Let X1, . . . , Xn be a random sample from the uniform distribution on the interval (θ, θ + 1), θ > 0. Find a sufficient statistic for θ.
Let X1, . . . , Xn be a random sample from the uniform distribution on...
Let X1, . . . , Xn be a random sample from the discrete uniform distribution on 1, 2, . . . , θ. Using the definition of sufficient statistic, show that X(n) is a sufficient statistic for θ.
2. Let X1, X2,. ., Xn be a random sample from a uniform distribution on the interval (0-1,0+1). . Find the method of moment estimator of θ. Is your estimator an unbiased estimator of θ? . Given the following n 5 observations of X, give a point estimate of θ: 6.61 7.70 6.98 8.36 7.26
Let X1, . . . , Xn be a random sample from a population with density 8. Let Xi,... ,Xn be a random sample from a population with density 17 J 2.rg2 , if 0<、〈릉 0 , if otherwise ( a) Find the maximum likelihood estimator (MLE) of θ . (b) Find a sufficient statistic for θ (c) Is the above MLE a minimal sufficient statistic? Explain fully.
7.6.4. Let X1, X2,... , Xn be a random sample from a uniform (0,) distribution. Continuing with Example 7.6.2, find the MVUEs for the following functions of (a) g(0)-?2, i.e., the variance of the distribution (b) g(0)- , i.e., the pdf of the distribution C) or t real, g(9)- , î.?., the mgf of the distribution. Example 7.6.2. Suppose X1, X2,... , Xn are iid random variables with the com- mon uniform (0,0) distribution. Let Yn - max{X1, X2,... ,...
Let X1,...,xn be a random sample from uniform distribution on the interval (0,). Find the method of moments estimator of . 273X 2X ох none of the answers provided here
Let X1, ..., Xn be IID observations from Uniform(0, θ). T(X) = max(X1, . . . Xn) is a sufficient statistic (additionally, T is the MLE for θ). Find a (1 − α)-level confidence interval for θ. [Note: The support of this distribution changes depending on the value of θ, so we cannot use Fisher’s approximation for the MLE because not all of the regularity assumptions hold.]
suppose X1 -> Xn is a random sample from a uniform distribution on the interval [0,theta]. let X1 = min {X1,X2,...Xn} and let Yn= nX1. show that Yn converges in distribution to an exponential random variable with mean theta.
3. Let X1, X2, . . . , Xn be a random sample from a distribution with the probability density function f(x; θ) (1/02)Te-x/θ. O < _T < OO, 0 < θ < 00 . Find the MLE θ
Let X1, ..., Xn be a random sample from a distribution with pdf 2πσχ (a) If σ and μ are both unknown, find a minimal sufficient statistic T. (b) If σ is known and μ is unknown, is T from last part a sufficient statistic? Is it a minimal sufficient statistic? Prove your answer. (c) Let V (II1 X)/m, what is the distribution of V? Are V andindependently distributed? Let X1, ..., Xn be a random sample from a distribution...
5. Suppose that X1, X2, , Xn s a random sample from a uniform distribution on the interval (9,8 + 1). (a) Determine the bias of the estimator X, the sample mean. (b) Determine the mean-square error of X as an estimator of θ. (c) Find a function, a, of that is an unbiased estimator of θ. Determine the mean-square error of θ.