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Abstract
Let be a sequence of independent random variables with common general error distribution
with shape parameter v>0, and let
denote the r-th largest order statistics of
. With different normalizing constants the distributional expansions and the uniform convergence rates of normalized powered order statistics
are established. An alternative method is presented to estimate the probability of the r-th extremes. Numerical analyses are provided to support the main results.
1. Introduction
The general error distribution is an extension of the normal distribution. The density of general error distribution, say , is given by
(1)
(1) where v>0 is the shape parameter, and
with
denoting the Gamma function. It is known that
is the standard normal density and
is the density of Laplace distribution.
For the general error distribution with parameter v (denoted by ), Nelson (Citation1991) used it in volatility models since
can capture the heavy tailedness of high frequency financial time series. Peng et al. (Citation2009) considered the tail behaviour of
. Peng et al. (Citation2010) established the uniform convergence rates of normalized extremes under optimal normalizing constants. Jia and Li (Citation2014) established the higher-order distributional expansions of normalized maximum. Aforementioned studies show that the optimal convergence rate is proportional to
, similar to the results of Hall (Citation1979) and Nair (Citation1981) on normal extremes. In order to improve the convergence rate of normal extremes, Hall (Citation1980) studied the asymptotics of
, the powered r-th largest order statistics, and showed that the distribution of normalized
converges to its limit at the rate of
under optimal normalizing constants, while the convergence rates are still the order of
in the case of
. For more details, see Hall (Citation1980). For more work on higher-order expansions of powered-extremes from normal samples, see Li and Peng (Citation2018) and Zhou and Ling (Citation2016). For other related work on distributional expansions of extremes, see Liao and Peng (Citation2012) for lognormal distribution, Liao et al. (Citation2014a, Citation2014b) for logarithmic general error distribution and skew-normal distribution, and Hashorva et al. (Citation2016), Liao and Peng (Citation2014, Citation2015), Liao et al. (Citation2016) and Lu and Peng (Citation2017) for bivariate Hüsler-Reiss models.
In extreme value theory and its applications, it is important to know the convergence rate of distribution of normalized maximum to its ultimate distribution, cf. Cao and Zhang (Citation2021), Hall (Citation1979, Citation1980), Leadbetter et al. (Citation1983) and Nair (Citation1981). For more advanced topics related to extreme value theory and its applications, see Zhang (Citation2021) for an excellent review. Motivation of this paper is to consider the higher-order expansions and the uniform convergence rate of powered order statistics from sample, and an alternative method to estimate the probability related to powered order statistics, which are the complete extensions the results given by Hall (Citation1980) for
case. Let
be independent and identically distributed random variables with common distribution function
following
with density given by (Equation1
(1)
(1) ). For positive integer r, let
denote the r-th largest order statistics of
and
for later use. It is known that the distributional convergence rate of normalized maximum may depend heavily on the normalizing constants, see Hall (Citation1979, Citation1980), Leadbetter et al. (Citation1983), Nair (Citation1981) and Resnick (Citation1987) for normal samples. For the
distribution and positive power index p, it is necessary to discuss how to find the normalizing constants
and
such that
(2)
(2) where
with
, and further work on distributional expansions and uniform convergence rates of
with different normalizing constants.
For v>0 with normalizing constants and
given by
(3)
(3) Peng et al. (Citation2009) showed that
(4)
(4) where M follows the Gumbel extreme value distribution
. With normalizing constants
and
given by
(5)
(5) we will show that (Equation2
(2)
(2) ) holds by replacing
and
by
and
, respectively, and investigate further its higher-order expansions and the uniform convergence rates. Similarly, with
the optimal convergence rates of
are derived under the following normalizing constants
(6)
(6) where constant
is the solution of the equation
(7)
(7) Note that for the normal case, it follows from (Equation6
(6)
(6) ) that
and
since v = 2 and
, which are just the normalizing constants given by Hall (Citation1980).
For the normal case, Hall (Citation1980) showed that the optimal convergence rate of is the order of
if we choose the normalizing constants
and
. For the powered r-th largest order statistics
from the
sample, it follows from (Equation6
(6)
(6) ) that the convergence rate can be improved if p = v. By Equation (3.1) of Lemma 1 in Jia and Li (Citation2014), for
and large x we have
(8)
(8) Similar to Hall (Citation1980), as p = v we choose the optimal normalizing constants
and
as follows.
(9)
(9) where
is given by (Equation7
(7)
(7) ). Note that if v = 2 and
, for the normal case,
and
are just the normalizing constants given by Hall (Citation1980).
The rest of this paper is organized as follows. Section 2 provides the main results and Section 3 presents some numerical analyses. Auxiliary lemmas are deferred to Section 4. Section 5 gives the proofs of the main results.
2. Main results
In this section, we provide the higher-order distributional expansions and the uniform convergence rates of powered extremes under different normalizing constants. Furthermore, we showed a method to estimate the probabilities of the extremes. Throughout this paper, let for positive integer r and
for
. Recall that the power index p is positive.
Theorem 2.1
Let be a sequence of independent random variables with common distribution
, and
denotes the r-th maximal term of
. Then,
if v = 1 and p = 1, with normalizing constants
and
given by
and
, we have
if v = 1 and
, with normalizing constants
and
given by
and
, we have
if
, with normalizing constants
and
given by (Equation5
(5)
(5) ), we have
if
and
, with normalizing constants
and
given by (Equation6
(6)
(6) ), we have
where
(10)
(10) and
(11)
(11)
if
and p = v, with normalizing constants
and
given by (Equation9
(9)
(9) ), we have
where
(12)
(12) and
(13)
(13)
Remark 2.1
Theorem 2.1 (i)–(ii) show the difference of the convergence rates for the powered-extremes of the Laplace distribution as p = 1 and , respectively. Meanwhile, it follows from (Equation3
(3)
(3) ) and (Equation5
(5)
(5) )–(Equation7
(7)
(7) ) that
and
as v = 1 since
, so it is not necessary to consider the case of v = 1 in Theorem 2.1 (iv)–(v).
Remark 2.2
For , with normalizing constants
and
given by (Equation6
(6)
(6) ), Theorem 2.1 (iv) shows that the convergence rate of
to the extreme value distribution
is proportional to
since
by (Equation7
(7)
(7) ), while it can be improved to the order of
with optimal choice of normalizing constants
and
given by (Equation9
(9)
(9) ) as p = 2, which coincides with the normal case studied by Hall (Citation1980).
Theorem 2.2
Let be a sequence of independent random variables with common distribution
, and
denotes the r-th maximal term of
. The following results hold.
If v = 1 and p = 1, with normalizing constants
and
given by
and
, then
is the order of
.
If v = 1 and
, with normalizing constants
and
given by
and
, then
is the order of
.
If
and p>0, with normalizing constants
and
given by (Equation5
(5)
(5) ), then
is the order of
.
If
and
, with normalizing constants
and
given by (Equation6
(6)
(6) ), then
is the order of
.
If
and p = v, with normalizing constants
and
given by (Equation9
(9)
(9) ), then
is the order of
.
Theorem 2.3
Let be defined by (Equation7
(7)
(7) ) and
If v>1 and
, then
(14)
(14) If 0<v<1,
and
, then
(15)
(15) where
.
Similarly, for the bounds of the r-th order statistics with r>1, we have
(16)
(16) where
(17)
(17) for v>1, and for
,
(18)
(18)
3. Numerical analyses
In this section, small numerical analyses are provided to support the main results. We compare the actual values of probability of powered order statistics with its higher order expansions provided by Theorem 2.1, and with two bounds based on Theorem 2.3.
First we compare the actual values of with the following asymptotics.
If v = 1 and p = 1, the second-order and the third-order asymptotics are respectively given by
and
.
If v = 1 and
, the second-order and the third-order asymptotics are respectively given by
and
.
If
, the second-order and the third-order asymptotics are respectively given by
and
.
If
and
, the second-order and the third-order asymptotics are respectively given by
and
.
If
and p = v, the second-order and the third-order asymptotics are respectively given by
and
.
For different parameters v and p, with sample size , Figure shows the relationship between the actual values and the three asymptotics mentioned above with parameter r = 1 and given interval
, which supports our findings. For the case of r>1, it is difficult to calculate the actual value directly, so we estimate the actual values by calculating the empirical distribution function. For given
, Figure shows the difference between the estimated actual values with the three asymptotics as r = 2, which may be acceptable.
Figure 1. Actual values and its approximations with n = 1000, r = 1, . The actual values compared with the first-order asymptotics, the second-order asymptotics and the third-order asymptotics.
![Figure 1. Actual values and its approximations with n = 1000, r = 1, x∈[−5,10]. The actual values compared with the first-order asymptotics, the second-order asymptotics and the third-order asymptotics.](/cms/asset/10195a2f-5675-40a2-80f1-59e31c4bd470/tstf_a_2146955_f0001_oc.jpg)
Figure 2. Actual values and its approximations with r = 2, . The actual values (estimated by empirical distribution) compared with the first-order asymptotics, the second-order asymptotics and the third-order asymptotics.
![Figure 2. Actual values and its approximations with r = 2, x∈[−6,6]. The actual values (estimated by empirical distribution) compared with the first-order asymptotics, the second-order asymptotics and the third-order asymptotics.](/cms/asset/06e62102-35ed-4ff4-b14c-766c89bdb167/tstf_a_2146955_f0002_oc.jpg)
To end this section, we compare the two bounds given by Theorem 2.3 with the actural values of extremes.
For , calculate
,
,
and relative error
, where
and
are given by (Equation14
(14)
(14) ) and (Equation15
(15)
(15) ), respectively. Tables show the results. For the case of 0<v<1, Table shows that for given v = 0.2, 0.5, with increasing sample size n,
decreases; and for given n, with increasing v,
decreases. For given n and v,
is a decreasing function of x. Table shows the fact that for given x = 5,
is a decreasing function of n and v. For similar facts for the case of v>1, see Tables and .
Table 1. Comparison of ,
and
along with n and x for given v = 0.2 and v = 0.5, respectively.
Table 2. Comparison of ,
and
along with n and 0<v<1 for given x = 5.
Table 3. Comparison of ,
and
along with n and x for given v = 2 and v = 5, respectively.
Table 4. Comparison of ,
and
along with n and v>1 for given x = 3.5.
For the r-th order statistics , calculate
,
,
and
with
and
given by (Equation17
(17)
(17) ) for v>1, and (Equation18
(18)
(18) ) for 0<v<1. Table shows that
is an increasing function of n and x for given v = 0.5 and r = 2, and it is irregular for the case of v = 5; the two bounds are closer to the actual values. For given x = 2.8 and n = 30, Table shows that
is an increasing function of r for given v. Tables also show the fact that two relative accurate bounds control the range of the probability of the extremes, and provide an alternative to linear interpolation. It is an effective method to estimate the probability of the extremes.
Table 5. Comparison of ,
and
along with n and x for given r = 2, and v = 0.5 and v = 5, respectively.
Table 6. Comparison of ,
and
along with v and
for given x = 2.8 and n = 30.
4. Auxiliary lemmas
Let C be a positive constant with values varying from place to place. In order to prove the main results, we need some auxiliary lemmas.
Lemma 4.1
Let denote the
distribution function with parameter v>0. We have the following results.
If v = 1 and p = 1, with normalizing constants
and
given by
and
,
(19)
(19)
If v = 1 and
, with normalizing constants
and
given by
and
,
(20)
(20)
If
, with normalizing constants
and
given by (Equation5
(5)
(5) ),
(21)
(21)
If
and
, with normalizing constants
and
given by (Equation6
(6)
(6) ),
(22)
(22)
If
and p = v, with normalizing constants
and
given by (Equation9
(9)
(9) ),
(23)
(23)
Proof.
(i) If v = 1 and p = 1, by (Equation1(1)
(1) ) the Laplace density
is given by
(24)
(24) By (Equation24
(24)
(24) ) and the values of
and
given by Theorem 2.1(i), we get
(ii) If v = 1 and
, note that
(25)
(25) The claimed result (Equation20
(20)
(20) ) follows from (Equation24
(24)
(24) ) and (Equation25
(25)
(25) ).
(iii) If , with normalizing constants
and
given by (Equation5
(5)
(5) ),
and
we have
(26)
(26) and
Further,
(27)
(27) Combining (Equation8
(8)
(8) ) and (Equation26
(26)
(26) )–(Equation27
(27)
(27) ), we derive (Equation21
(21)
(21) ).
(iv) If and
, with normalizing constants
and
, we have
By using arguments similar to (Equation26
(26)
(26) )–(Equation27
(27)
(27) ), we have
(28)
(28) and by (Equation7
(7)
(7) ),
(29)
(29) Further,
(30)
(30) Combining (Equation8
(8)
(8) ) and (Equation28
(28)
(28) )–(Equation30
(30)
(30) ), we derive the desired result (Equation22
(22)
(22) ).
(v) If and p = v, with normalizing constants
and
given by (Equation9
(9)
(9) ), let
By arguments similar to (Equation26
(26)
(26) )–(Equation27
(27)
(27) ), we can get
(31)
(31) and by (Equation7
(7)
(7) ),
(32)
(32) Further,
(33)
(33) Combining (Equation8
(8)
(8) ) and (Equation31
(31)
(31) )–(Equation33
(33)
(33) ), we derive (Equation23
(23)
(23) ).
Lemma 4.2
Let be a sequence of i.i.d. random variables with common distribution
with parameter v>0 and
denotes the r-th maximal term of
for
. Assume that there exists positive constant
such that when
,
(34)
(34) where
.
Proof.
By arguments similar to Hall (Citation1980), we have
since
(35)
(35) The proof is completed.
Lemma 4.3
Let satisfy the assumptions of Lemma 4.2 and
denotes the r-th largest order statistics of
. The following results hold.
If v = 1 and
, with normalizing constants
and
given by
and
, let
, and then
(36)
(36)
If
, with normalizing constants
and
given by (Equation5
(5)
(5) ), let
, and then
(37)
(37)
If
and
, with normalizing
and
given by (Equation6
(6)
(6) ), let
, and then
(38)
(38)
If
and p = v, with normalizing
and
given by (Equation9
(9)
(9) ), let
, and then
(39)
(39)
Proof.
Note that (40)
(40) and
(41)
(41) If v = 1 and
, note that
and
for any
. It follows from (Equation40
(40)
(40) ) and (Equation41
(41)
(41) ) that (Equation36
(36)
(36) ) holds. If
, and
, from (Equation40
(40)
(40) ) and (Equation41
(41)
(41) ) we can get (Equation37
(37)
(37) ). If
and
, recall that
and
, and then (Equation38
(38)
(38) ) follows. If
, p = v, and
, from (Equation40
(40)
(40) ) and (Equation41
(41)
(41) ) we can get (Equation39
(39)
(39) ).
The following is about the Mills' type inequalities of GED.
Lemma 4.4
For 0<v<1, as
we have
For v>1, for all x>0 we have
Proof.
Note that assertion (ii) is just Lemma 2.2 of Peng et al. (Citation2009). We only show that assertion (i) holds. For as 0<v<1,
implies that
(42)
(42) and
(43)
(43) It follows from (Equation42
(42)
(42) ) that
Similarly, by (Equation43
(43)
(43) ),
Hence, the assertion follows.
Recall that and for large enough n,
and
.
Lemma 4.5
If v = 1 and
, let
and
, and then
If
and p>0, let
and
, and then
where
, and
.
If
and
, let
and
, and then
If
and p = v, let
and
, and then
Proof.
(i) If v = 1 and , then
. It follows from Lemma 1 of Hall (Citation1980) and (Equation20
(20)
(20) ) that
and
Combining the above results, we can get (i).
(ii) In case of v>1, . By Lemma 1 of Hall (Citation1980), Lemma 4.4 and some tedious calculation, we have
and
which implies (ii) for the case v>1.
If 0<v<1, it follows from Lemma 1 of Hall (Citation1980) and Lemma 4.4 that
and
which implies (ii) for the case 0<v<1.
(iii) If , in case of v>1,
. It follows from Lemma 1 of Hall (Citation1980) and Lemma 4.4 that
since
as
. Next,
which implies (iii) for the case v>1.
If , in case of 0<v<1, it follows from Lemma 1 of Hall (Citation1980) and Lemma 4.4 that
and
which implies (iii) for the case of 0<v<1.
(iv) If p = v, the proof is similar to the case of (iii). If v>1 and , we have
and
Therefore, the result in (iv) for the case of v>1 can be proved.
If p = v, in case of 0<v<1,
and
Hence the result of (iv) can be proved for 0<v<1.
Lemma 4.6
If v = 1 and
, let
and
, and then
If
, let
and
, and then
If
and
, let
and
, and then
If
and p = v, let
and
, and then
Proof.
(i) We can get the assertion of (i) since
(ii) In case of v>1,
and for the case 0<v<1,
Hence, we can get (ii).
(iii) If , in case of v>1,
and for the case 0<v<1,
Therefore, we can get (iii).
(iv) If p = v, in case of v>1
and for the case of 0<v<1,
Then, we complete the proof of (iv).
Let
where
and
.
Lemma 4.7
If v = 1 and
, let
and
, and then
If
and p>0, let
and
, and then
If
and
, let
and
, and then
If
and p = v, let
and
, and then
Proof.
(i) If v = 1 and , since
, then
and
Therefore, we can get (i).
(ii) If v>1, since ,
and
In case of 0<v<1, noting that
, we have
and
Therefore, we can get (ii).
(iii) If , in case of v>1 and noting that
, we have
and
If
, in case of 0<v<1, noting that
, we have
and
Therefore, we can get (iii).
(iv) If p = v, in case of v>1, noting that , we have
and
If p = v, in case of 0<v<1, noting that
, we have
and
The desired result (iv) follows.
5. Proofs
Proof
Proof of Theorem 2.1
(i) Let with
and
given by Theorem 2.1 (i). By using (Equation19
(19)
(19) ) and (Equation35
(35)
(35) ), and some tedious calculation we have
(44)
(44) Hence, the desired results follow from (Equation44
(44)
(44) ).
(ii) Let with
and
given by Theorem 2.1(ii). By using (Equation20
(20)
(20) ) we have
(45)
(45) It follows from Lemma 4.3 and (Equation45
(45)
(45) ) that
(46)
(46) Hence, following (Equation60
(60)
(60) ) we get the desired results.
(iii) Let with
and
given by (iii) of Theorem 2.1. By using (Equation21
(21)
(21) ) we have
(47)
(47) It follows from Lemma 4.3 and (Equation47
(47)
(47) ) that
(48)
(48) Therefore, following (Equation61
(61)
(61) ), we get the desired results.
(iv) Let with
and
given by (iv) of Theorem 2.1 and
,
are given by (Equation10
(10)
(10) ) and (Equation11
(11)
(11) ). By using (Equation22
(22)
(22) ) we have
(49)
(49) It follows from Lemma 4.3 and (Equation49
(49)
(49) ) that
(50)
(50) implying the desired results.
(v) Let with
and
given by Theorem 2.1(v) and
and
be given by (Equation12
(12)
(12) ) and (Equation13
(13)
(13) ). By using (Equation23
(23)
(23) ) we have
(51)
(51) It follows from Lemma 4.3 and (Equation51
(51)
(51) ) that
(52)
(52) Hence, by using (Equation52
(52)
(52) ), we derive the desired results.
Proof
Proof of Theorem 2.2
The lower bounds are from Theorem 2.1. The rest is to derive the upper bounds. By the arguments similar to Hall (Citation1980) and some tedious calculations, we have (53)
(53) For
,
(54)
(54) Note that
(55)
(55) (i) If v = 1 and p = 1, by (Equation19
(19)
(19) ) and (Equation53
(53)
(53) )–(Equation55
(55)
(55) ), we have
and
Therefore, combining Theorem 2.1 (i), the assertion of (i) can be proved.
(ii) If v = 1 and , by Lemma 4.5 (i) for
we have
(56)
(56) For
, by the arguments similar to those used in (Equation56
(56)
(56) ), it follows from Lemma 4.6(i) that
It follows from (Equation53
(53)
(53) )–(Equation55
(55)
(55) ) that
as
.
Suppose that , in view of (Equation54
(54)
(54) ) and Lemma 4.7(i),
It follows from (3.19) that
(57)
(57) Combining with Lemma 4.3(i) and Theorem 2.1(ii), we complete the proof of (ii).
(iii) If , from Lemma 4.1 (iii), Lemma 4.5 (ii) and the similar arguments used in (Equation56
(56)
(56) ), for
recall that
, and we have
Next, from Lemma 4.6(ii), for
,
Therefore for
, it follows from (Equation53
(53)
(53) )–(Equation55
(55)
(55) ) that
.
Note that for the case , Lemma 4.5 (ii) also implies that
and in view of (Equation55
(55)
(55) ) and Lemma 4.7(ii), we can get
Now, from (Equation54
(54)
(54) ) and by the similar arguments used in (Equation57
(57)
(57) ), we have
Hence, combining with Lemma 4.3(ii), Theorem 2.1(iii) and (Equation53
(53)
(53) )–(Equation55
(55)
(55) ), (iii) can be proved.
(iv) If and
, by Lemma 4.5 (iii) and the similar arguments used in (Equation56
(56)
(56) ), for
, we have
Next, from Lemma 4.6 (iii), for
,
Therefore for
, it follows from (Equation53
(53)
(53) )–(Equation55
(55)
(55) ) that
.
Note that for the case , Lemma 4.5 (iii) also implies that
and in view of (Equation55
(55)
(55) ) and Lemma 4.7 (iii), we can get
Now, from (Equation54
(54)
(54) ) and by the similar arguments used in (Equation57
(57)
(57) ), we have
Hence, combining with Lemma 4.3 (iii), Theorem 2.1 (iv) and (Equation53
(53)
(53) )–(Equation55
(55)
(55) ), (iv) can be proved.
(v) If and p = v, for
, it follows from Lemmas 4.5-4.6 (iv) and (Equation53
(53)
(53) ) that
and combining (Equation54
(54)
(54) )–(Equation55
(55)
(55) ), we can get
.
Note that for , by Lemma 4.5 (iv) and the similar arguments used in (Equation56
(56)
(56) ),
and in view of (Equation55
(55)
(55) ) and Lemma 4.7 (iv),
Finally, by (Equation54
(54)
(54) ) and the similar arguments used in (Equation57
(57)
(57) ), we have
Combining with Lemma 4.3(iv) and Theorem 2.1(v), we finish the proof of (v).
Proof
Proof of Theorem 2.3
For the r-th largest order statistics, we have
(58)
(58) and for any 0<z<n,
(59)
(59) We first consider the bounds of
as v>1. It follows from (Equation59
(59)
(59) ) and Lemma 4.4(ii) that
For the lower bound, note that
(60)
(60) with
By (Equation59
(59)
(59) ) and (Equation60
(60)
(60) ) we have
(61)
(61) where
. Noting that for
and
, we have
(62)
(62) It follows from (Equation61
(61)
(61) ) and (Equation62
(62)
(62) ) that
(63)
(63) which is the desired lower bound.
For the bounds of the r-th order statistics as v>1, by (Equation58(58)
(58) ), (Equation59
(59)
(59) ) and Lemma 4.4(ii), we have
For the lower bound, by (Equation63
(63)
(63) ), (Equation59
(59)
(59) ) and Lemma 4.4(ii), we have
The remainder is to consider the bounds as 0<v<1. For the bounds of
as 0<v<1, it follows from (Equation59
(59)
(59) ) and Lemma 4.4(i) that
and
where
.
For the bounds of the r-th order statistics, it follows from (Equation59(59)
(59) ) and Lemma 4.4(i) that
By similar arguments used in (Equation63
(63)
(63) ), one can show that
Thus, the proof is completed.
Acknowledgements
The authors would like to thank the Editor-in-Chief, the Associated Editor and the two referees for careful reading and comments which greatly improved the paper.
Disclosure statement
No potential conflict of interest was reported by the author(s).
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