Bayesian estimation and classification for two logistic populations with a common location

Abstract

The problems of estimation and classification for two logistic populations with a common location and different scale parameters are considered. The MLEs of the associated parameters are derived by solving a system of non-linear equations numerically as they do not have closed-form expressions. The asymptotic confidence intervals and bootstrap confidence intervals are derived numerically. Bayes estimators for the associated parameters using Lindley’s approximation method with respect to three types of priors, namely the vague prior, Jeffrey’s prior and conjugate prior, are also derived numerically. Further, Bayes estimators using the Markov chain Monte Carlo (MCMC) method that uses the Metropolis-Hastings algorithm are also derived. Moreover, using these MCMC samples, the highest posterior density (HPD) credible confidence intervals are also derived for the associated parameters. The point estimators are compared through their bias and mean squared error, whereas the interval estimators are compared through coverage probabilities and expected lengths using the Monte-Carlo simulation method numerically. Based on these estimators, certain classification rules are derived to classify a new observation into one of the two logistic populations under the same model set-up. The expected probability of misclassification for each classification rule is computed numerically to evaluate their performances. Finally, two real-life examples are considered where the datasets have been satisfactorily modeled by using the logistic distribution with a common location, and the estimation and classification methodologies have been demonstrated.

Appendix

The likelihood function for our model is given by

$$\begin{aligned} l(\mu ,\sigma _1,\sigma _2|data)=\prod _{i=1}^{m} \frac{1}{\sigma _{1}} \frac{\exp \{-(\frac{x_{i}-\mu }{\sigma _{1}})\}}{\big [1+\exp \{-(\frac{x_{i}-\mu }{\sigma _{1}})\}\big ]^2} \prod _{j=1}^{n} \frac{1}{\sigma _{2}} \frac{\exp \{-(\frac{y_{j}-\mu }{\sigma _{2}})\}}{\big [1+\exp \{-(\frac{y_{j}-\mu }{\sigma _{2}})\}\big ]^2}. \end{aligned}$$

(A.1)

The proof of existence and uniqueness of the MLE of the model parameters can be done easily by extending the idea of Antle et al. (1970) who proved the results for a single logistic population that involves two parameters. We prove the following theorem that guarantees the MLEs of the model parameters exist and also unique, provided not all the $x_i$s and $y_j$s are equal, that is at least two of the x values must be different and similarly at least two of the y values are different. If all the $x_i$s are equal or/and all the $y_j$s are equal, and say that $x_i=\mu$ and/or $y_j=\mu ,$ then the likelihood function $l(\mu ,\sigma _1,\sigma _2|data)$ goes to infinity, as either of the $\sigma _1$ or $\sigma _2$ tends to 0.

Theorem A.1. If the function $l(\mu ,\sigma _{1},\sigma _{2})$ is given by (A.1), then it is quasi-concave. Moreover, if $x_i$s and $y_j$s in (A.1) are not all equal, then the function $l(\mu ,\sigma _{1},\sigma _{2})$ has a unique maximum.

Proof

Consider a plane $\mu =a\sigma _{1}+b\sigma _{2}+k$ on the parameter space $\{(\mu ,\sigma _1,\sigma _2):-\infty<\mu <\infty ,\sigma _1>0,\sigma _2>0\}$ where a,  b and k are any constants. On this plane we take a line $\sigma _1=\eta \sigma _2,$ for any constant $\eta >0,$ which implies $\mu =d\sigma _{2}+k.$ Now on the line $\mu =d\sigma _{2}+k,$ we will show that the likelihood function is quasi concave. The likelihood function on this line is

$$\begin{aligned} l(\sigma _2)=\prod _{i=1}^{m}\frac{1}{\eta \sigma _{2}} \frac{\exp \big \{-\big (\frac{x_{i}-d\sigma _{2}-k}{\eta \sigma _{2}}\big )\big \} }{\Big [1+\exp \big \{-\big (\frac{x_{i}-d\sigma _{2}-k}{\eta \sigma _{2}}\big )\big \}\Big ]^2} \prod _{j=1}^{n} \frac{1}{\sigma _{2}} \frac{\exp \big \{-\big (\frac{y_{j}-d\sigma _{2}-k}{\sigma _{2}}\big )\big \}}{\Big [1+\exp \big \{-\big (\frac{y_{j}-d\sigma _{2}-k}{\sigma _{2}}\big )\big \}\Big ]^2}. \end{aligned}$$

(A.2)

Let $\alpha _{i}=x_{i}-k,$ $\beta _{j}=y_{j}-c,$ $k_{1} = \exp {(-d/\eta )}$ and $k_{2}=\exp {(-d)}.$ With these notations, the above likelihood function reduces to

$$\begin{aligned} l(\sigma _2)= \prod _{i=1}^{m} \frac{k_{1}}{\eta \sigma _{2}} \frac{\exp \big \{-\big (\frac{\alpha _{i}}{\eta \sigma _{2}}\big )\big \} }{\big [k_{1}+\exp \big \{-\big (\frac{\alpha _{i}}{\eta \sigma _{2}}\big )\big \}\big ]^2} \prod _{j=1}^{n} \frac{1}{\sigma _{2}} \frac{\exp \big \{-\big (\frac{\beta _{j}}{\sigma _{2}}\big )\big \}}{\big [k_{2}+\exp \big \{-\big (\frac{\beta _{j}}{\sigma _{2}}\big )\big \}\big ]^2}. \end{aligned}$$

(A.3)

Taking logarithm on both sides, one gets

$$\begin{aligned} L(\sigma _2)&= -m\log {k_{1}}-m\log {(\eta \sigma _{2})} - \sum _{i=1}^{m} \frac{\alpha _{i}}{\eta \sigma _{2}} - 2\sum _{i=1}^{m}\log {\big (k_{1}+\exp \big \{-\big (\frac{\alpha _{i}}{\eta \sigma _{2}}\big )\big \}\big )} \nonumber \\&\quad - n\log {k_{2}} - n\log {\sigma _{2}} - \sum _{j=1}^{n} \frac{\beta _{j}}{\sigma _{2}}- 2\sum _{i=1}^{n}\log {\Big (k_{2}+\exp \big \{-\big (\frac{\beta _{j}}{\sigma _{2}}\big )\big \}\big )}. \end{aligned}$$

(A.4)

Consider the derivative of L : 

$$\begin{aligned} L'(\sigma _2)&= \frac{1}{\sigma _{2}^2}\Bigg [-(m+n)\sigma _{2} + \sum _{i=1}^{m} {\frac{\alpha _{i}}{\eta }\Bigg (1-\frac{2}{k_{1}\exp \big \{\frac{\alpha _{i}}{\eta \sigma _{2}}\big \} +1}\Bigg )}+ \sum _{j=1}^{n} {\beta _{j}\Bigg (1-\frac{2}{k_{2}\exp \big \{\frac{\beta _{j}}{\sigma _{2}}\big \} +1}\Bigg )}\Bigg ] \nonumber \\&= \frac{1}{\sigma _{2}^{2}} G(\sigma _{2}). \end{aligned}$$

(A.5)

The sign of the derivative will only depend on the sign of $G(\sigma _{2}).$ Further differentiating $G(\sigma _{2}),$ we have

$$\begin{aligned} G'(\sigma _2)= -(m+n) -2\sum _{i=1}^{m} {\Big (\frac{\alpha _{i}}{\eta \sigma _{2}}\Big )^2 \frac{k_{1}\exp \big \{\frac{\alpha _{i}}{\eta \sigma _{2}}\big \}}{\Big [{k_{1}\exp \big \{\frac{\alpha _{i}}{\eta \sigma _{2}}\big \} +1}\Big ]^2}} -2\sum _{j=1}^{n} \Big (\frac{\beta _{j}}{\sigma _{2}} \Big )^2 \frac{k_{2}\exp \big \{\frac{\beta _{j}}{\sigma _{2}}\big \}}{\Big [{k_{2}\exp \big \{\frac{\beta _{j}}{\sigma _{2}}\big \} +1}\Big ]^2}. \end{aligned}$$

It is clearly seen that $G'(\sigma _2)<0 ~\forall ~ \sigma _{2}>0,$ which implies that $G(\sigma _{2})$ is decreasing in the interval $(0,\infty ).$ It is easy to observe that $\lim G(\sigma _{2}) = -\infty$ as $\sigma _{2} \rightarrow \infty .$ Next, we will find the limit of $G(\sigma _{2})$ when $b\rightarrow 0.$ To do so, let us define the following functions as

$$\begin{aligned} Z_{1}(\sigma _{2}) = 1-\frac{2}{k_{1}\exp \big \{\frac{\alpha _{i}}{\eta \sigma _{2}}\big \} +1} ~ \text {and}~ Z_{2}(\sigma _{2}) = 1-\frac{2}{k_{2}\exp \big \{\frac{\beta _{j}}{\sigma _{2}}\big \} +1}. \end{aligned}$$

(A.6)

To obtain the limit, we consider the following four cases:

• Case-1 $\alpha _{i}>0$ and $\beta _{j} >0.$ In this case we have $\lim Z_{1}(\sigma _{2}) \rightarrow 1$ and $\lim Z_{2}(\sigma _{2}) \rightarrow 1$ as $\sigma _{2}$ tends to 0.

• Case-2 $\alpha _{i}<0$ and $\beta _{j} <0.$ In this case, we have $\lim Z_{1}(\sigma _{2}) \rightarrow -1$ and $\lim Z_{2}(\sigma _{2}) \rightarrow -1$ as $\sigma _{2}$ tends to 0.

• Case-3 $\alpha _{i}>0$ and $\beta _{j} <0.$ In this case, we have $\lim Z_{1}(\sigma _{2}) \rightarrow 1$ and $\lim Z_{2}(\sigma _{2}) \rightarrow -1$ as $\sigma _{2}$ tends to 0.

• Case-4 $\alpha _{i}<0$ and $\beta _{j} >0.$ In this case we have $\lim Z_{1}(\sigma _{2}) \rightarrow -1$ and $\lim Z_{2}(\sigma _{2}) \rightarrow 1$ as $\sigma _{2}$ tends to 0.

It is easily seen that for all the above cases (Cases-1-4), $\lim G(\sigma _{2}) \ge 0$ as $\sigma _{2}$ tends to 0. The equality will hold if and only if all $\alpha _{i}$ and $\beta _{j}$ are zero. It now follows that if all $\alpha _{i}$ and $\beta _{j}$ are 0,  that is $x_i=y_j=\mu ,$ then the function $L(\sigma _{2}),$ is strictly decreasing, but if at least one of the $\alpha _{i}$s and $\beta _{j}$s are not zero, then there exists a $\sigma _{2}^{0},$ such that $L(\sigma _{2}),$ is increasing for $0<\sigma _{2}<\sigma _{2}^{0}$ and decreasing in the interval $\sigma _{2}^{0}< \sigma _{2} < \infty .$ This proves the theorem. $\square$