ACOGIMIENTO RESIDENCIAL

= 1 1000 (4.6)

All simulations were implemented in the statistical software R and the code for simulation scenarios based on lognormal distribution is available in Appendix IV (Algorithm-A IV-1).

4.5 Results and discussions

4.5.1 Data from lognormal distribution

Figure 4.2 and Figure 4.3 illustrate the MSE values provided by the traditional statistical methods after substitution of the censored values together with those provided by the MLE, rROS, and KM methods. These figures represent simulation scenarios based on the data sets generated from lognormal distributions with µ = 1, 2,…,10 and σ=0.5, 1.9, 3.3, with 50% censoring. The plots for all other scenarios of σ are available in Figure-A IV-1 and Figure-A- IV-2 of Appendix IV. Note that, in Figure 4.2 and Figure 4.3, the y-axis is in log-scale, whereas the x-axis is in linear scale. In general, the substitution-based method does not consistently perform better or worse than other methods (i.e., MLE, rROS, and KM) across all simulation scenarios. For example, for the mean estimation, Figure 4.2b shows that the substitution-based method has comparable or smaller MSEs compared to other estimators as

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long as the simulated data are generated from μ equal to 1 and 2 and σ=1.9. However, the substitution-based method provides larger MSEs for any combination of μ>2 and σ=1.9. For the standard deviation estimation (Figure 4.3b), the performance of the substitution-based method is similar or better than other methods in scenarios where the simulated data are generated from μ=1,2,3,4 and σ=1.9. When the μ of the data generating distribution exceeds 5, the performance of the substitution-based method starts to deteriorate. The same observations can be made for any given σ in this study (see the plots in Appendix IV).

a) b) c)

Figure 4.2 The MSEs of different methods in estimating the mean of lognormal distribution with μ=1,2,…,10 and a) σ=0.5, b) σ =1.9, c) σ=3.3

a) b) c)

Figure 4.3 The MSEs of different methods in estimating the standard deviation of lognormal distribution with μ=1,2,…,10 and a) σ=0.5, b) σ =1.9, c) σ=3.3

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As shown in Figure 4.2 and Figure 4.3, depending on the characteristics of the simulated data (i.e., mean and standard deviation of the data generating distributions), substituting the censored observations may or may not lead to good estimates. This result is probably due to misspecification of the shape of the original distribution of data after substituting the censored observations with a constant value. To further investigate the reason for this behavior of the substitution-based method, let us focus on the simulation scenarios of σ=1.9 with 50% censoring. Figure 4.4 shows the distribution of data after substitution with DL/2 superimposed on the distribution of uncensored data in the following three situations:

1) Substitution of the censored values results in estimates that are equivalent to those provided by MLE, rROS, and KM (μ=2, σ=1.9);

2) Substitution of the censored values leads to slight over/under estimation (μ=5, σ=1.9); 3) Substitution of the censored values clearly results in poor estimates (μ=10, σ=1.9).

a) b) c)

Figure 4.4 The distributions of original and substituted data generated from lognormal distributions with σ=1.9 and different values a) = 2, b) = 5, and c) = 10

Noticeable is that, in the case of μ=2 (Figure 4.4a), the shape of the distribution of uncensored and substituted data is almost similar. However, when μ increases (Figure 4.4b and Figure 4.4c), substituting the censored values introduces a peak at the substituted value, leading to an incorrect characterization of the shape of the distribution. In fact, as observed in Figure 4.4c, substitution of censored values generates a bimodal distribution, which is far from the shape of the original distribution. We also investigate whether the distribution

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remains lognormal after the substitution of censored observations visually by the quantile- quantile (Q-Q) plots and formally by the Shapiro-Wilk test. The Q-Q plots for the substituted data for the three aforementioned scenarios (i.e., lognormal distribution, = 2,5,10 and = 1.9, 50% censoring), are shown in Figure 4.5. All Q-Q plots in this Figure include clusters of horizontal points of substituted values, changing the initial distribution of data sets. However, the impact of substitution is more pronounced in some simulation scenarios. For example, the Q-Q plot of the substituted data simulated from μ=2 (Figure 4.5a) appears roughly linear, indicating that the data set after substitution of the censored observation may be still lognormal. As μ increases, substantial deviation from linearity indicates that the substituted data no longer follow the lognormal distribution (Figure 4.5b and Figure 4.5c). A Shapiro- Wilk test provides p-values smaller than = 0.05, rejecting the normality of log- transformed data in all three scenarios.

a) b) c)

Figure 4.5 The Q-Q plots of substituted data generated from lognormal distribution with

σ=1.9 and different μ values a) = 2, b) = 5, and c) = 10

To provide a better demonstration of the shortcomings of the substitution-based method in estimating the mean and standard deviation compared to the alternative methods, Figure 4.6 illustrates MSEs of the substitution-based method for different combinations of μ and σ. The following can be inferred from Figure 4.6:

1. Related to the mean estimation, Figure 4.6a shows that, for a given σ, substitution produces larger MSEs as μ increases. Moreover, for any μ>4, substitution produces larger MSEs as σ decreases;

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2. Obtaining good estimates of the standard deviation (i.e., estimates with small MSEs) is largely influenced by the underlying σ of data (Figure 4.6b). For example, when the underlying σ of the simulated data is 0.5, the MSE values increase as μ increases. On the other hand, when the underlying σ of the simulated data is 1.9, the MSEs initially decrease (up to μ =4) and then start increasing as μ becomes larger. This implies that the performance of traditional methods after substitution of the censored values is difficult to predict before knowing the distributional characteristics of the data.

a) b)

Figure 4.6 The MSEs of the substitution method in estimating a) the mean and b) standard deviation for different combinations of μ and σ of lognormal distribution Figure 4.7a and Figure 4.7b illustrate the MSEs produced by MLE in estimating the mean and standard deviation of lognormal distributions with different combinations of the parameters μ and σ. For conciseness, only the results obtained from the MLE method are depicted since plots produced by the rROS and KM methods are similar. Plots relative to the rROS and KM methods are illustrated in Figure-A IV-3 and Figure-A IV-4 of Appendix IV. Comparison of Figure 4.7a and Figure 4.7b implies that MSEs of MLEs of the mean are approximately constant over different values of μ, whereas this behavior is not clearly observed for estimating the standard deviation. In fact, large MSEs are obtained at the left- end of the curves in Figure 4.7b, corresponding to moderate and highly skewed distributions (CV>1). This behavior is not surprising as Singh et al. (2006) and Shoari, Dubé & Chenouri

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(2015) agreed that the performance of estimators in the case of moderately to highly skewed data sets differs from that of the mildly skewed data sets.

a) b)

Figure 4.7 The MSEs of the MLE method in estimating a) the mean and b) standard deviation for different combinations of μ and σ of lognormal distribution