Let Ψˆ be the ML estimate of Ψ obtained in (a) above. Plot the fitted two-component normal mixture density f(w; Ψˆ ) on top of a histogram of the n = 75 data points. Choose the number of bins N for the histogram by consideration of n ≈ 2 N−1 and/or using the formula, bin width ≈ 2 × Sample IQR n1/3 , to guide in the choice of the number of bins N.
Question
Let Ψˆ be the ML estimate of Ψ obtained in (a) above. Plot the fitted two-component normal mixture density f(w; Ψˆ ) on top of a histogram of the n = 75 data points. Choose the number of bins N for the histogram by consideration of n ≈ 2 N−1 and/or using the formula, bin width ≈ 2 × Sample IQR n1/3 , to guide in the choice of the number of bins N.
Solution
To answer this question, we need to follow several steps:
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Calculate the ML estimate of Ψ (Ψˆ): This step depends on the data and the method used in part (a) of your question, which is not provided here. However, the ML (Maximum Likelihood) estimate is typically calculated by setting the derivative of the likelihood function to zero and solving for Ψ.
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Calculate the fitted two-component normal mixture density f(w; Ψˆ): This is a function that represents a mixture of two normal distributions. The form of the function is f(w; Ψˆ) = p*N(w; μ1, σ1) + (1-p)*N(w; μ2, σ2), where N(w; μ, σ) is the normal distribution with mean μ and standard deviation σ, and p is the proportion of the first component in the mixture. The parameters μ1, σ1, μ2, σ2, and p are estimated by Ψˆ.
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Plot the histogram of the data: This can be done using software like Python (matplotlib, seaborn libraries) or R (ggplot2 library). The number of bins N can be chosen using the provided formulas.
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The first formula suggests that n (the number of data points) should be approximately equal to 2^(N-1). Solving for N gives N ≈ log2(n) + 1.
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The second formula suggests that the bin width should be approximately equal to 2 times the sample IQR (interquartile range) divided by the cube root of n. The number of bins N is then approximately equal to the range of the data divided by the bin width.
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Plot the fitted density on top of the histogram: This can also be done using Python or R. The function f(w; Ψˆ) is plotted as a smooth curve on the same plot as the histogram.
Please note that the exact implementation of these steps depends on the specific data and software used.
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