1. Let T-Σ-iz, where Z1,Zo, replacement from the set {1,2,... , N Show that ,Žm are...
Suppose that X = (Xi, X2, , X.) and Y-X,,Y2, , Ym) are random samples from continuous distributions F and G, respectively.Wilcoxon's two-sample test statistic W - W(X, Y) is defined to beRi where R, is the rank of Y in the combined sample 1. Let T Σǐn i Zi where Zi,Z2, ,Zm are numbers sampled at random without replacement from the set {1,2,..., N} Show that E(Z) = (N + 1)/2 and hence E(T) m(N + 1)/2 Show that...
Hello can someone help me answer this please Suppose that X (X1,X2, . . . , X.) ald Y-(Yİ,%, , Y,n) are random samples from continuous distributions F and G, respectively. Wilcoxon's two-sample test statistioc W- W(X,Y) is defined to be X/Ri where Ri is the rank of Y in the combined sample. 1. Let T Z, where Z, Z2,, Zm are numbers sampled at random without replacement from the set {1,2,...,N) Show that E(Z) = (N + 1)/2 and...
Suppose that X = (Xi, X2, . . . , Xn) and Y = (y,Y2, . . . ,Yn) are random samples from continuous distributions F and G, respectively. Wilcoxon's two-sample test statistic W = W(X,Y) is defined to be Σ-ngi Ri where Ri is the rank of in the combined sample. 2 where U is the number of pairs (Xi,Y) with Xiくy, In other words n m U=ΣΣΊ, , where 1,,-ĺ0 otherwise. i,ji 3. Continuing from Question 2 show...
1. You have two independent samples, Xi,..., Xn and Yi,... , Ym drawn from populations with continuous distributions. Suppose the two samples are combined and the combined set of values are put in increasing order. Let Vr- 1 if the value with rank r in the combined sample is a Y and V,-0 if it is an X, for r-1, . . . ,N, where N-m+ n. Show that, if the two populations are the same then mn E(V) TES...
1. You have two independent samples, X1,... , Xn and Y,... , Ym drawn from populations with continuous distributions. Suppose the two samples are combined and the combined set of values are put in increasing order. Let Vr-1 if the value with rank r in the combined sample is a Y and V0 if it is an X, for r-1...., N, where N-m+n Show that, if the two populations are the same then mn The general linear rank statistic is...
1, and let σ be a permutation of {1, , n). Recall that for each integer m a) Let n 1, we denote ơm--σ ο . . . o σ. Show that n times b) Let 21, and let be a permutation of..,n consisting of a unique cycle of length n. Deduce from the previous question that there exists i e (1,..., n) such that i +c() )+22(n1). 1, and let σ be a permutation of {1, , n). Recall...
. Let Yı, . . . , Ý, be a sample from N(0, σ*) distribution. Show that both Gi (Yi, . . . , X,; σ) = nHare pivots. j-1 72 and G (1) Recall the confidence interval based on Gi that we derived in class. (2) Let Z be N(0, 1) random variable. Find the expectation and variance of |Z. (3) If n is large, what is the approzimate distribution of (4) Use (3) to construct an approximate confidence...
Question 5 Suppose we have the following two samples , rini from No(21, Σ), Sample l: r1 1, Sample 2: T21 , . . . , z2n2 from MgWa, Σ 2). Two new = C2, + d for all 1-1,2 and j = 1, 2, . . . ,n, where C is a p x p nonsingular matrix and d is a p x 1 vector. Based on Samples and 2, the T2-statistic for testing μι μ2 is denoted as...
Please show every step, thank you. Let Xi ~ N(μ, σ?), where ơỈ are known and positive for i-1, are independent. Let /- (a) Find the mean and variance of μ. (b) Compare μ to X,-n-Σί.i Xi as an estimator of μ. , n, and Xi, X, , E-1(1/o .m be the MLE of μ. Let Xi ~ N(μ, σ?), where ơỈ are known and positive for i-1, are independent. Let /- (a) Find the mean and variance of μ....
Suppose that X1, X2, . . . , Xn is an iid sample of N (0, σ2 ) observations, where σ 2 > 0 is unknown. Consider testing H0 : σ 2 = σ 2 0 versus H1 : σ 2 6= σ 2 0 ; where σ 2 0 is known. (a) Derive a size α likelihood ratio test of H0 versus H1. Your rejection region should be written in terms of a sufficient statistic. (b) When the null...