Let Y1, Y2, …, Yn denote a random sample from the...

Let $Y_{1}, Y_{2}, \ldots, Y_{n}$ denote a random sample from the probability density function $$f(y | \theta)=\left\{\begin{array}{l} \theta y^{\theta-1} \\0\end{array}\right.$$, $$0<y<1, \theta>0$$, elsewhere. a. Show that this density function is in the (one-parameter) exponential family and that $\left.\sum_{i=1}^{n}-\ln \left(Y_{i}\right) \text { is sufficient for } \theta \text { . (See Exercise } 9.45 .\right)$ b. If $W_{i}=-\ln \left(Y_{i}\right),$ show that $W_{i}$ has an exponential distribution with mean $1 / \theta$. c. Use methods similar to those in Example 9.10 to show that $2 \theta \sum_{i=1}^{n} W_{i}$ has a $\chi^{2}$ distribution with 2n df. d. Show that $$E\left(\frac{1}{2 \theta \sum_{i=1}^{n} W_{i}}\right)=\frac{1}{2(n-1)}$$. [Hint: Recall Exercise 4.112.] e. What is the MVUE for $\theta ?$

Let $Y_{1}, Y_{2}, \ldots, Y_{n}$ denote a random sample from the probability density function $$f(y | \theta)=\left\{\begin{array}{l}
\theta y^{\theta-1} \\0\end{array}\right.$$, $$0<y<1, \theta>0$$, elsewhere.
a. Show that this density function is in the (one-parameter) exponential family and that $\left.\sum_{i=1}^{n}-\ln \left(Y_{i}\right) \text { is sufficient for } \theta \text { . (See Exercise } 9.45 .\right)$
b. If $W_{i}=-\ln \left(Y_{i}\right),$ show that $W_{i}$ has an exponential distribution with mean $1 / \theta$.
c. Use methods similar to those in Example 9.10 to show that $2 \theta \sum_{i=1}^{n} W_{i}$ has a $\chi^{2}$ distribution with 2n df.
d. Show that $$E\left(\frac{1}{2 \theta \sum_{i=1}^{n} W_{i}}\right)=\frac{1}{2(n-1)}$$. [Hint: Recall Exercise 4.112.]
e. What is the MVUE for $\theta ?$

Key Concepts

Minimum Variance Unbiased Estimator (MVUE)

The MVUE is an estimator that, among all unbiased estimators, has the smallest variance. The existence of an MVUE is often guaranteed under conditions using methods such as the Lehmann–Scheffé theorem, which relies on the presence of a complete sufficient statistic. Finding the MVUE involves deriving an estimator from the sufficient statistic that is unbiased for the parameter of interest and exhibits the lowest possible variance.

Chi-Squared Distribution

The chi-squared distribution is a special case of the gamma distribution that is widely used in hypothesis testing and in the construction of confidence intervals. It arises naturally when summing the squares of independent standard normal random variables and can be obtained through suitable scaling of gamma-distributed variables. Understanding its properties is essential for constructing test statistics in many inferential procedures.

Gamma Distribution

The gamma distribution generalizes the exponential distribution and arises as the sum of independent, exponentially distributed random variables. With shape and scale parameters, it has a flexible form that can model a variety of data behaviors. Recognizing that the sum of exponentials follows a gamma distribution is crucial in deriving results that involve the sum of transformed observations, such as relating them to a chi-squared distribution.

Exponential Family

A family of distributions is said to be in the exponential family if its probability density (or mass) function can be written in the form f(y|?) = h(y) exp{?(?)T(y) ? A(?)}, where ?(?) is the natural parameter, T(y) is the sufficient statistic, and A(?) is the log-partition function. Recognizing that a density belongs to this family allows one to exploit many desirable statistical properties such as the existence of sufficient statistics that are functions of the data, conjugate priors in Bayesian analysis, and simplified estimation procedures.

Transformation of Variables

Transforming a random variable involves applying a function to it in order to derive a new variable whose distribution may be easier to work with. In this problem, the transformation from Y to W = -ln(Y) changes the original density into one that follows an exponential distribution. Such transformations are commonly used in statistical inference to obtain tractable forms or to match standard distributions with known properties.

Sufficient Statistic

A statistic is sufficient for a parameter if it captures all the information in the sample that is relevant for inference about that parameter. In the context of the exponential family, the statistic T(y) appears naturally in the exponent of the density function and is sufficient due to the factorization theorem. This concept simplifies data analysis by reducing the dimensionality of the data without sacrificing any information about the parameter of interest.

Exponential Distribution

The exponential distribution is a continuous probability distribution often used to model the time between independent events that occur at a constant average rate. It is characterized by a memoryless property and has the density function f(w) = ? exp(-?w) for w ? 0, with mean 1/?. Its simple form and well-understood properties make it a common building block in reliability theory and survival analysis.

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