Statistical interview question: Explain the rate of convergence as it relates to statistics

The central limit theorem (CLT) states that a population with mean (mu) and standard deviation (sigma) and taking samples of the population with replacement, the distribution of the sample means will be approximately normally distributed.

The three rules of the central limit theorem are:-

1. The sample must be random,
2. The samples should be unrelated to each other, and
3. One sample should not impact the others.

Convergence refers to the tendency of a sequence of values, such as means or proportions, to approach a specific target or limiting values as the size increases. It is a fundamental concept in probability theory and statistics, particularly in the context of the CLT.

The speed of convergence in statistical distributions, particularly in the context of the CLT, refers to how quickly a sequence of random variables approaches a target distribution, often the normal distribution, as the sample size increases. This rate of convergence is crucial for understanding how well sample statistics approximate population parameters and how many data points are needed to achieve a desired level of accuracy.

It is important to note that a lower convergence rate indicates a faster convergence.

One question that can be asked is this: Which of the two distributions will converge faster, exponential or distribution?

The best way to answer this question is to write a function in python calculating the convergence of both the exponential and uniform distributions.

I have the code for the function to calculate the convergence of an exponential distribution, which accepts the sample size, number of samples list, distribution, and scale as arguments. This function uses the libraries of numpy and scipy to enable it to execute.

I wrote a short program to define the variables of sample size and number of samples list as inputs to the function:-

I then modified the function to accept a uniform distribution as the argument:-

I modified the short program using the uniform distribution as the input argument:-

I used pandas to create a dataframe that lists the exponential convergence rates in one column and the uniform convergence rates in the second. As can be seen in the dataframe below, the convergence rates of the uniform distribution are lower than those of the exponential distribution. This means that uniform distribution will converge faster than the exponential distribution.

Convergence is one topic that needs careful attention in the field of statistics and even machine learning. I personally had been studying it for quite a while before writing this post because I wanted to be sure that I had conveyed my thoughts accurately.