William Feller's An Introduction to Probability Theory and Its Applications, PDF

By William Feller

ISBN-10: 0471257087

ISBN-13: 9780471257080

Significant alterations during this version comprise the substitution of probabilistic arguments for combinatorial artifices, and the addition of recent sections on branching tactics, Markov chains, and the De Moivre-Laplace theorem.

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Additional info for An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition

Example text

Rao showed little interest and asked him to approach Professor D. Basu. Basu immediately found the role of “sufficiency” in this context. It is thus that “sufficiency” came to be appreciated in the context of finite population sampling. Basu (1958), as a matter of fact, observed that while y¯ = n1 nK=1 yK with yK as the y - value for the unit chosen on the Kth draw out of n draws by SRSWR from a population U = (1, . . , i, . . , N ) has 1 σ2 2 ,σ = Ep (¯ y ) = Y¯ and Vp (¯ y) = n N N 1 1 (yi − Y¯ )2 , y¯v = v yi , i∈sv the sample mean of the v(1 ≤ v ≤ n) distinct units in the part sv of the set of v distinct units in the sample s found in n draws has Ep (¯ yv ) = Y¯ and Vp (¯ yv ) = E Writing S2 = 1 N −1 1 v − 1 N (yi − −Y )2 , Q = N σ2 .

Every statistic t = t (d) which is a function of d has the effect of Partitioning the Data Space into Partition Sets which are “mutually non-overlapping” with their union coinciding with the Data Space. If t1 and t2 are two statistics, such that every partition set induced by t1 is contained in at least one partition set induced by t2 , then t2 induces a thicker partitioning while t1 induces a thinner partitioning than t2 . ’ A desirable statistic is one that induces the thickest partitioning without losing any relevant information, A sufficient statistic sacrifices no information of relevance.

As soon as one tries to appy Basu’s proof starting with an HLUE one must fail because it cannot be applied without crossing the limit of a homogeneous class which rules out an added term to an HLUE involving Y − free constant ingredients. Now let us show that a Uni-cluster Design admits a UMV member in the HLUE class of estimators for Y . For this we need to first enunciate and prove a Complete Class Theorem below invoking certain virtues of sufficient statistics in the context of survey sampling we have already discussed.

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An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition by William Feller


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