If _j, an unbiased estimator of _j, is also a consistent estimator of _j, then when the sample size tends to infinity: A)the distribution of _j collapses to a single value of zero. B)the distribution of _j diverges away from a single value of zero. C) the distribution of _j collapses to the single point _j. D) the distribution of _j diverges away from _j.

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This question is using notation and terminology from statistical inference. The symbol 11eab06d_2507_9890_beed_f5bfb5255f5c_TB2133_11 _j represents a parameter of the population, which is unknown and fixed. An estimator is a random variable that takes on different values depending on the sample that is used to compute it. An estimator is unbiased if its expected value is equal to the parameter it is estimating. An estimator is consistent if it approaches the true parameter value as the sample size increases. The question tells us that an unbiased estimator of the parameter 11eab06d_2507_9890_beed_f5bfb5255f5c_TB2133_11 _j is also a consistent estimator of the parameter 11eab06d_2507_9891_beed_a707c99d9773_TB2133_11 _j. This means that as the sample size tends to infinity, the estimator approaches the true parameter value of both parameters. Since the two parameters are different, it's possible that their true values are different, and the estimator might approach different values for the two parameters. The question asks what happens to the distribution of another parameter, 11eab06d_2507_e6b4_beed_bd4d616fa060_TB2133_11 _j, as the sample size tends to infinity. It gives us two options: either the distribution collapses to a single point (option C), or the distribution diverges away from a single point (option D). To answer this question, we need to remember that the estimator is a random variable, which means that it has a distribution. As the sample size tends to infinity, the estimator approaches the true parameter value, which means that its distribution becomes more and more concentrated around the true value. Option C says that the distribution collapses to a single point, which means that the estimator becomes more and more precise as the sample size increases, until it is almost certain to take on a specific value (the true parameter value). Option D says that the distribution diverges away from a single value, which doesn't make sense in this context - if the estimator is consistent, it approaches the true parameter value, not some other value. Therefore, the best choice is C, and the explanation is that as the sample size tends to infinity, the distribution of 11eab06d_2507_e6b4_beed_bd4d616fa060_TB2133_11 _j collapses to the single point 11eab06d_2507_e6b5_beed_d5548108e8c2_TB2133_11 _j, because the estimator is consistent and becomes more and more precise as the sample size increases. If 11eab06d_2508_34d8_beed_37d62d2463c0_TB2133_11 _j, an unbiased estimator of 11eab06d_2508_34d9_beed_f94e5c7a6a91_TB2133_11 _j, is also a consistent estimator of 11eab06d_2508_34da_beed_f9bce1f3dfd4_TB2133_11 _j, then when the sample size tends to infinity the distribution of 11eab06d_2508_5beb_beed_b3c070dea107_TB2133_11 _j collapses to the single point 11eab06d_2508_5bec_beed_b3e5173b9fce_TB2133_11 _j.

If j, an Unbiased Estimator of j, Is

If _j, an Unbiased Estimator of _j, Is