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Chapter 10: Problem 2

Bestimmen Sie im statistischen Produktmodell \(\left(\mathbb{R}^{n}, \mathscr{B}^{n}, u_{[0, \hat{v}]}^{\otimes n}: \vartheta>0\right.\) ) die G眉tefunktion des Tests mit Annahmebereich \(\left\\{\frac{1}{2}<\max \left\\{X_{1}, \ldots, X_{n}\right\\} \leq 1\right\\}\) f眉r das Testproblem \(H_{0}: \vartheta=1\) gegen \(H_{1}: \vartheta \neq 1\)

Short Answer

Expert verified
The G眉tefunktion is \(\beta(\vartheta) = \left(\frac{\min(1, \vartheta) - \frac{1}{2\vartheta}}{\vartheta}\right)^n - \left(\frac{1}{2\vartheta}\right)^n\).

Step by step solution

01

Define the Statistical Framework

In the statistical product model \[ (\mathbb{R}^{n}, \mathscr{B}^{n}, u_{[0, \hat{v}]}^{\otimes n}: \vartheta > 0) \]we consider a test to determine the hypothesis \(H_0: \vartheta = 1\) against \(H_1: \vartheta eq 1\). Here, \(u_{[0, \hat{v}]}\) is a uniform distribution on the interval \([0, \hat{v}]\) or more precisely for the product model, \([0, \vartheta]\) in each component. \(n\) denotes the sample size, and the sample vectors \(X = (X_1, X_2, \ldots, X_n)\) are given.
02

Define the Acceptance Region

The acceptance region of the test is specified as\[\left\{\frac{1}{2}<\max\{X_1, \ldots, X_n\} \leq 1\right\}\]This means we accept \(H_0\) if the maximum observed value in the sample is between \(\frac{1}{2}\) and \(1\), inclusive of \(1\) but exclusive of \(\frac{1}{2}\).
03

Calculate Function of Acceptance

The function that determines acceptance under the hypothesis \(H_0\), called the power function or G眉tefunktion, is \[\beta(\vartheta) = P_{\vartheta}\left(\frac{1}{2}<\max\{X_1, \, \ldots, \, X_n\} \leq 1\right)\]To compute this, we need to find the probability that all sample values are less than or equal to 1 and at least one is greater than \(\frac{1}{2}\).
04

Compute Probability Under \(H_0\)

For \(H_0\) where \(\vartheta = 1\), the probability simplifies to computing the survival function of the maximum:\[1 - P_{H_0}(\max\{X_1, \ldots, X_n\} \leq \frac{1}{2})\]The probability that each \(X_i\) is less than or equal to \(\frac{1}{2}\) is \(\frac{1}{2}\). Since they are independent, \[P_{H_0}(\max\{X_1, \ldots, X_n\} \leq \frac{1}{2}) = \left(\frac{1}{2}\right)^n\]The power function or G眉tefunktion thus is:\[\beta(1) = 1 - \left(\frac{1}{2}\right)^n\]
05

Generalize to Any \(\vartheta\)

In general, for any \(\vartheta eq 1\), the same reasoning applies but the scaling of the interval changes. The probability that each \(X_i\) falls within the specified bounds needs to be calculated accordingly for the hypothesis. For general \(\vartheta\),\[\beta(\vartheta) = \left(\frac{\min(1, \vartheta) - \frac{1}{2\vartheta}}{\vartheta}\right)^n - \left(\frac{1}{2\vartheta}\right)^n\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Uniform Distribution
A uniform distribution is a fundamental type of probability distribution. It implies that all outcomes are equally likely within a specified interval. For example, for a uniform distribution across the interval \[ [0, \vartheta] \], every point between \([0, \vartheta]\) has the same chance of being selected. This kind of distribution is often visualized as a level line, which means that the probability distribution function is constant over the interval.
  • This model is simple and extensively used in testing hypotheses because it assumes an equal likelihood for all values.
  • In the problem provided, the uniform distribution operates over a product model interval \([0, \vartheta]\).
  • Having uniform distribution means that as each sample point is calculated, the entirety of the interval is covered with equal probability, making it ideal for hypothesis testing.
Power Function
The power function, denoted usually as \( \beta(\vartheta) \), is a crucial aspect of hypothesis testing because it reflects how well a test can detect differences or effects. It tells us the probability of correctly rejecting the null hypothesis \(H_0\) when the alternative hypothesis \(H_1\) is true.
  • To put it simply, power function is the probability that the test correctly detects an effect when there is one.
  • When \( \vartheta = 1 \), we calculate \( \beta(\vartheta) \) based on the given condition over uniform distribution, representing the test鈥檚 sensitivity under different parameter values.
  • This power function is influenced by the sample size \(n\) and the choice of acceptance region, highlighting the test's overall effectiveness.
Acceptance Region
The acceptance region is where we decide whether to accept or reject the null hypothesis. It鈥檚 essentially a defined range of values that, if the test statistic falls within, lead to acceptance of \(H_0\). In our scenario, the acceptance region is articulated as \{\frac{1}{2} < \max\{X_1, \ldots, X_n\} \leq 1\}.
  • This specific range implies that the maximum value found within the sample data must lie between 0.5 and 1. This particular setting underscores a controlled approach to deciding when \(H_0\) should be accepted.
  • Choosing an acceptance region critically impacts the likelihood of Type I and Type II errors 鈥� errors due to incorrect rejection or acceptance of the hypothesis.
  • The formulation of an acceptance region can help maintain a robust test, aligned with practical data analysis requirements.
Statistical Product Model
A statistical product model refers to a scenario in which multiple random variables or components are considered together, often in the context of a statistical distribution. The exercise involves such a model \( (\mathbb{R}^n, \mathscr{B}^n, u_{[0, \hat{v}] }^{\otimes n}: \vartheta > 0) \). Here, the product symbol \(\otimes\) denotes the combination of several independent uniform distributions.
  • The model here comprises several independent observations drawn from the same distribution, providing a permutation of statistical views.
  • Each \(X_i\) within the model is independently and identically distributed over \([0, \vartheta]\), which builds an analytical framework for assessing statistical variances.
  • Analyzing this product model provides insight into how changes in one part of the model impact the whole, driving statistical inference closer to the actual data scenario.

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Most popular questions from this chapter

Zeigen Sie direkt (d. h. ohne Benutzung von Satz (10.19)), dass der einseitige \(t\)-Test im zweiparametrigen GauB'schen Produktmodell unverfalscht ist. Benutzen Sie dazu den Satz (9.17) von Student.

Sei \(\left(E, \mathscr{E} ; Q_{0}, Q_{1}\right)\) ein statistisches Standardmodell mit einfacher Hypothese und Alternative und strikt positiven Dichten \(e_{0}, \varrho_{1}\). F眉r den Log-Likelihood-Quotienten \(h=\) \(\log \varrho_{1} / \varrho 0\) existiere die Varianz \(v_{0}=\mathbb{V}_{0}(h) .\) Im zugeh枚rigen unendlichen Produktmodell sei \(R_{n}\) der Likelihood-Quotient nach \(n\) Beobachtungen. Zeigen Sie: Der Neyman-Pearson Test zu einem vorgegebenen \(0<\alpha<1\) hat einen Ablehnungsbereich der Gestalt $$ \left\\{\log R_{n}>-n H\left(Q_{0} ; Q_{1}\right)+\sqrt{n v_{0}} \Phi^{-1}(1-\alpha)\left(1+\delta_{n}\right)\right\\} $$ mit \(\delta_{n} \rightarrow 0\) f眉r \(n \rightarrow \infty\). (Hinweis: Bestimmen Sie das asymptotische Niveau der Tests mit konstantem \(\delta_{n}=\delta \neq 0 .\) )

Zeigen Sie, dass im zweiparametrigen GauB'schen Produktmodell kein gleichma脽ig bester Test f眉r das einseitige Testproblem \(H_{0}: m \leq 0\) gegen \(H_{1}: m>0\) existiert.

Zusammenhang von Konfidenzbereichen und Tests. Sei \(\left(\mathfrak{x}, \mathscr{F}, P_{\vartheta}: \vartheta \in \Theta\right.\) ) ein statistisches Modell. Zeigen Sie: (a) Ist \(C: X \rightarrow \mathfrak{P}(\Theta)\) ein Konfidenzbereich zum Irrtumsniveau \(\alpha\) und \(\vartheta_{0} \in \Theta\), so ist \(\left\\{\vartheta_{0} \notin C(\cdot)\right\\}\) der Ablehnungsbereich eines Tests von \(H_{0}: \vartheta=\theta_{0}\) gegen \(H_{1}: \vartheta \neq \vartheta_{0}\) zum Niveau \(\alpha\). (b) Ist umgekehrt fur jedes \(\vartheta_{0} \in \Theta\) ein nichtrandomisierter Test fur \(H_{0}: \vartheta=\vartheta_{0}\) gegen \(H_{1}: \vartheta \neq \vartheta_{0}\) zum Niveau \(\alpha\) gegeben, so l盲sst sich daraus ein Konfidenzbereich zum Irrtumsniveau \(\alpha\) gewinnen.

Bei einer Razzia findet die Polizei bei einem Gl眉cksspieler eine M眉nze, von der ein anderer Spieler behauptet, dass, \(\mathrm{Zahl}^{4}\) mit einer Wahrscheinlichkeit von \(p=0.75\) statt mit \(p=0.5\) erscheint. Aus Zeitgr眉nden kann die M眉nze nur \(n=10 \mathrm{Mal}\) 眉berpr眉ft werden. W盲hlen Sie Nullhypothese und Alternative gem盲脽 dem Rechtsgrundsatz 鈥濱n dubio pro reo" und geben Sie einen zugeh枚rigen besten Test zum Irrtumsniveau \(\alpha=0.01\) an. (Taschenrechner verwenden!)
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