Theorem 1. 7. is coilsistent in MS if i. \( 1 \operatorname{ar}\left(1_{. .}\right) \rightarrow 0 \) as \( n \rightarrow \chi \).ard 2. Tn as unbrused or nsymptotically unduased. Example 11. Sirow Ihat. 1. \( X \) is consistent in MS for \( \mu \) 2. \( S^{-2} \) is conssstent in MS for \( \sigma^{2} \) where \( \mu \) and \( \sigma^{2} \) are the population uean and population variance respectivels. Rennar\% 1. Consistent in mean square implies comstent in probability, low the converse is not alsays the Theorem 2. If \( T_{n} \) is consastent in \( M S \) then it is consistent in probability. (A sufficient condition but not neressary)
Added by S. W.
Close
Step 1
Consistency in Mean Square (MS): A sequence of estimators is said to be consistent in mean square if the mean square error (MSE) of the estimators tends to zero as the sample size tends to infinity. The MSE is the sum of the variance of the estimator and the Show more…
Show all steps
Your feedback will help us improve your experience
Adi S and 97 other Physics 101 Mechanics educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
1- Let X1, X2, ... be an iid sequence of rv's each with finite mean μ, finite variance σ² > 0 and mgf M(t) defined for |t| < h with h > 0. (a) Define the sample variance Sₙ² and show that the sequence {Sₙ²} provides a consistent estimator for σ². (b) Show that for all t with 0 < |t| < h, M(t) > e^tμ. (c) Show that for |t| < h/2 the sequence {1/n ∑ᵢ₌₁ⁿ e^tXᵢ} converges in probability and the sequence {[nM(t) - ∑ᵢ₌₁ⁿ e^tXᵢ] / [√(n(M(2t) - M(t)²))]} converges in distribution. In each case, describe the limit.
Adi S.
Breanna O.
Suppose that Xi, Xz, Xn is a random sample from a Bernoulli distribution. For this distribution, E (X;) = 0 and V (X;) = 0(1 - 0). Let $} = Ex ~nX and $} S} (where X = Zx) be competing (n _ 1)2 (n _ I)(n _ 3) estimators of the unknown V(X;) = 0(1 - 0) . Given: V (S?) p4(0) - [0(1 _ 0)]2 n3 where /4(0) is the fourth central moment of the distribution, and hence does not depend on the sample size n_ (a) Prove that: (i) E (S}) = 0(1 _ 0); and (ii) E (S) = "= 40( _ 0) (b) Determine: V (X); and (ii) V (S3) _ (c) Determine: MSE (X); (ii) MSE (S}); and (iii) MSE (S) _ (d) Which is the better estimator of 0(1 = 0) between S? and S2 in terms of the mean square error? Justify your answer: (e) Suppose that $; is an estimator of 0(1 _ 0) based on a random sample of size n . Another equivalent definition of the consistency of S; as an estimator of 0(1 _ 0) is that lim MSE (S) = 0. V_OC Show that S} and S? are consistent estimators of 0(1
Recommended Textbooks
University Physics with Modern Physics
Physics: Principles with Applications
Fundamentals of Physics
Watch the video solution with this free unlock.
EMAIL
PASSWORD