The Rayleigh density function is given by ReferenceA

QUESTION:

The Rayleigh density function is given by

                                                        \(f(y)=\left\{\begin{array}{cl}\left(\frac{2 y}{\theta}\right) e^{-y^{2} / \theta}, & y>0 \\0 & \text { elsewhere }\end{array}\right.\)

In Exercise 6.34(a), you established that \(Y^{2}\)has an exponential distribution with mean \(\theta\). If \(Y_{1}, Y_{2}, \ldots, Y_{n}\)denote a random sample from a Rayleigh distribution, show that \(W_{n}=\frac{1}{n} \sum_{i=1}^{n} Y_{i}^{2}\) is a consistent estimator for \(\theta\).

Equation Transcription:

  {

Text Transcription:  

f(x) = {(frac{2 y}{theta}) e^{-y^{2} /theta}, & y > 0 0 & elsewhere }

Y^2

theta

Y_1, Y_2, …., Y_n

W_n = frac{1}{n} sum_{i=1}^{n} Y_{i}^{2}

theta