ON A UNIFORMLY INTEGRABLE FAMILY OF POLYNOMIALS DEFINED ON THE UNIT INTERVAL

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Uniformly Integrable Family of Polynomials Alexandre Leblanc and

Brad C. Johnson vol. 8, iss. 3, art. 67, 2007

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ON A UNIFORMLY INTEGRABLE FAMILY OF POLYNOMIALS DEFINED ON THE UNIT INTERVAL

ALEXANDRE LEBLANC AND BRAD C. JOHNSON

338 Machray Hall, Department of Statistics University of Manitoba

Canada R3T 2N2.

EMail:{alex_leblanc,brad_johnson}@umanitoba.ca

Received: 27 October, 2006

Accepted: 24 July, 2007

Communicated by: G.P.H. Styan

2000 AMS Sub. Class.: Primary: 05A20, 26A48; Secondary: 60C05.

Key words: Uniform integrability, Bernstein polynomials, Probability inequalities, Combina- torial inequalities, Completely monotonic functions.

Abstract: In this short note, we establish the uniform integrability and pointwise convergence of an (unbounded) family of polynomials on the unit interval that arises in work on statistical density estimation using Bernstein polynomials. These results are proved by first establishing/generalizing some combinatorial and probability inequalities that rely on a new family of completely monotonic functions.

Acknowledgements: This research was partially supported by the Natural Sciences and Engineering Council of Canada. The authors would also like to thank John F. Brewster and James C. Fu for insightful discussions.

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1. Introduction

LetP_n,k(x) : [0,1] → [0,1]denote the probability of exactlyk successes inn independent Bernoulli trials with success probabilityx, i.e.

Pn,k(x) = Pr{Bin(n, x) = k}= n

k

x^k(1−x)^n−k,

and, for integersr, s≥1, define the family of functions{S_n,r,s}^∞_n=1by

(1.1) S_n,r,s(x) :=√

n

X

k=0

P_rn,rk(x)P_sn,sk(x).

This family of polynomials arises in the context of statistical density estimation based on Bernstein polynomials. Specifically, the caser = s = 1 has been con- sidered by many authors (for example, Babu et al. [3], Kakizawa [5] and Vitale [8]) and the caser = 1ands = 2 has been considered by Leblanc [6]. Issues linked to uniform integrability and pointwise convergence of{S_n,1,1} and{S_n,1,2}have also been addressed by these authors. However, the generalization to anyr, s≥1has not yet been considered. In the present paper we will establish the following result.

Theorem 1.1. Letr, sbe fixed positive integers. Then (i) 0≤S_n,r,s(x)≤√

nforx∈[0,1]andS_n,r,s(0) =S_n,r,s(1) =√ n.

(ii) {S_n,r,s}^∞_n=0 is uniformly integrable (w.r.t. Lebesgue measure) on[0,1].

(iii) S_n,r,s(x)→gcd(r, s)[rs(r+s)2πx(1−x)]^−1/2 forx∈(0,1)asn → ∞.

For the case r = s = 1, Babu et al. [3, Lemma 3.1] contains a proof of (iii).

Leblanc [6, Lemma 3] gives a proof of Theorem 1.1 whenr = 1 and s = 2. The

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proof herein generalizes (but follows along the same lines as) these previous results.

As an application of Theorem1.1 we have, for any functionf that is bounded on [0,1],

(1.2) lim

n→∞

Z 1 0

Sn,r,s(x)f(x)dx = gcd(r, s) prs(r+s)

Z 1 0

f(x)

p2πx(1−x)dx,

the latter integral generally being easier to evaluate (or approximate). This simple consequence of Theorem 1.1 plays an important role in assessing the performance of nonparametric density estimators based on Bernstein polynomials. Kakizawa [5], for example, went to great lengths to establish (1.2) for the caser=s= 1.

In establishing Theorem1.1, we first show that, for all0≤k ≤nandx∈[0,1], (see Corollary2.3)

(1.3) P_n,k(x)≥P_2n,2k(x)≥P_3n,3k(x)≥ · · · .

The proof of this inequality is based on a class of completely monotonic functions and hence is of general interest. Using completely different methods, Leblanc and Johnson [7] previously showed that{P₂^j_n,2^j_k(x)}^∞_j=0 is decreasing inj and hence (1.3) is a generalization of this earlier result.

The remainder of this paper is organized as follows. In Section 2we introduce a new family of completely monotonic functions and obtain some necessary combinatorial and probability inequalities. In Section3, we prove Theorem1.1. Finally, in Section4, we highlight the fact that the results in Section2can be used to obtain other interesting inequalities.

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2. Preliminary Results

Recall that a real valued function f is said to be completely monotonic on(a, b)if and only if(−1)ⁿf⁽ⁿ⁾(x) ≥ 0 for all x ∈ (a, b)and integers n ≥ 0 (cf. Feller [4, XIII.4]). We begin with the following lemma.

Lemma 2.1. Let{a_k}^m_k=1 and{b_k}^m_k=1 be real numbers such thata₁ ≥ a₂ ≥ · · · ≥ a_mandb₁ ≥b₂ ≥ · · · ≥b_m ≥0and letψ denote the digamma function. Define

φ_δ(x) :=

m

X

k=1

a_kψ(b_kx+δ), x >0, δ ≥0.

Ifδ ≥1/2andPm

k=1a_k ≥0, thenφ⁰_δ is completely monotonic on(0,∞)and hence φ_δis increasing and concave on(0,∞).

The proof follows along the same lines as that in Alzer and Berg [2], who show that φ₀ is completely monotonic (and hence decreasing and convex) if and only if Pa_k = 0andP

a_klnb_k≥0.

Proof. Letx > 0andδ ≥ 1/2and recall that the integral representation ofψ⁽ⁿ⁾ is (cf. Abramowitz and Stegun [1, pp. 260])

ψ⁽ⁿ⁾(x) = (−1)ⁿ⁺¹ Z ∞

0

tⁿe^−xt

1−e^−tdt, n = 1,2, . . . . Therefore, forn= 1,2, . . .,

(−1)ⁿ⁺¹φ⁽ⁿ⁾_δ (x) = (−1)ⁿ⁺¹

m

X

k=1

a_kbⁿ_kψ⁽ⁿ⁾(b_kx+δ) (2.1)

=

m

X

k=1

a_k Z ∞

0

(bkt)ⁿe^−xb^k^t e^δt(1−e^−t)dt.

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The substitution(s)u=b_ktyield (2.2) (−1)ⁿ⁺¹φ⁽ⁿ⁾_δ (x) =

Z ∞ 0

uⁿ⁻¹e^−ux

m

X

k=1

a_kη(u/b_k)du,

where η(x) = xe^−δx(1−e^−x)⁻¹ > 0. A little calculus shows that, forδ ≥ 1/2, η is strictly decreasing on(0,∞)and hence, for everyu > 0, {η(u/b_k)}^m_k=1 is decreasing [note that, if b_k = 0, there is no difficulty in taking η(u/b_k) = η(∞) = limx→∞η(x) = 0, since these terms vanish in (2.1)]. Since{a_k}^m_k=1 is also decreasing, Chebyshev’s inequality for sums yields

m

X

k=1

a_kη(u/b_k)≥ 1 m

m

X

k=1

a_k

! _m X

k=1

η(u/b_k)

! . We see that, if Pm

k=1ak ≥ 0, the integrand in (2.2) is non-negative and hence (−1)ⁿ⁺¹φ⁽ⁿ⁾_δ ≥ 0 on (0,∞). We conclude that φ⁰_δ is completely monotonic on (0,∞)and, in particular,φ_δis increasing and concave on(0,∞)wheneverδ≥1/2 andP

a_k ≥0.

Lemma 2.2. Letn, k, j be integers such that0≤k ≤nandj ≥1and define

Q_n,k(j) =

(j −1)n (j−1)k

jn

jk

= Γ((j−1)n+ 1)Γ(jk+ 1)Γ(j(n−k) + 1) Γ(jn+ 1)Γ((j −1)k+ 1)Γ((j−1)(n−k) + 1).

ThenQ_n,k(j)is decreasing inj and

j→∞lim Qn,k(j) = k

n k

n−k n

n−k

.

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Proof. The limit is easily verified using Stirling’s formula, thus we need only show thatQ_n,k(j)is decreasing inj. TreatingQ_n,k(j)as a continuous function in j and differentiating we obtain

Q⁰_n,k(j) = Q_n,k(j) (

k

q_j(k)−q_j(n)

+ (n−k)

q_j(n−k)−q_j(n) )

, whereq_j(x) = ψ(jx+ 1)−ψ(jx−x+ 1). Now, takingδ = 1,a₁ = 1,a₂ =−1, b₁ =j andb₂ =j−1in Lemma2.1, we have thatq_j(x)is increasing on(0,∞)and henceQ⁰_n,k(j)≤0for allj ≥1sinceQ_n,k(j)>0always.

Remark 1. In light of Lemma2.1, we may define, forj ≥1andδ >0, Q_n,k,δ(j) = Γ((j−1)n+δ)

Γ((j−1)n+δ) Γ((j−1)k+δ)

, Γ(jn+δ)

Γ(jk+δ) Γ(j(n−k) +δ). The same arguments in the proof of Theorem2.2 show thatQ_n,k,δ(j)is decreasing inj for allδ≥1/2and has the same limiting value of(k/n)^k(1−k/n)^n−k.

Corollary 2.3. Let 0 ≤ k ≤ n. Then {P_jn,jk(x)}^∞_j=1 is decreasing inj for every fixedx∈[0,1].

Proof. P(j−1)n,(j−1)k(x) ≥ Pjn,jk(x)if and only ifQn,k(j) ≥ x^k(1−x)^n−k and we have, by Lemma2.2,

Q_n,k(j)≥(k/n)^k(1−k/n)^n−k= sup

x∈[0,1]

x^k(1−x)^n−k, which completes the proof.

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3. Proof of Theorem 1.1

We now give a proof of Theorem1.1. First note that (i) holds since

n

X

k=0

Prn,rk(x)Psn,sk(x)≤

n

X

k=0

Prn,rk(x)≤

rn

X

k=0

Prn,k(x) = 1,

with equality if and only ifx = 0,1. Similarly, (ii) holds since{S_n,1,1}^∞_n=1 is uniformly integrable on [0,1] (cf. [6]) and, by Corollary 2.3, we have S_n,r,s(x) ≤ S_n,1,1(x)for allx∈[0,1].

To prove (iii), let U₁, . . . , U_n and V₁, . . . , V_n be two sequences of independent random variables such thatU_i is Binomial(r, x)andV_i is Binomial(s, x). Now, de- fineW_i =r⁻¹U_i−s⁻¹V_i so thatW_i has a lattice distribution with spangcd(r, s)/rs (cf. Feller [4]). We can writeS_n,r,s(x)in terms of theW_i as

Sn,r,s(x)

√n =

n

X

k=0

P_rn,rk(x)P_sn,sk(x) = P

n

X

i=1

Ui

r =

n

X

i=1

Vi

s

!

=P

n

X

i=1

W_i = 0

! .

Now, define the standardized variables W_i^∗ = W_i√ rs/p

(r+s)x(1−x) so that Var(W_i^∗) = 1 and note that these also have a lattice distribution, but with span gcd(r, s)/p

rs(r+s)x(1−x). Theorem 3 of Section XV.5 of Feller [4] now leads to

n→∞lim

S_n,r,s(x)

√n = lim

n→∞P

√1 n

m

X

i=1

W_i^∗ = 0

!

= gcd(r, s)φ(0) pnrs(r+s)x(1−x), whereφcorresponds to the standard normal probability density function. The result now follows from the fact thatφ(0) = 1/√

2π.

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4. Concluding Comments

We conclude by pointing out the fact that Lemma 2.2 also leads to some other interesting combinatorial and discrete probability inequalities. For example, since Q_n,k(j)is decreasing, we immediately obtain

(j −1)n (j−1)k

(j+ 1)n (j + 1)k

≥ jn

jk 2

.

Indeed, since Q_n,k(j −m+ 1) ≥ Q_n,j(j +m)for m = 1, . . . , j, we see that the sequence{A_m}^j_m=1 defined by

(4.1) A_m =

(j+m)n (j+m)k

(j−m)n (j−m)k

is increasing.

Finally, Corollary2.3trivially leads to a similar family of inequalities for “num- ber of failure” negative binomial probabilities. LetH_n,kbe the probability of exactly n failures(n ≥ 0)before thekth success(k ≥ 1)in a sequence of i.i.d. Bernoulli trials with success probabilityp∈[0,1]so that, forj = 1,2, . . .,

H_jn,jk=

jn+jk−1 jk−1

p^jk(1−p)^jn= k

n+kP_j(n+k),jk.

Hence, as a direct consequence of Corollary 2.3, we have that {H_jn,jk}^∞_j=1 is also decreasing.

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References

[1] M. ABRAMOWITZANDI.A. STEGUN, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Stan- dards, Applied Mathematics Series 55, 4th printing, Washington (1965).

[2] H. ALZERAND C. BERG, A class of completely monotonic functions, II, Ra- manujan J., 11(2) (2006), 225–248.

[3] G.J. BABU, A.J. CANTYANDY.P. CHAUBEY, Application of Bernstein poly- nomials for smooth estimation of a distribution and density function, J. Statisti- cal Planning and Inference, 105 (2002), 377–392.

[4] W. FELLER, An Introduction to Probability Theory and Its Applications. Vol- ume II, Second Edition. John Wiley and Sons, New York, (1971).

[5] Y. KAKIZAWA, Bernstein polynomial probability density estimation, J. Non- parametric Statistics, 11 (2004), 709–729.

[6] A. LEBLANC, A bias-corrected approach to density estimation using Bernstein polynomials, (submitted for publication).

[7] A. LEBLANC AND B.C. JOHNSON, A family of inequalities related to bino- mial probabilities. Department of Statistics, University of Manitoba. Tech. Re- port, 2006-03.

[8] R.A. VITALE, A Bernstein polynomial approach to density function estimation, Statistical Inference and Related Topics, 2 (1975), 87–99.