Approximation Theorems of Mathematical Statistics (Wiley by Robert J. Serfling

By Robert J. Serfling

This paperback reprint of 1 of the simplest within the box covers a large variety of restrict theorems invaluable in mathematical facts, in addition to tools of facts and methods of program. The manipulation of "probability" theorems to acquire "statistical" theorems is emphasised.

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And n = 1,2,. . Assume that the limits ak = limn ap) exist (finite), each k. Then (i) the limits { a k }are the moments o f a distributionfunction F; (ii) Vthe F gioen by (i) is unique, then F, =+F. 4. This result provides a convergence of moments criterion for convergence in distribution. 13, which provides conditions under which the moments {ak}determine a unique F. The following result, due to Scheff6 (1947) provides a convergence of densities criterion. ) Theorem C (Scheffk). ) be a sequence of densities of absolutely continuous distribution functions, with limn f,(x) = f(x), each real x.

The following result is quite useful. Theorem (Pblya), f'f F, * F and F is continuous, then lim supIF,(t) ,-+a - F(t)I = 0. 12). For generalities, see Ranga Rao (1962). 4 Convergence in Distribution for Perturbed Random Variables A common situation in mathematical statistics is that the statistic of interest is a slight modification of a random variable having a known limit distribution. A fundamental role is played by the following theorem, which was developed by Slutsky (1925) and popularized by CramCr (1946).

7975. On the other hand, Esden (1956) has determined the following “asymptotically best ’* constant: More generally, independent summands not necessarily identically distributed are also treated in Berry and Essten’s work. For this case the righthand side of (*) takes the form where C is a universal constant. ) summands, has recently been carried out by Landers and Rogge (1976). For t sufficiently large, while n remains fixed, the quantities G,(t) and @(t) each become so close to 1 that the bound given by (*) is too crude.

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