Question

Let \(X_{1}, X_{2}, \ldots, X_{n}\) be a random sample from a distribution with pdf \(f(x ; \theta)=\) \(\theta x^{\theta-1}, 0

Short answer

The complete sufficient statistic for \(\theta\) is \(\ln(X_1 X_2...X_n)\). The sequential probability ratio test for \(H_0: \theta=2\) against \(H_1: \theta=3\) involves calculating the likelihood ratio \(\frac{L(\theta = 2)}{L(\theta = 3)}\) and comparing it to certain thresholds based on error rates \(\alpha\) and \(\beta\).

Step-by-step solution

Expression of the joint pdf

We begin by writing the probability density function of the \(n\) i.i.d random variables, \(X_1, X_2, ..., X_n\) given by the equation: \(f(x_1, x_2, ..., x_n ; \theta) = \theta^n (x_1 x_2 ... x_n)^{\theta -1}\)

Identification of the Sufficient Statistic

By the Factorization Theorem, we note that the second part of our pdf can be treated as a function of \(\theta\). Thus, we have \(\ln(X_1 X_2...X_n)\) as a complete sufficient statistic for \(\theta\)

Setting up the Hypothesis to be Tested

For part (b), we have two hypotheses \(H_0 : \theta = 2\) and \(H_1 : \theta = 3\), able to define \(\alpha\) and \(\beta\) as \(\frac{1}{10}\). In this case, \(\alpha\) and \(\beta\) refer to the type I and type II errors, respectively.

Performing the Sequential Probability Ratio Test (SPRT)

The expression for the Sequential Probability Ratio Test (SPRT) is obtained as \(\frac{L(\theta = 2)}{L(\theta = 3)}\), where \(L(\theta)\) denotes the likelihood function of \(\theta\). We stop taking samples when this ratio exceeds \(B = \frac{1-\beta}{\alpha}\) or falls below \(A = \frac{\beta}{1 - \alpha}\). We reject \(H_0\) when the ratio falls below \(A\), and we accept \(H_0\) when the ratio crosses \(B\).

Key concepts

Complete Sufficient Statistic

In the realm of statistics, a complete sufficient statistic is a powerful concept that simplifies the process of estimating parameters in a probability distribution. It consolidates all the relevant information from the sample data into a single statistic that's sufficient for making estimations about the population parameter. Completeness means that no other statistic that can be calculated from the sample provides any additional information about the parameters of the probability distribution.

A statistic 'T' is sufficient for a parameter \theta if the conditional distribution of the data given the statistic T does not depend on the parameter \theta. It's complete if for every measurable function g, the equation E(g(T)) = 0 implies that P(g(T) = 0) = 1 for all values of the parameter, assuming E(g(T)) exists. In the given exercise, the logarithm of the product of the sample data, \( ln(X_1 X_2...X_n) \), is shown to encapsulate all necessary information about the parameter \theta, hence it's a complete sufficient statistic.

Probability Density Function

The probability density function (pdf) is an equation that represents the relative likelihood of a continuous random variable to take on a given value. It's fundamental in the field of statistics as it describes the distribution of random variables in a continuum. The pdf is a non-negative function, and the area under the curve between two values corresponds to the probability that the random variable falls within that interval.

For our specific exercise, the pdf is given by \( f(x ; \theta)=\theta x^{\theta-1} \), but only for the interval \(0 < x < 1\); the probability is zero elsewhere. This particular function tells us how the probability is distributed across the interval \(0, 1\) for different values of the parameter \theta.

Likelihood Function

The likelihood function plays a central role in statistical inference, especially in parameter estimation. It is defined for a set of parameter values given a sample of data and is derived from the probability density function. However, instead of the variables, the likelihood function treats the sample data as given and the parameters as variables.

This function is useful for finding the parameter values that maximize the likelihood of obtaining the observed data. In the context of the provided exercise, the ratio of the likelihood functions for \( \theta = 2 \) and \( \theta = 3 \) is used in the Sequential Probability Ratio Test to decide whether to reject the null hypothesis \(H_0: \theta=2\) or not.

Type I and Type II Errors

Making a decision in hypothesis testing can lead to two kinds of errors:

• Type I error: This occurs when the null hypothesis is true, but we incorrectly reject it. It is denoted by \( \alpha \) and is known as the level of significance of the test.
• Type II error: This happens when the null hypothesis is false, but we fail to reject it. It is represented by \( \beta \) and relates to the power of the test, which is \(1 - \beta\).

The chosen values of \( \alpha \) and \( \beta \) can affect the conclusion of our hypothesis test and are predetermined based on the desired confidence level of the test. In the problem, both \( \alpha \) and \( \beta \) are set to \( \frac{1}{10} \) implying a 90% confidence level for both avoiding a Type I error and correctly rejecting a false null hypothesis.

Factorization Theorem

The factorization theorem provides a method to find a sufficient statistic for a parameter \( \theta \). It asserts that a statistic T(X) is sufficient for \( \theta \) if and only if the joint pdf or pmf of the sample \( X \) can be factorized into two functions, such that:

• One function \( h(x) \) depends only on the data X and not on \( \theta \) (and is not affected by different values of \( \theta \) ).
• The other function \( g(T(x), \theta) \) depends on the sample data only through the statistic T(X) and the parameter \( \theta \) .

In simpler terms, if the likelihood of the sample data can be expressed as the product of these two functions, where one captures all the necessary information about \( \theta \) and the other contains none, then we've isolated a sufficient statistic for the parameter. In the exercise's solution, the factorization theorem helps identify \( ln(X_1 X_2...X_n) \) as a sufficient statistic.