- When $p$ is $1$ or $0$, the randomized estimate degenerates to an estimator without randomized sampling.

We shall analyze only discuss the situation in which $p \in (\frac{1}{2},1)$

We shall only discuss situations in which $p \in (\frac{1}{2},1)$

(the situation in which $p \in (0,\frac{1}{2})$ is symmetric).

(a situation in which $p \in (0,\frac{1}{2})$ is symmetric).

From expressions {eq}`eq:five` and {eq}`eq:seven` we can deduce that:

- The MSE of $\hat{\pi}$ decreases as $p$ increasing.

- The MSE of $\hat{\pi}$ decreases as $p$ increases.

## Comparing Two Survey Designs

In our non-randomized response method, we suppose that:

- Members of Group A tells the truth with probability of $T_a$ while the members of Group B tells the truth with probability of $T_b$

- Members of Group A tells the truth with probability $T_a$ while the members of Group B tells the truth with probability $T_b$

- $Y_i$ is $1$ or $0$ according to whether the sample's $i\text{th}$ member's report is in Group A or not.

Then we can estimate $\pi$ as:

\text{MSE Ratio}=\frac{\text{Mean Square Error Randomized}}{\text{Mean Square Error Regular}}

$$

We can compute MSE Ratios for different surveys and survey designs associated with different parameter values.

We can compute MSE Ratios for different survey designs associated with different parameter values.

The following Python code computes the objects we want to stare at in order to make comparisons

The following Python code computes objects we want to stare at in order to make comparisons

under different values of $\pi_A$ and $n$:

```{code-cell} ipython3

We can generate MSE Ratios theoretically using the above formulas.

We can also perform a Monte Carlo simulation of the MSE Ratio.

We can also perform Monte Carlo simulations of a MSE Ratio.

```{code-cell} ipython3

cp1 = Comparison(0.6, 1000)

df1_mc

```

The theoretical calculations do a good job of predicting the Monte Carlo results.

The theoretical calculations do a good job of predicting Monte Carlo results.

We see that in many situations, especially when the bias is not small, the MSE of the randomized-sampling methods is smaller than that of the non-randomized sampling method.

{doc}`This QuantEcon lecture <util_rand_resp>` describes some alternative randomized response surveys.

That lecture presents the utilitarian analysis of those alternatives conducted by Lars Ljungqvist

That lecture presents a utilitarian analysis of those alternatives conducted by Lars Ljungqvist

{cite}`ljungqvist1993unified`.