scholarly journals Order Statistics and Benford's Law

2008 ◽  
Vol 2008 ◽  
pp. 1-19 ◽  
Author(s):  
Steven J. Miller ◽  
Mark J. Nigrini
Keyword(s):  
Order Statistics ◽  
Taylor Series ◽  
Pareto Distribution ◽  
Taylor Series Expansion ◽  
Benford’S Law ◽  
Exponential Behavior ◽  
Second Derivatives ◽  
Benford's Law ◽  
Standard Exponential Distribution ◽  
Uniformly Bounded

Fix a baseB>1and letζhave the standard exponential distribution; the distribution of digits ofζbaseBis known to be very close to Benford's law. If there exists aCsuch that the distribution of digits ofCtimes the elements of some set is the same as that ofζ, we say that set exhibits shifted exponential behavior baseB. LetX1,…,XNbe i.i.d.r.v. If theXi's are Unif, then asN→∞the distribution of the digits of the differences between adjacent order statistics converges to shifted exponential behavior. If insteadXi's come from a compactly supported distribution with uniformly bounded first and second derivatives and a second-order Taylor series expansion at each point, then the distribution of digits of anyNδconsecutive differencesandallN−1normalized differences of the order statistics exhibit shifted exponential behavior. We derive conditions on the probability density which determine whether or not the distribution of the digits of all the unnormalized differences converges to Benford's law, shifted exponential behavior, or oscillates between the two, and show that the Pareto distribution leads to oscillating behavior.

Place probabilities in normal order statistics models for horse races

1981 ◽  
Vol 18 (04) ◽  
pp. 839-852 ◽  
Author(s):  
R. J. Henery
Keyword(s):  
Order Statistics ◽  
Series Expansion ◽  
Taylor Series ◽  
Upper Bound ◽  
Taylor Series Expansion ◽  
Distribution Functions ◽  
Normal Order ◽  
Increasing Failure Rate ◽  
Normal Populations ◽  
Independent Observations

Independent observations X 0, X 1…, X N+1 are drawn from each of N populations whose distribution functions F(x – θi ) have means θ i , 0 ≦ i < N, and we wish to calculate the probability P k;N that X 0 is the k th largest observation. For normal populations an approximation is given for P K;N based on a Taylor series expansion in the θ 's. If F(x) has an increasing failure rate, as is the case for the normal, an upper bound can be given for the ‘win' probability P 1;N Some moment relations for normal order statistics are also given.

Place probabilities in normal order statistics models for horse races

1981 ◽  
Vol 18 (4) ◽  
pp. 839-852 ◽  
Author(s):  
R. J. Henery
Keyword(s):  
Order Statistics ◽  
Series Expansion ◽  
Taylor Series ◽  
Upper Bound ◽  
Taylor Series Expansion ◽  
Distribution Functions ◽  
Normal Order ◽  
Increasing Failure Rate ◽  
Normal Populations ◽  
Independent Observations

Independent observations X0, X1…, XN+1 are drawn from each of N populations whose distribution functions F(x – θi) have means θ i, 0 ≦ i < N, and we wish to calculate the probability Pk;N that X0 is the k th largest observation. For normal populations an approximation is given for PK;N based on a Taylor series expansion in the θ 's. If F(x) has an increasing failure rate, as is the case for the normal, an upper bound can be given for the ‘win' probability P1;N Some moment relations for normal order statistics are also given.

Investigation of Accuracy of a Calibrated Estimator of a Ratio by Modelling

2005 ◽  
Vol 10 (4) ◽  
pp. 333-342
Author(s):  
V. Chadyšas ◽  
D. Krapavickaitė
Keyword(s):  
Mean Square Error ◽  
Taylor Series ◽  
Random Sampling ◽  
Finite Population ◽  
Taylor Series Expansion ◽  
Simple Random Sampling ◽  
Population Parameter ◽  
Mean Square ◽  
Parameter Ratio ◽  
Using Data

Estimator of finite population parameter – ratio of totals of two variables – is investigated by modelling in the case of simple random sampling. Traditional estimator of the ratio is compared with the calibrated estimator of the ratio introduced by Plikusas [1]. The Taylor series expansion of the estimators are used for the expressions of approximate biases and approximate variances [2]. Some estimator of bias is introduced in this paper. Using data of artificial population the accuracy of two estimators of the ratio is compared by modelling. Dependence of the estimates of mean square error of the estimators of the ratio on the correlation coefficient of variables which are used in the numerator and denominator, is also shown in the modelling.

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