Given a random sample of size n from a Poisson population, use the method of maximum likelihood to obtain an estimator for the parameter λ
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Given a random sample of size n from a Poisson population, use the method of maximum likelihood to obtain an estimator for the parameter λ
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- Consider a random sample drawn from the normal distribution with mean μ and variance o². The maximum likelihood estimator maximizes the value of the mean and variance of the normal distribution in order to estimate the unknown population mean. True FalseDerive the maximum likelihood estimator (MLE) of p and identify whether the MLE is a linear estimator or not.Consider the following regression model: Yi = B1 + B2X2i + ß3X3i + ß4X4i + ui Using the model above show that the maximum likelihood estimator for the variance, var (ui||X2i, X3i, B4X4i), is biased (be sure to comment of the nature of the bias).
- Given an iid random sample {X,,X,,..., X,}, consider the following likelihood function for the parameter p: L(p) = I[pX:(1– p)'–X; i=1This column contains data x1, . . . , xn, which is assumed to be a sample from a normal distribution N(µ, σ 2 ). For this sample, find the maximum likelihood estimates µˆ and σˆ.The lifetime of an electronical component is to be determined; it is assumed that it is an ex- ponentially distributed random variable. Randomly, users are asked for feedback for when the component had to be replaced; below you can find a sample of 5 such answers (in months): 19,23,21,22,24. Fill in the blanks below. (a) Using the method of maximum likelyhood, the parameter of this distribution is estimated to 2 = WRITE YOUR ANSWER WITH THREE DECIMAL PLACES in the form N.xxx. DO NOT ROUND. (b) Let L be the estimator for the parameter of this distribution obtained by the method of moments (above), and let H be the estimator for the parameter of this distribution obtained by the method of maximum likelyhood. What comparison relation do we have between L and M in this situation? Use one of the symbols to fill in the blank. L M
- For a random variable, its hazard function also referred to as the instantaneous failure rate is defined as the instantaneous risk (conditional probabilty) that an event of interest will happen in a narrow span of time duration. For a discrete random variable X, its hazard function is defined by the formula hX(k) =P(X=k+ 1|X > k) =pX(k)1−FX(k). For a Poisson distribution with λ= 4.2, find hX(k) and use R to plot the hazard function.The differentiation approach to derive the maximum likelihood estimator (mle) is not appropriate in all the cases. Let X₁, X2,,X₁ be a random sample of size n from the population of X. Consider the probability function of X fe-(2-0), if 0Suppose ?1, ?2, ..., ?? is a random sample from(4) Consider n i.i.d. samples of X ~ N(µ,0²). Find the maximum likelihood estimate of o?.Let X11 X12, Xini and X21, X22, X2n2 be two independent random samples of size n₁ and n₂ from two normal populations N(₁, 2) and N(2, 2) respectively. (a) Derive the maximum likelihood estimators (mle's) of all the parameters in the first population (X₁). Using analogy, state the mle's of the parameters of the second population. (b) Find the pooled estimator of the common variance when it is assumed that of = = ². Suggest an unbiased estimator of o². |Assume that you want to estimate an unknown parameter by using a noisy received signal for which you know the mean value and the covariance matrix of the noise but not its probability density function (pdf). You also know that the relationship between the unknown parameter and the received signal is linear. Would you use the BLUE estimation method, or the Maximum Likelihood estimation method? Justify your answer.
