On an inequality for the normal distribution arising in bioequivalence studies

For (μ,σ2) ≠ (0,1), and 0 < z < ∞, we prove that where φ and Φ are, respectively, the p.d.f. and the c.d.f. of a standard normal random variable. This inequality is sharp in the sense that the right-hand side cannot be replaced by a larger quantity which depends only on μ and σ. In other words, for any given (μ,σ) ≠ (0,1), the infimum, over 0 < z < ∞, of the left-hand side of the inequality is equal to the right-hand side. We also point out how this inequality arises in the context of defining individual bioequivalence.

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Some Inequalities for Stop-Loss Premiums

Astin Bulletin ◽

10.1017/s0515036100011405 ◽

1977 ◽

Vol 9 (1-2) ◽

pp. 75-83 ◽

Cited By ~ 64

Author(s):

H. Bühlmann ◽

B. Gagliardi ◽

H. U. Gerber ◽

E. Straub

Keyword(s):

Continuous Function ◽

Random Variable ◽

Left Hand ◽

Right Hand ◽

Image Position ◽

The Right ◽

Stop Loss

In this paper any given risk S (a random variable) is assumed to have a (finite or infinite) mean. We enforce this by imposing E[S−] < ∞.Let then v(t) be a twice differentiate function withand let z be a constant with o ≤ z ≤ 1.We define the premium P as followsor equivalentlyNotation: v−(∞) = ∞.The definitions (1) and (equivalently) (2) are meaningful because of theLemma: a) E[v(S − zQ)] exists for all Q∈(− ∞, + ∞).b) The set{Q∣ − ∞ < Q < + ∞, E[v(S−zQ)]>v((1−z)Q)} is not empty.Proof: a) b) Because of a) E[v(S−zQ)] is always finite or equal to + ∞ If v(− ∞) = − ∞ then E[v(S − zQ)] > v((1 − z)Q) is satisfied for sufficiently small Q. The left hand side of the inequality is a nonincreasing continuous function in P (strictly decreasing if z > 0), while the right hand side is a nondecreasing continuous function in Q (strictly increasing if z > 1).If v(− ∞) = c finite then E[v(S − zQ)] > c(otherwise S would need to be equal to − ∞ with probability 1) and again E[v(S − zQ)] > v((1 − z)Q) is satisfied for sufficiently small Q.

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Asymptotic Approximation of an Integral Involving the Normal Distribution*

Canadian Mathematical Bulletin ◽

10.4153/cmb-1986-028-x ◽

1986 ◽

Vol 29 (2) ◽

pp. 167-176 ◽

Cited By ~ 1

Author(s):

J. P. McClure ◽

R. Wong

Keyword(s):

Distribution Function ◽

Normal Distribution ◽

Positive Constant ◽

Asymptotic Approximation ◽

Cumulative Distribution Function ◽

Random Variable ◽

Normal Random Variable ◽

Cumulative Distribution ◽

Standard Normal Random Variable ◽

Standard Normal

AbstractAn asymptotic approximation is obtained, as k → ∞, for the integralwhere Φ is the cumulative distribution function for a standard normal random variable, and L is a positive constant. The problem is motivated by a question in statistics, and an outline of'the application is given. Similar methods may be used to approximate other integrals involving the normal distribution.

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A Fourth Century Head in Central Museum, Athens

The Journal of Hellenic Studies ◽

10.2307/624069 ◽

1895 ◽

Vol 15 ◽

pp. 194-201 ◽

Cited By ~ 1

Author(s):

E. F. Benson

Keyword(s):

Fourth Century ◽

Present Condition ◽

Left Hand ◽

Right Hand ◽

Image Position ◽

The Right

There is among the fourth century works in the Central Museum at Athens a head found at Laurium. It is made of Parian marble but it has been completely discoloured by slag or refuse from the lead mines, and is now quite black. In its present condition it is quite impossible to obtain a satisfactory photograph of it, and the reproduction given of it in the figure is from a cast.It has been published, as far as I am aware, only in M. Kavvadias' catalogue. There it is described as a head of the Lykeian Apollo. This identification rests solely on a passage of Lucian, who mentions a statue of the Lykeian Apollo in the gymnasium at Athens.He says of it ( 7)—It will be seen from a glance at the photograph that the grounds for this identification are very slender. The left hand with the bow does not exist, and the only reason for supposing therefore that this is a head of the Lykeian Apollo consists in the fact that the right hand of the statue rests on the head. This in itself seems insufficient and, among other reasons, it is I think rendered impossible by the phrase For the hand is not idly resting, it is not a tired hand; the posture of the fingers is firm and energetic.

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An inequality relating means

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100018417 ◽

1944 ◽

Vol 40 (3) ◽

pp. 253-255

Author(s):

J. Bronowski

Keyword(s):

Arithmetic Mean ◽

Arithmetic Means ◽

Geometric Means ◽

Left Hand ◽

Right Hand ◽

Image Position ◽

The Right

1. Let a, b be positive constants; and let y1, y2, …, yn be real exponents, not all equal, having arithmetic mean y defined by(here, and in what follows, the summation ∑ extends over the values i = 1, 2, …, n). Then it is clear thatsince the right-hand sides are the geometric means of the positive numbers whose arithmetic means stand on the left-hand sides. I know of no results, however, which relate the ratios and and I have had occasion recently to require such results. This note gives an inequality between these ratios, subject to certain restrictions on a and b.

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Precise Rates in Log Laws for NA Sequences

Discrete Dynamics in Nature and Society ◽

10.1155/2007/89107 ◽

2007 ◽

Vol 2007 ◽

pp. 1-11

Author(s):

Yuexu Zhao

Keyword(s):

Random Variables ◽

Random Variable ◽

Stationary Sequence ◽

Normal Random Variable ◽

Negatively Associated ◽

Standard Normal Random Variable ◽

Na Random Variables ◽

Standard Normal ◽

Na Sequences

LetX1,X2,…be a strictly stationary sequence of negatively associated (NA) random variables withEX1=0, setSn=X1+⋯+Xn, suppose thatσ2=EX12+2∑n=2∞EX1Xn>0andEX12<∞,if−1<α≤1;EX12(log|X1|)α<∞, ifα>1. We provelimε↓0ε2α+2∑n=1∞((logn)α/n)P(|Sn|≥σ(ε+κn)2nlogn)=2−(α+1)(α+1)−1E|N|2α+2, whereκn=O(1/logn)and N is the standard normal random variable.

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A New Fragment of Greek Music in Cairo

The Journal of Hellenic Studies ◽

10.2307/627422 ◽

1931 ◽

Vol 51 (1) ◽

pp. 91-100 ◽

Cited By ~ 2

Author(s):

J. F. Mountford

Keyword(s):

Extreme Right ◽

Bottom Edge ◽

Left Hand ◽

Right Hand ◽

Formed Part ◽

Image Position ◽

The Right

The papyrus numbered 59533 in the Catalogue Général of the Cairo Museum is a mere scrap (13 cm. × 12 cm.), on one side of which is written a fragmentary text with suprascript musical signs:The writing is along the fibres, that is to say, on the recto if the scrap originally formed part of a roll; the verso is blank; and the papyrus has been folded horizontally. The right-hand and the top seem to be the original edges of a sheet or roll; the left-hand is clearly defective, and though the bottom edge may be the original edge of a sheet, it is not the bottom of a roll. The writing is carried to the extreme right-hand edge, without a margin. Below the text there is some scribbling which seems to have no particular significance and is probably to be taken as a probatio pennae rather than as a signature.

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A Berry-Esseen bound for the lightbulb process

Advances in Applied Probability ◽

10.1017/s0001867800005176 ◽

2011 ◽

Vol 43 (03) ◽

pp. 875-898 ◽

Cited By ~ 1

Author(s):

Larry Goldstein ◽

Haimeng Zhang

Keyword(s):

Conditional Expectation ◽

Spectral Decomposition ◽

Random Variable ◽

Stein’S Method ◽

Normal Random Variable ◽

The Past ◽

Terminal Time ◽

Standard Normal Random Variable ◽

Standard Normal ◽

Size Bias

In the so-called lightbulb process, on daysr= 1,…,n, out ofnlightbulbs, all initially off, exactlyrbulbs, selected uniformly and independent of the past, have their status changed from off to on, or vice versa. WithXthe number of bulbs on at the terminal timen, an even integer, and μ =n/2, σ2= var(X), we have supz∈R| P((X- μ)/σ ≤z) - P(Z≤z) | ≤nΔ̅0/2σ2+ 1.64n/σ3+ 2/σ, whereZis a standard normal random variable and Δ̅0= 1/2√n+ 1/2n+ e−n/2/3 forn≥ 6, yielding a bound of orderO(n−1/2) asn→ ∞. A similar, though slightly larger bound, holds for oddn. The results are shown using a version of Stein's method for bounded, monotone size bias couplings. The argument for evenndepends on the construction of a variableXson the same space asXthat has theX-size bias distribution, that is, which satisfies E[Xg(X)] = μE[g(Xs)] for all bounded continuousg, and for which there exists aB≥ 0, in this caseB= 2, such thatX≤Xs≤X+Balmost surely. The argument for oddnis similar to that for evenn, but one first couplesXclosely toV, a symmetrized version ofX, for which a size bias coupling ofVtoVscan proceed as in the even case. In both the even and odd cases, the crucial calculation of the variance of a conditional expectation requires detailed information on the spectral decomposition of the lightbulb chain.

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