Question

(6) The sequence of random variable are independent of each other and they follow the normal distribution

X1, ..., xn(n > 1)

  $(\mu \epsilon \mathbb{R})$. However, the actual value of $X_{i}$ were not observed, instead we only observed if each $X_{i}$ is either greater than or

equal to 0, or less than 0.

     

And you can use the fact that there is the inverse function that  is continuous.

Answer the following questions.

$\cdot$   Find the maximum likelihood estimator $\widehat{\mu }_{n}$ of ${\mu }$ .

$\cdot$   When $n\rightarrow \infty$ , show $\widehat{\mu }_{n} \overset{p}{\rightarrow} \mu$ ,   where $\overset{p}{\rightarrow}$ represents conversion of probability.

Answer 1

*1.-Xn NAM, 1),n, MER po at each xi it, fex) = be Ź Cray?, cocaco - Uh< tx. Likelihood frietim of Xive-sty js, LIM2L) = f(xi.my jeź Â (*;-432 log-likelihood frictim of X.-Xn . t(m) = log1(212) = -29628)-Ź Ź Cli-4032 likelihood separation 20 5) Ź Ź cximây = 0 » în = “ Å xi = ☆ Again, drews one E so at înza of mis în = F = 5 Exi of Mis înai. Hence maximum likelihood estimator Now, since, xe... Xn D N(MI) Therefore în = 3 = ÁŽK; ~ N(M, A Now, PCIE-M18+) for any eso - 1 PIX-MISED :: PC-ST-ME E) : 1. POSE { va(2-1) vel . 1 P((-m rot) + PSU (8-ms-t) -- Bline) + PEME) = 1- P(0) - Is(-a) Com As now =1-1+0 E-3(0) = 1 ¢ B1-0)-o] = 0 FENNM, A T l :) In G D- M) WN(O) $6.) be the cdf of standard normal C distribution Therefore, PCIE-M1 >c) so as nosas for any €20. Hence, în = X R M as nosas ie în convergence in probability as nga. D