er-von Mises and chi-square statistics", Ann. Stat., vol. 18, pp. 1491-1495, 1990. [http://dx.doi.org/10.1214/aos/1176347764] |
[3] | T. Inglot, and T. Ledwina, "Moderately large deviations and expansions of large deviations for some functionals of weighted empirical process", Ann. Probab, vol. 21, no. 3, pp. 1691-1705, 1993. |
[4] | W.C.M. Kallenberg, "Intermediate efficiency, theory and examples", Ann. Stat., vol. 11, no. 1, pp. 170-182, 1983. |
[5] | W.C.M. Kallenberg, "On moderate deviation theory in estimation", Ann. Stat., vol. 11, no. 2, pp. 498-504, 1983. [http://dx.doi.org/10.1214/aos/1176346156] |
[6] | W.C. Kallenberg, and T. Ledwina, "On local and non-local measures of efficiency", Ann. Stat., vol. 15, pp. 1401-1420, 1987. [http://dx.doi.org/10.1214/aos/1176350601] |
[7] | T. Ledwina, T. Inglot, and W.C. Kallenberg, "Strong moderate deviation theorems", Ann. Probab., vol. 20, pp. 987-1003, 1992. [http://dx.doi.org/10.1214/aop/1176989814] |
[8] | W-D. Richter, "Remark on moderate deviations in the multidimensional central limit theorem", Math. Nachr., vol. 122, pp. 167-173, 1985. [http://dx.doi.org/10.1002/mana.19851220117] |
[9] | W-D. Richter, "Eine geometrische und eine asymptotische Methode in der Statistik. Universitàt Bremen, Mathematik-Arbeitspapiere Preprint 44", Bremen-Rostock Statistik Seminar, vol. 1992, pp. 2-10, 1995. |
[10] | W-D. Richter, "Skewness-kurtosis adjusted confidence estimators and significance tests", J. Stat. Distrib. Appl., 2016. (arXiv:1504.02553) [http://dx.doi.org/10.1186/s40488-016-0042-3] |
[11] | T.A. Wood, "Bootstrap relative errors and subexponential distributions", Bernoulli, vol. 6, no. 5, pp. 809-834, 2000. [http://dx.doi.org/10.2307/3318757] |
[12] | N.N. Amosova, "On the probabilities of moderate deviations for sums of independent random variables. Teor", Veroyatnost. i Primenen, vol. 24, no. 4, pp. 858-865, 1979. |
[13] | W-D. Richter, "Moderate deviations in special sets of R^{k}", Math. Nachr., vol. 113, pp. 339-354, 1983. [http://dx.doi.org/10.1002/mana.19831130130] |
[14] | W-D. Richter, "Multidimensional narrow integral domains of normal attraction", In: Limit Theorems in Probability Theory and Related Fields, TU Dresden: Germany, 1987, pp. 163-184. |
[15] | X. Fan, I. Grama, and Q. Liu, "A generalization of Cramér large deviations for martingales", Comptes Rendus Mathematique, vol. 352, no. 10, pp. 853-858, 2015. arXiv: 1503.06627v1 |
[16] | I.A. Ibragimov, and Y.W. Linnik, Independent and Stationary Sequence, Russian edition 1965 Walters, Nordhoff. Translation from, 1971. |
[17] | Yu.V. Linnik, "Limit theorems for sums of independent variables taking into account large deviations", I-III. Theor. Probab Appl., vol. 6, pp. 131-148, 1961. 345-360; 7 (1962), 115-129. |
[18] | S.V. Nagaev, "Some limit theorems for large deviations", Theory Probab. Appl., vol. 10, pp. 231-254, 1965. |
[19] | W.-D. Richter, "Probabilities of large deviations of sums of independent identically distributed random vectors", Lith. Math J., vol. 19, no. 3, pp. 333-343, 1979. [http://dx.doi.org/10.1007/BF00969969] |
[20] | L.V. Osipov, "Multidimensional limit theorems for large deviations", Theory Probab. Appl, vol. 20, pp. 38-56, 1975. [http://dx.doi.org/10.1137/1120004] |
[21] | W-D. Richter, "Multidimensional domains of large deviations", In: Colloquia Mathematica Societatis János Bolyai, vol. 57. 1990. Limit Theorems in Probability and Statistics, Pecs: Hungary, 1989, pp. 443-458. |
[22] | C. Ittrich, D. Krause, and W-D. Richter, "Probabilities and large quantiles of noncentral generalized chi-square distributions", Statistics, vol. 34, pp. 53-101, 2000. [http://dx.doi.org/10.1080/02331880008802705] |
[23] | R.A. Fisher, and E.A. Cornish, "The percentile points of distributions having known cumulants", Technometrics, vol. 2, no. 2, pp. 209-225, 1960. [http://dx.doi.org/10.1080/00401706.1960.10489895] |
[24] | L.N. Bolschev, and N.V. Smirnov, Tables of Mathematical Statistics (in Russian), Nauka: Moscow, 1983. |
[25] | C. Field, and E. Ronchetti, Small Sample Asymptotics, vol. 13. Institute of Mathematical Statistics Lecture Notes- Monograph Series: Hayward California, 1990. |
[26] | J.C. Fu, C.M. Len, and C.Y. Peng, "A numerical comparison of normal and large deviation approximation for tail probabilities", J.Japan. Statist. Soc, vol. 20, no. 1, pp. 61-67, 1990. |
[27] | J.L. Jensen, Saddlepoint Approximations, Clarendon Press: Oxford, 1995. |
[28] | V.V. Petrov, "Asymptotic behaviour of probabilities of large deviations", Theory Probab. Appl., vol. 13, pp. 408-420, 1968. [http://dx.doi.org/10.1137/1113050] |