Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

$Generalized Neyman-Pearson Lemma. Let f_(0)(x),f_(1)(x),cdots,f_(k)(x) be k+1 probability density functions. Let phi_(0) be a test function of the form phi_(0)(x)={[1","," if ",f_(0)(x) > sum_(j=1)^(k)a_(j)f_(j)(x)],[gamma(x)","," if ",f_(0)(x)=sum_(j=1)^(k)a_(j)f_(j)(x)],[0","," if ",f_(0)(x) < sum_(j=1)^(k)a_(j)f_(j)(x)]:} where a_(j) >= 0 for j=1,cdots,k. Show that phi_(0) maximizes int phi(x)f_(0)(x)dx among all phi,0 <= phi <= 1, such that int phi(x)f_(j)(x)dx <= intphi_(0)(x)f_(j)(x)dx,quad j=1,2,cdots,k$

Generalized Neyman-Pearson Lemma. Let $f_{0}(x), f_{1}(x), \cdots, f_{k}(x)$ be $k+1$ probability density functions. Let $\phi_{0}$ be a test function of the form $\newline$ $\phi_{0}(x)=\left\{\begin{array}{lll} 1, & \text { if } & f_{0}(x)>\sum_{j=1}^{k} a_{j} f_{j}(x) \\ \gamma(x), & \text { if } & f_{0}(x)=\sum_{j=1}^{k} a_{j} f_{j}(x) \\ 0, & \text { if } & f_{0}(x)<\sum_{j=1}^{k} a_{j} f_{j}(x) \end{array}\right.$ $\newline$ where $a_{j} \geq 0$ for $j=1, \cdots, k$ . Show that $\phi_{0}$ maximizes $\newline$ $\int \phi(x) f_{0}(x) d x$ $\newline$ among all $\phi, 0 \leq \phi \leq 1$ , such that $\newline$ $\int \phi(x) f_{j}(x) d x \leq \int \phi_{0}(x) f_{j}(x) d x, \quad j=1,2, \cdots, k$

View step-by-step help

Home

Math Problems

Grade 6

Solve one-step multiplication and division equations: word problems

Full solution

Q. Generalized Neyman-Pearson Lemma. Let

f_{0}(x), f_{1}(x), \cdots, f_{k}(x)

k+1

probability density functions. Let

\phi_{0}

be a test function of the form

\newline

\phi_{0}(x)=\left\{\begin{array}{lll} 1, & \text { if } & f_{0}(x)>\sum_{j=1}^{k} a_{j} f_{j}(x) \\ \gamma(x), & \text { if } & f_{0}(x)=\sum_{j=1}^{k} a_{j} f_{j}(x) \\ 0, & \text { if } & f_{0}(x)<\sum_{j=1}^{k} a_{j} f_{j}(x) \end{array}\right.

\newline

where

a_{j} \geq 0

for

j=1, \cdots, k

. Show that

\phi_{0}

maximizes

\newline

\int \phi(x) f_{0}(x) d x

\newline

among all

\phi, 0 \leq \phi \leq 1

, such that

\newline

\int \phi(x) f_{j}(x) d x \leq \int \phi_{0}(x) f_{j}(x) d x, \quad j=1,2, \cdots, k

Understand Problem & Set Equation: Step $1$ : Understand the problem and set up the equation. $\newline$ We need to show that $\phi_0$ maximizes $\int \phi(x)f_0(x)dx$ under the constraints $\int \phi(x)f_j(x)dx \leq \int \phi_0(x)f_j(x)dx$ for $j=1,2,\ldots,k$ .
Analyze Test Function: Step $2$ : Analyze the test function $\phi_0$ . $\newline$ \phi_0(x) = \begin{cases} $\newline1 & \text{if } f_0(x) > \sum_{j=1}^k a_j f_j(x) \\ $\newline$ \gamma(x) & \text{if } f_0(x) = \sum_{j=1}^k a_j f_j(x) \\ $\newline$ 0 & \text{if } f_0(x) < \sum_{j=1}^k a_j f_j(x) $\newline$ \end{cases}$ $\newline$ This function is designed to maximize $f_0(x)$ whenever possible by assigning the highest weight ( $1$ ) when $f_0(x)$ is greater than the weighted sum of other densities.
Consider Other Test Function: Step $3$ : Consider any other test function $\phi$ that satisfies the constraints. $\newline$ For any $\phi$ such that $0 \leq \phi \leq 1$ and $\int \phi(x)f_j(x)dx \leq \int \phi_0(x)f_j(x)dx$ , we need to compare $\int \phi(x)f_0(x)dx$ to $\int \phi_0(x)f_0(x)dx$ .
Use Properties & Constraints: Step $4$ : Use the properties of $\phi_0$ and the constraints. $\newline$ Since $\phi_0$ is maximized at points where $f_0(x)$ is highest relative to the other $f_j(x)$ , any other $\phi$ that is less than or equal to $\phi_0$ at these points will result in a lower or equal integral value. This is because $\phi$ cannot exceed $1$ and must adhere to the constraints for each $f_j(x)$ .
Conclude Proof: Step $5$ : Conclude the proof. $\newline$ Given that $\phi_0$ is constructed to maximize $f_0(x)$ under the constraints, and any deviation from $\phi_0$ in the form of a lower value would decrease $\int \phi(x)f_0(x)dx$ , it follows that $\phi_0$ indeed maximizes the integral among all permissible $\phi$ .