tailieunhanh - Handbook of mathematics for engineers and scienteists part 161

Handbook of mathematics for engineers and scienteists part 161. Tài liệu toán học quốc tế để phục vụ cho các bạn tham khảo, tài liệu bằng tiếng anh rất hữu ích cho mọi người. | 1088 Mathematical Statistics . Sample variances. The statistic 1 n a2 a2 - m 2 - V X -m 2 n i 1 is called the sample variance variance of the empirical distribution of the sample X1 . Xn. Suppose that ak E X and ik E X1 - a1 k . If a4 to then the sample variance has the properties of asymptotic unbiasness and consistency. The statistic 1n s2 H - m 2 V Xi - m 2 s 2 1 i 1 is called the adjusted sample variance and s is called the sample mean-square deviation of the sample X1 . Xn. They satisfy the formulas E a2 n-1 a2 E s2 a2 Var s2 - - 3a4Y n n n-4 The statistic s2 is an unbiased estimator of the variance i2- . Characteristics of asymmetry and excessî The number Yi 71 72 -17 Xi - m 3 u2 3 2 na3 is called the sample asymmetry coefficient of a random sample X1 . Xn. The number Y - 3 -L Xi - m 4 - 3 u 2 na4 r 2 i 1 is called the sample excess coefficient of a random sample X1 . Xn. The sample excess coefficient is used in criteria for testing the hypothesis H0 Y2 0 which implies that the distribution of the random variable Xi differs from the normal distribution. The statistics Y1 and Y2 are consistent estimators of asymmetry and excess. . Statistical Estimation . Estimators and Their Properties . Notion of estimator. A statistical estimator 0 or simply an estimator of an unknown parameter 0 for a sample X1 . Xn is a function 0 0 X1 . Xn depending only on the sample X1 . Xn. An estimator is a random variable and varies depending on the sample. Just as any random variable it has a distribution function Eg x . The distribution law of the statistic 0 0 X1 . Xn can be found by well-known methods of probability theory. Estimators producing separate points in the space of parameters or parametric functions to be estimated are called point estimators. Estimators producing sets of points in the space of parameters or parametric functions to be estimated are called interval estimators. .