Let be independent random variables, where ~, Is sufficient for ?
Here,
So,
So,
Now, Joint PDF of X1,X2,.....,Xn is given by:
This can be written as:
where is independent of
Thus, by Fisher Neyman criteria, is sufficient for .
Let be independent random variables, where ~, Is sufficient for ? We were unable to transcribe...
Let be independent random variables, where ~, Find a sufficient statistic for . We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Let be independent random variables, where ~, . Find a sufficient statistics for . We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageuni form(i) We were unable to transcribe this imageWe were unable to transcribe this image
Let be a sequence of independent random variables with and . Show that in probability, We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
3. Let ,..., be independent random sample from N(), where is unknown. (i) Find a sufficient statistic of . (ii) Find the MLE of . (iii) Find a pivotal quantity and use it to construct a 100(1–)% confidence interval for . We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable...
Let be a sequence of random variables, and let Y be a random variable on the same sample space. Let An(ϵ) be the event that |Yn − Y | > ϵ. It can be shown that a sufficient condition for Yn to converge to Y w.p.1 as n → ∞ is that for every ϵ > 0, (a) Let be independent uniformly distributed random variables on [0, 1], and let Yn = min(X1, . . . , Xn). In class,...
Let , ... be independent random variables with mean zero and finite variance. Show that We were unable to transcribe this imageWe were unable to transcribe this image
Let be independent, identically distributed random variables with . Let and for , . (a) Show that is a martingale. (b) Explain why satisfies the conditions of the martingale convergence theorem (c) Let . Explain why (Hint: there are at least two ways to show this. One is to consider and use the law of large numbers. Another is to note that with probability one does not converge) (d) Use the optional sampling theorem to determine the probability that ever attains...
Let , be independent N(0,1) distributed random variables. Define and . Without using calculus, show that . We were unable to transcribe this imageWe were unable to transcribe this imageW1 = x + x x1 - x x} + Xž We were unable to transcribe this image
Let be a random sample from , where is an unknown parameter. Show that is a sufficient statistics for , where is the sample variance. We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image2 We were unable to transcribe this imageWe were unable to transcribe this image
Suppose is a random sample from , where and . (a) Find a minimal sufficient statistic for . (b) Find a complete statistic for . (c) Show that is independent of , where . 7l We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageに! Х-Л. We were unable to transcribe this image