Estimation of the Mean and Standard Deviation by Order Statistics

Sarhan, A. E.

doi:10.1214/aoms/1177728788

Cited by 78 publications

(26 citation statements)

References 7 publications

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Order By: Relevance

“…The symmetric matrix V, given below, is the variance matrix of the ordered observations in samples of size n from an exponential distribution as given in Sarhan [7]. The inverse is required when estimation of population parameters is desired.…”

Section: Examplesmentioning

confidence: 99%

Matrix Inversion, Its Interest and Application in Analysis of Data

Greenberg¹,

Sarhan²

1959

Journal of the American Statistical Association

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Matrix inversion is used in the least squares analysis of data to estimate parameters and their varianoea and covariances. When the data come from the analysis of variance, analysis of covariance, order statistics, or the fitting of response-surfaces, the matrix to be inverted usually falls into a structured pattern that simplifies its inversion.One class of patterned matrices is characterized by non-eingular symmetrical arrangemepts in which linear combinations of the first (r -1) rows provide the right-hand portion, starting with the elements on the principal diagonal, of the rth and remaining rows. That is: for r ;ai ~j, with v;; =v;; for all i ~j. The inverses of matrices of this clase contain a non-null principal diagonal, and immediately adjacent to the principal diagonal, (r -1) non-niJll superdiagonals and (r -1) non-null subdiagonals. All other elements are zero. These inverses are called diagonal matrices of type r. That is, a matrix is diagonal of type r if a;, ~o for I i -il ~rand a;; ~a# for all i ~j. When r ~2, the inverse is easily written in terms of BH and Vif· A general procedure for obtaining the inverse when r =3 is given.The results for r = 2 are illustrated by a problem in order statistics using the two-parameter exponential distribution.Patterned matrices also are amenable to partitioning and this is another convenient device to find the exact inverse quickly. In fitting a response-surface to data, for example, a complicated-looking matrix can be abbreviated and simplified by selective partitioning. When this has been done, the exact inverse can be found by equating the product of the matrix and its inverse to the elements of the identity matrix. This procedure is illustrated with some data from a problem in the fitting of a respon&e-surface.When the matrix has no special pattern, as in the usual regression problem, the recommended procedure for matrix inversion is the modified square root method.

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Section: Examplesmentioning

confidence: 99%

Matrix Inversion, Its Interest and Application in Analysis of Data

Greenberg¹,

Sarhan²

1959

Journal of the American Statistical Association

Self Cite

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show abstract

“…I N RECENT literature, linear combinations of the sample ordered values are used to provide estimates of the parameters of certain distributions [3,4,5,10]. These estimates and the general class of statistics which are derived from order statistics are termed systematic by Mosteller [6].…”

Section: Introductionmentioning

confidence: 99%

“…These estimates and the general class of statistics which are derived from order statistics are termed systematic by Mosteller [6]. The efficiencies of these estimates and of some other linear estimates have been discussed for certain symmetric distributions [3,7,10].…”

Section: Introductionmentioning

confidence: 99%

“…It has been shown, in the case of the rectangular population [5,10], that the two extreme values (with equal numerical weights) provide us with the best linear estimate of the mean and standard deviation. In Sections 2 and 3 below it is shown that the two extreme sample elements will have the largest weights in the best linear estimates of the mean and standard deviation of a given skewed distribution with finite range.…”

Section: Introductionmentioning

confidence: 99%

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