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Suppose that a statistic U has a probability density function that is positive over the interval a ≤ u ≤ b and suppose that the density depends on a parameter θ that can range over the interval α1 ≤ θ ≤ α2. Suppose also that g(u) is continuous for u in the interval [a, b]. If E[g(U) | θ] = 0 for all θ in the interval [α1, α2] implies that g(u) is identically zero, then the family of density functions { fU (u | θ), α1 ≤ θ ≤ α2} is said to be complete. (All statistics that we employed in Section 9.5 have complete families of density functions.) Suppose that U is a sufficient statistic for θ, and g1(U) and g2(U) are both unbiased estimators of θ. Show that, if the family of density functions for U is complete, g1(U) must equal g2(U), and thus there is a unique function of U that is an unbiased estimator of θ. Coupled with the Rao-Blackwell theorem, the property of completeness of fU (u | θ), along with the sufficiency of U, assures us that there is a unique minimum-variance unbiased estimator (UMVUE) of θ.
You are planning to test a vaccine for a virus that has no vaccine. Since the disease is usually not serious, you will expose 100 volunteers to the virus. After some time, you will record whether or not each volunteer has been infected.
A client has an investment portfolio with mean value is equal to 4500 USD, with a standard deviation of 300 USD. What is the probability that the value of her portfolio is between 3900 USD AND 5400 USD?
The resort's historical climate data suggests that the probability of a good winter with lots of snow during any given year is 40%.
a. What is meant by the trend component of a time series? How is a linear trend different from a nonlinear trend? b. Determine the actual percentage growth in the median selling price between the base year and year 10.
What is the probability that my sister has blue eyes? (As stated in Problem 42, an individual who receives a blue-eyed gene from each parent will have blue eyes, whereas one who receives one blue-eyed and one brown-eyed gene will have brown eyes.)
Show that when n = 2, the multiplicative utility function is equivalent to the two- attribute multilinear utility function. (Hint: This is an algebra problem. You must show that one utility function is a scaled version of the other.
suppose the conditional distribution of y given x x is gamma with ? 4 and ? 1x and the marginal distribution of x is
What is the difference between an independent and a dependent variable? Does a regression model imply causation? Why or why not?
a directory of personal computer retail outlets in the united states contains 12080 alphabetized entries. explain how
Beginning in March 1981, R. Foster Winans was a Wall Street Journal reporter and one of the writers of the "Heard on the Street" column (the "Heard" column), a widely read and influential column in the Journal.
a mean of 20 minutes and a standard deviation of 3 minutes. how long does it take the slowest 10% of the students to walk from the dorms to the chemistry lab.
in a normal distribution what z-score value separates the highest 40 of the scores from the rest of the
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