Asymptotic approximation of inverse moments of nonnegative random variables

Statistics and Probability Letters 79 (2009) 1366–1371

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Statistics and Probability Letters journal homepage: www.elsevier.com/locate/stapro

Asymptotic approximation of inverse moments of nonnegative random variables Tiee-Jian Wu a,∗ , Xiaoping Shi b , Baiqi Miao b,1 a

Department of Statistics, National Cheng-Kung University, Tainan 70101, Taiwan

b

Department of Finance and Statistics, University of Science and Technology of China, Hefei, Anhui 230026, China

article

info

a b s t r a c t Let {Zn , n ≥ 1} be a sequence of independent nonnegative r.v.’s (random variables) with finite second moments. It is shown that under a Lindeberg-type condition, the α th inverse moment E {a + Xn }−α can be asymptotically approximated by the inverse of the α th moment {a + EXn }−α where a > 0, α > 0, and {Xn } are the naturally-scaled partial sums. Furthermore, it is shown that, when {Zn } only possess finite rth moments, 1 ≤ r < 2, the preceding asymptotic approximation can still be valid by using different norming constants which are the standard deviations of partial sums of suitably truncated {Zn }. © 2009 Elsevier B.V. All rights reserved.

Article history: Received 5 March 2008 Received in revised form 2 December 2008 Accepted 23 February 2009 Available online 28 February 2009 MSC: primary 60F99 secondary 62G20

1. Introduction Suppose {Zn , n ≥ 1} are independent nonnegative r.v.’s. Let us denote Xn =

n X

Zi /σn

and σn2 =

i=1

n X

Var Zi .

(1.1)

i=1

The main purpose of this paper is to prove that under suitable conditions the following equivalence relation holds, namely, E {a + Xn }−α ∼ {a + EXn }−α ,

n→∞

(1.2)

for every real constants a > 0 and α > 0. Here and below, cn ∼ dn means cn dn → 1. Relation (1.2) shows that the exact α th inverse moment (left-hand side of (1.2)), usually difficult to compute, can be asymptotically approximated by the inverse of the α th moment (right-hand side of (1.2)), usually much easier to compute. The inverse moments appear in many practical applications. For example, they may be applied in Stein estimation and post-stratification (see Wooff (1985) and Pittenger (1990)), evaluating risks of estimators and powers of tests (see Marciniak and Wesolowski (1999) and Fujioka (2001)). In addition, they appear in certain problems of reliability (see Gupta and Akman (1998)) and life testing (see Mendenhall and Lehman (1960)), insurance and financial mathematics (see Ramsay (1993)), complex systems (see Jurlewicz and Weron (2002)), and others. Under certain asymptotic-normality condition relation (1.2) is established in Theorem 2.1 of Garcia and Palacios (2001). But, unfortunately, that theorem is not true under the suggested assumptions, as pointed out by Kaluszka and Okolewski (2004). The latter authors establish (1.2) for α
∗

Corresponding author. E-mail address: [email protected] (T.-J. Wu).

1 Part of the research was done while the third author was visiting the National Center for Theoretical Sciences (South), Tainan, Taiwan. 0167-7152/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2009.02.010

T.-J. Wu et al. / Statistics and Probability Letters 79 (2009) 1366–1371

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very recently by Hu et al. (2007) while their condition EZn3 < ∞ and Zn satisfies L3 is relaxed to EZnc < ∞ and Zn satisfies Lc for some c > 2. In this paper, we establish (1.2) for every α > 0 while the aforementioned condition is further relaxed to the condition EZn2 < ∞ and Zn satisfies a Lindeberg-type condition (see Theorem 1 and Remark 1). This significantly increases the applicability of (1.2). Moreover, if it is only assumed that EZnr < ∞ for some 1 ≤ r < 2, then (along with other suitable assumptions) we establish E {a + Xn0 }−α ∼ {a + EXn0 }−α ,

n→∞

(1.3)

for every real a > 0 and α > 0, where Xn0 =

n X

Zi /σ¯ n

i =1

and

σ¯ n2 =

n X

Var {Zi I [Zi ≤ Mn ]}

(1.4)

i =1

with I [·] denoting the indicator function and Mn suitable positive constants satisfying Mn → ∞ as n → ∞ and Mn ≤ C0 n{2−δ}/{4−r } ,

n≥1

(1.5)

for some 0 < δ < r /2 and a finite positive constant C0 not depending on n (see Theorem 2). The result (1.3) is new in the literature. It holds with norming constant different from the standard deviation σn in (1.2) and without assuming EZn2 < ∞, hence has broad applicability. Finally, we show that for the special case that {Zn } are i.i.d., the condition EZ12 < ∞ alone is sufficient for (1.2), and the condition EZ1r < ∞, 1 ≤ r < 2 for (1.3). This result is apparently superior to existing ones. In Section 2 we state the main results. Theorems 1 and 2 and Corollary 1 deal with the independent but not identically distributed case, and Corollary 2 the i.i.d. case. Theorem 3 deals with the speed of convergence of the relative error in (1.2). At the end of Section 2 we give numerical examples to demonstrate the validity of (1.2). Section 3 is devoted to proofs. Our method of proof includes tail-truncation of {Zn } and applying Bernstein’s inequality to establish exponential bounds on tail probabilities of suitably normalized partial sums of truncated {Zn }. 2. Main results For the rest, all r.v.’s are assumed to be non-degenerated (to avoid triviality) and all inequalities are required to hold for all sufficiently large n. We shall first state our results, then make several remarks to discuss the conditions imposed, and finally give numerical examples. For clarity, we shall state each of our results in a self-contained fashion. Theorem 1. Suppose {Zn } is a sequence of independent, nonnegative and non-degenerated r.v.’s such that EZn2 < ∞ for all n and EXn → ∞, where Xn is defined by (1.1). Furthermore, assume that there exists a finite positive constant C1 not depending on n such that sup1≤i≤n EZi /σn ≤ C1 and

σn−2

n X

E {Zi2 I [Zi > ησn ]} → 0,

n→∞

(2.1)

i=1

for some η > 0. Then relation (1.2) holds for every real a > 0 and α > 0. Theorem 2. Suppose {Zn } is a sequence of independent, nonnegative and non-degenerated r.v.’s. Furthermore, suppose EZnr ≤ C2 Pn for some 1 ≤ r < 2, {Zn } is u.i. (uniformly integrable), n−1 i=1 EZi ≥ C3 and n−1/2 σ¯ n ≥ C4 for some positive constants {Mn } satisfying (1.5), where σ¯ n is defined in (1.4) and C2 , C3 and C4 are finite positive constants not depending on n. Then for Xn0 defined by (1.4), relation (1.3) holds for every real a > 0 and α > 0. The following result is an immediate consequence of Theorem 1: Corollary 1. Suppose {Zn } is a sequence of independent, nonnegative and non-degenerated r.v.’s. Furthermore, Suppose {Zn2 } is Pn u.i., n−1 i=1 EZi ≥ C5 and n−1/2 σn ≥ C6 , where C5 and C6 are finite positive constants not depending on n, then for Xn defined by (1.1), relation (1.2) holds for every real a > 0 and α > 0. The following result is an immediate consequence of Theorems 1 and 2: Corollary 2. Suppose {Zn } is a sequence of nonnegative, non-degenerated and i.i.d. r.v.’s. Then for every real a > 0 and α > 0: (i) If EZ12 < ∞, then for Xn defined by (1.1), relation (1.2) holds. (ii) If EZ1r < ∞ for some 1 ≤ r < 2, then for Xn0 defined by (1.4) with {Mn } being any sequence of positive constants satisfying (1.5), relation (1.3) holds. The next result gives the convergence rate of relative error in the relation (1.2).

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Theorem 3. For every real a > 0 and α > 0, the following three assertions hold: (i) Under the conditions of Theorem 1,


|E {a + Xn }−α /{a + EXn }−α − 1| = O α{a + EXn }−1 ,

n → ∞.

(2.2)

(ii) Under the conditions of Corollary 1,

|E {a + Xn }−α /{a + EXn }−α − 1| = O(n−1/2 ),

n → ∞.

(2.3)

(iii) Under the conditions of Corollary 2. (i), (2.3) holds. Remark 1. In Theorem 1, the condition sup1≤i≤n EZi /σn ≤ C1 (denote this condition by (A) herein) is not stringent because typically σn = O(n1/2 ) and then sup1≤i≤n EZi = O(n1/2 ). Also, (A) and (2.1) together is analogous to the Lindeberg condition:

σn−2

n X

E {(Zi − EZi )2 I [|Zi − EZi | > λσn ]} → 0,

n→∞

(2.4)

i =1

for all λ > 0. To see this, suppose (A) and (2.1) hold. Denote the left-hand sides of the → signs in (2.1) and (2.4) by Rn (η) and Sn (λ), respectively. Pick any λ ≥ max{η, C1 }. By noting that I [|Zi − EZi | > λσn ] = I [Zi − EZi > λσn ] ≤ I [Zi > ησn ] for all 1 ≤ i ≤ n and that

σn−2

n n X X (EZi )2 EI [Zi > ησn ] ≤ C12 E {(Zi /ησn )2 I [Zi > ησn ]}, i =1

(2.5)

i=1

we get Sn (λ) ≤ 2Rn (η) + 2C12 η−2 Rn (η) → 0, i.e., (2.4) holds for all λ ≥ max{η, C1 }. Conversely, suppose (A) and (2.4) hold. Pick any η > 21/2 C1 . By noting that I [Zi > ησn ] ≤ I [Zi − EZi > (η − C1 )σn ] and (2.5), we get Rn (η) ≤ 2Sn (η − C1 ) + 2C12 η−2 Rn (η). Thus (1 − 2C12 η−2 )Rn (η) → 0, i.e., (2.1) holds for all η > 21/2 C1 . Finally, we note that under (A), the condition (2.1) is weaker than the Lyapunov condition Lc , c > 2 because (2.4) is well known to be so. Remark 2. In Theorem 2 and Corollary 1, both the average mean and average variance of Z1 , . . . , Zn (average variance of truncated Zi ’s in the case of Theorem 2) are required to be eventually bounded away from 0. Also, the required u.i. of {Znr }, r ≥ 1 is ensured by either (a) P [Znr ≥ τ ] ≤ P [|Y | ≥ τ ], all τ > 0 for some r.v. Y with E |Y | < ∞, or (b) EZnr +d ≤ C7 for some d > 0 where C7 is a finite positive constant not depending on n or (c) EZnr < ∞, Zn → Z in law and EZnr → EZ r for some nonnegative r.v. Z with EZ r < ∞ (cf., e.g., pp. 32–33 of Billingsley (1968)). Remark 3. The condition in Theorem 2 is stated in the present form for simplicity. Evidently, there are some redundancies there. Specifically, if EZnr ≤ C2 for some r > 1, then {Zn } is u.i., and the latter is redundant. On the other hand, if {Zn } is u.i., then EZnr ≤ C2 for r = 1, and the latter is redundant. Remark 4. Evidently, (1.2) (similarly, for (1.3)) is equivalent to E {a + bXn }−α ∼ {a + bEXn }−α for every b > 0. We conclude this section by giving three numerical examples to demonstrate the validity of the approximation (1.2) (and (2.3)). Let Z1 , . . . , Zn be i.i.d. Denote the relative error Tn (α) = E {1 + Xn }−α /{1 + EXn }−α − 1. Then for n = 20, 40, 80, 160, respectively: (i) Tn (1) = .034, .019, .01, .005, Tn (2) = .106, .058, .031, .016 and Tn (3) = .225, .12, .063, .033 when Z1 ∼ Exponential(β), any β > 0, (ii) Tn (1) = .037, .02, .01, .005, Tn (2) =.121, .062, .032, .017 and Tn (3) = .271, .13, .066, .034 when Z1 ∼ Bernoulli (.5), and (iii) Tn (1) = .033, .018, .01, .005, Tn (2) = .10, .056, .031, .016 and Tn (3) = .209, .115, .062, .033 when Z1 ∼ Inverse Gaussian (µ, µ), any µ > 0. Evidently, for any n, when α increases the relative error increases and the accuracy of the approximation (1.2) decreases. On the other hand, for any α , when n increases the relative error decreases (to 0) and the accuracy of (1.2) increases fairly quickly. 3. Proofs In what follows, C denotes a finite generic positive constant not depending on n. Proof of Theorem 1. Let us decompose Xn as Xn = Un + Vn ,

(3.1)

where Un = σn−1

n X i=1

Zi I [Zi ≤ ησn ],

Vn = σn−1

n X i=1

Zi I [Zi > ησn ],

(3.2)

T.-J. Wu et al. / Statistics and Probability Letters 79 (2009) 1366–1371

and denote µ ˜ n = EUn and σ˜ n2 = EVn ≤ σn−1

n X

Pn

i=1

1369

Var {Zi I [Zi ≤ ησn ]}. Since EXn → ∞ and

E {Zi (Zi /ησn )I [Zi > ησn ]} → 0

(3.3)

i=1

by (2.1), it follows that µ ˜ n ∼ EXn . Therefore, (1.2) will be proved if we show that E {a + Xn }−α ∼ {a + µ ˜ n }−α .

(3.4)

Now, applying Jensen’s inequality to the convex function φ(x) = (a + x)

−α

yields E {a + Xn }

−α

≥ {a + EXn }

lim inf{a + µ ˜ n }α E {a + Xn }−α ≥ lim inf{a + µ ˜ n }α {a + EXn }−α = 1. n→∞

−α

. Hence (3.5)

n→∞

By (2.1) and (3.3), the condition sup1≤i≤n EZi /σn ≤ C1 and after simple computation (recalling the notation used in Remark 1),

|σ˜ n2 σn−2 − 1| ≤ Rn (η) + 2C1 EVn → 0

(3.6)

and so σ˜ n ∼ σn . Next, let us pick any t with 0 < t < 1 and write E {a + Xn }−α = Q1 + Q2 ,

(3.7)

where Q1 = E {{a + Xn }−α I [Un < µ ˜n −µ ˜ tn ]},

(3.8)

Q2 = E {{a + Xn }−α I [Un ≥ µ ˜n −µ ˜ tn ]}. By the inequalities Q2 ≤ E {(a + Xn )−α I [Xn ≥ µ ˜n −µ ˜ tn ]} ≤ {a + µ ˜n −µ ˜ tn }−α and the fact µ ˜ n → ∞, we get lim sup{a + µ ˜ n }α Q2 ≤ 1.

(3.9)

n→∞

Put Yi = σn−1 Zi I [Zi ≤ ησn ]. Then −Un = and the fact µ ˜ n → ∞, we obtain

Pn

i=1

(−Yi ) with −η ≤ −Yi ≤ 0. Applying Bernstein’s inequality and using (3.6)

aα Q1 ≤ P [Un − EUn < −µ ˜ tn ] = P [−Un + EUn > µ ˜ tn ]

 2t ≤ exp −µ ˜ n /{2σn−2 σ˜ n2 + (2ηµ ˜ tn /3)} ≤ exp{−C µ ˜ tn }.

(3.10)

It follows that {a + µ ˜ n }α Q1 = o(1). This, together with (3.7) and (3.9), implies lim sup{a + µ ˜ n }α E {a + Xn }−α ≤ 1.

(3.11)

n→∞

From (3.5) and (3.11), we immediately conclude (3.4), and therefore (1.2).



0

Proof of Theorem 2. Let us decompose Xn as Xn0 = Un0 + Vn0 ,

(3.12)

where Un0 = σ¯ n−1

n X

Zi I [Zi ≤ Mn ],

Vn0 = σ¯ n−1

i=1

n X

Zi I [Zi > Mn ],

(3.13)

i=1

and denote µ ¯ n = EUn0 . Since {Zi } is u.i. and Mn → ∞, n− 1

n X

E {Zi I [Zi > Mn ]} ≤ sup E {Zi I [Zi > Mn ]} → 0.

(3.14)

i ≥1

i=1

This, along with the assumption that n−1 i=1 EZi ≥ C3 , implies µ ¯ n ∼ EXn0 . Therefore, in order to prove (1.3) it suffices to show that E {a + Xn0 }−α ∼ {a + µ ¯ n }−α . Arguing as in (3.5), we obtain

Pn

lim inf{a + µ ¯ n }α E {a + Xn0 }−α ≥ 1. n→∞

The inequalities σ¯ n2 ≤ on Mn , lead to

Pn

i=1

(3.15)

E {Zi2−r Zir I [Zi ≤ Mn ]} ≤ nMn2−r supi≥1 EZir , together with the fact µ ¯ n ∼ EXn0 and the assumption

µ ¯ n ≥ CEXn0 ≥ CC3 nσ¯ n−1 ≥ Cn1/2 Mn−1+{r /2} → ∞.

(3.16)

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Clearly, the quantity ξ = {r − 2δ}/{r + (2 − r )δ} is strictly between 0 and 1, and we can pick any pair (t0 , t ) with ξ ≤ t0 < t < 1. Simple computation shows that (2 − δ)/(4 − r ) ≤ (1 + t0 )/{2 + (2 − r )t0 }, hence Mn ≤ C0 n(1+t0 )/{2+(2−r )t0 } . This, together with (3.16) and the assumption on σ¯ n , yields Mn (µ ¯ tn0 σ¯ n )−1 ≤ CMn{2+(2−r )t0 }/2 n−(1+t0 )/2 ≤ C .

(3.17)

Let us write E {a + Xn0 }−α = Q10 + Q20 ,

(3.18)

where Q10 = E {{a + Xn0 }−α I [Un0 < µ ¯n −µ ¯ tn ]},

(3.19)

Q20 = E {{a + Xn0 }−α I [Un0 ≥ µ ¯n −µ ¯ tn ]}. By arguments similar to those in deriving (3.9), we have lim sup{a + µ ¯ n }α Q20 ≤ 1.

(3.20)

n→∞

Put Yi0 = σ¯ n−1 Zi I [Zi ≤ Mn ]. Then −Un0 = (3.16) and (3.17), we obtain

Pn

i =1

(−Yi0 ) with −Mn σ¯ n−1 ≤ −Yi ≤ 0. Applying Bernstein’s inequality and using

aα Q10 ≤ P [Un0 − EUn0 < −µ ¯ tn ] ≤ exp −3µ ¯ 2t ¯ tn σ¯ n−1 )} n /{6 + (2Mn µ





  ≤ exp −3µ ¯ tn−t0 /{C + 2Mn (µ ¯ tn0 σ¯ n )−1 } ≤ exp −C µ ¯ nt −t0 . α

(3.21)

It follows that {a + µ ¯ n } Q1 = o(1). This, together with (3.18) and (3.20), implies 0

lim sup{a + µ ¯ n }α E {a + Xn0 }−α ≤ 1.

(3.22)

n→∞

The proof follows from (3.15) and (3.22) immediately.



< ∞ for all 1 ≤ s ≤ 2. Therefore, sup1≤i≤n EZi = O(σn ) and → ∞. Since σn → ∞, we can argue as in (3.14) and obtain {Zi2 I [Zi > ησn ]} → 0, i.e., condition (2.1) is satisfied by picking

Zn2

Proof of Corollary 1. The u.i. of implies supn≥1 EZns n 2 2 O n . The latter implies EXn Cn1/2 n i=1 EZi −2 (recalling the notation in Remark 1) Rn C6 supi≥1 E

{ }

σ

≤

P

=

( )

≥

(η) ≤

any η > 0. The proof follows immediately from Theorem 1.



Proof of Corollary 2. (i) is an immediate consequence of Corollary 1. Next, put Wn = Z1 I [Z1 ≤ Mn ]. Since Mn → ∞, we have Wn → Z1 a.s. (almost surely). The fact EZ1 < ∞ implies EWn → EZ1 and, consequently, (Wn − EWn )2 → (Z1 − EZ1 )2 a.s.. By Fatou’s Lemma, lim inf n−1 σ¯ n2 = lim inf Var Wn ≥ E {lim inf(Wn − EWn )2 } = Var Z1 > 0. n→∞

n→∞

n→∞

Now, (ii) may be concluded from Theorem 2 quickly.



Proof of Theorem 3. (i) By Taylor’s expansion (φ is defined below (3.4))

φ(Xn ) = φ(EXn ) + φ 0 (ξn )(Xn − EXn )

(3.23)

where ξn is between Xn and EXn . Using Cauchy–Schwartz inequality, the fact {φ (x)} is decreasing in x ≥ 0 and (1.2), we have 0

2

{E φ(Xn ) − φ(EXn )}2 ≤ E {φ 0 (ξn )}2 Var Xn = E {φ 0 (ξn )}2   = E {φ 0 (ξn )}2 I [Xn ≤ EXn ] + E {φ 0 (ξn )}2 I [Xn > EXn ] ≤ E {φ 0 (Xn )}2 + {φ 0 (EXn )}2 ∼ 2{φ 0 (EXn )}2 = 2α 2 (a + EXn )−2(α+1) .

(3.24)

1/2

This leads to (2.2) quickly. (ii) (2.3) follows from (2.2) and the fact EXn ≥ Cn , as established in the proof of Corollary 1. (iii) The desired result follows form (2.2) and the fact EXn = n1/2 EZ1 (Var Z1 )−1/2 .  Acknowledgments The authors would like to express their sincere thanks to an associate editor and two referees for many valuable suggestions that significantly improved the quality and presentation of the article. Research of the first, second and third authors is supported by National Science Council of Taiwan grant NSC-95-2118-M006-003-MY3, University of Science and Technology of China Graduate Innovation Fund KD2006063 and National Natural Science Foundation of China grant 10471135, respectively.

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